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Some Concepts and Properties of Point Set Topology
ADDENDA A
Some Concepts and Properties of Point Set Topology A.1 Relation and Mapping A.1.1 Relation Definition 1.1.1 X and Y are any two sets. fðx; yÞjx ˛ X; y ˛ Yg is called a Cartesian product of X and Y, denoted by X Y, where ðx; yÞ is a pair of ordered elements. x is the first coordinate of ðx; yÞ, and y is the second coordinate of ðx; yÞ. X is a set of the first coordinates of X Y, and Y is a set of the second coordinates of X Y. Definition 1.1.2 X and Y are two sets. For any R3 X Y, R is called a relation from X to Y. Assume that R is a relation from X to Y. If ðx; yÞ ˛ R, then x and y are Rrelevant, denoted by xRy. Set fxjdy; ðx; yÞ ˛ Rg is called the domain of R, denoted by DðRÞ. Set fyjðx; yÞ ˛ R; x ˛ DðRÞg is called the range of R, denoted by RðRÞ. For A3X, letting fyjdx ˛ A; ðx; yÞ ˛ Rg ¼ RðAÞ, RðAÞ is called a set of images (or image) of A. For B3Y, letting fxjdy ˛ B; ðx; yÞ ˛ Rg ¼ R1 ðBÞ, R1 ðBÞ is called the preimage of B. Definition 1.1.3 For R3X Y; S3Y Z, letting T ¼ fðx; zÞjdy ˛ Y; ðx; yÞ ˛ R; ðy; zÞ ˛ Sg, T is called the composition of R and S, denoted by T ¼ S+R. For R3X Y, letting R1 ¼ fðy; xÞjðx; yÞ ˛ Rg3Y X, R1 is called the inverse of R. Proposition 1.1.1 For R3X Y; S3Y Z and T3Z U, we have
333
334 Addenda A (1) ðR1 Þ1 ¼ R (2) ðS+RÞ1 ¼ R1 +S1 (3) T+ðS+RÞ ¼ ðT+SÞ+R (4) cA; B3X, RðAWBÞ ¼ RðAÞWRðBÞ and RðAXBÞ3RðAÞXRðBÞ (5) ðS+RÞðAÞ ¼ SðRðAÞÞ Note that in (4) RðAXBÞ3RðAÞXRðBÞ rather than RðAXBÞ ¼ RðAÞXRðBÞ.
A.1.2 Equivalence Relation Definition 1.2.1 Assume that R is a relation from X to X (or a relation on X) and satisfies (1) DðXÞ ¼ fðx; xÞjx ˛ Xg3R (Reflexivity) (2) R ¼ R1 (Symmetry) (3) R+R3R (Transitivity) R is called an equivalence relation on X. Assume that R is an equivalence relation on X. For cx ˛ X, letting ½xR ¼ fyjyRx; y ˛ Xg, ½xR is an Requivalent set of x. Definition 1.2.2 For ca; Aa 3X, if Aa XAb ¼ B; a 6¼ b and W Aa ¼ X, then fAa g is a partition of x. a
Proposition 1.2.1 R is an equivalence relation on X. Then, f½xR jx ˛ Xg is a partition of X.
A.1.3 Mapping and OneeOne Mapping Definition 1.3.1 F is a relation from X to Y. For cx ˛ X, if there exists a unique y ˛ Y such that ðx; yÞ ˛ F, then F is called a mapping from X to Y, denoted by F : X/Y. If RðFÞ ¼ Y, F is called surjective, where RðFÞ is the range of F. For x1 ; x2 ˛ X, if x1 6¼ x2 0Fðx1 Þ 6¼ Fðx2 Þ, F is called 1-1 mapping. Proposition 1.3.1 f : X/Y is a mapping. For cA; B3X, we have f ðAWBÞ ¼ f ðAÞWf ðBÞ f ðAXBÞ ¼ f ðAÞXf ðBÞ A3f f 1 ðAÞ
Some Concepts and Properties of Point Set Topology 335 If A3B, then f ðAÞ3f ðBÞ. For cA; B3Y, we have f 1 ðAWBÞ ¼ f 1 ðAÞWf 1 ðBÞ f 1 ðAXBÞ ¼ f 1 ðAÞXf 1 ðBÞ f f 1 ðAÞ 3A; f 1 AC ¼ f 1 ðAÞC
If A3B, then f 1 ðAÞ3f 1 ðBÞ. If f is surjective, then cA3X, f ðAC ÞIf ðAÞC . If f is a 1-1 mapping, then f ðAC Þ3f ðAÞC . Where, AC is the complement of A. f 1 is the inverse of f. If f is surjective and 1-1 mapping, then A ¼ f 1 ðf ðAÞÞ and f ðAC Þ ¼ f ðAÞC . Definition 1.3.2 Assume that X is a Cartesian product of X1 ; X2 ; /; Xn . Let x ¼ ðx1 ; x2 ; /; xn Þ ˛ X. Define pi : X/Xi ; pi ðxÞ ¼ xi . pi is the projection of X on Xi , or a set of the i-th coordinates.
A.1.4 Finite Set, Countable Set and Uncountable Set Definition 1.4.1 A and B are two sets. If there exists a 1-1 surjective mapping from A to B, A and B are called equinumerous. Any set that is not equinumerous to its proper subsets is a finite set. A set that is equinumerous to the set N of all natural numbers is a countable set. An infinite set that is not equinumerous to the set N of all natural numbers is an uncountable set. Theorem 1.4.1 (Bernstein) If A and the subset of B are equinumerous, and B and the subset of A are also equinumerous, A and B are equinumerous.
A.2 Topology Space A.2.1 Metric Space X is a non-empty set. d : X X/R is a mapping, where R is a real set. cx; y; z ˛ X, d satisfies: (1) dðx; yÞ 0 and dðx; yÞ ¼ 05x ¼ y
336 Addenda A (2) dðx; yÞ ¼ dðy; xÞ (3) dðx; zÞ dðx; yÞ þ dðy; zÞ Then, d is a distance function on X and ðX; dÞ is a metric space. Definition 2.1.2 ðX; dÞ is a metric space. For x ˛ X; cε > 0, fyjdðx; yÞ < ε; y ˛ Xg ¼ Bðx; εÞ. Bðx; εÞ is called a spherical neighborhood with x as its center and ε as its radius, or simply ε neighborhood. Proposition 2.1.1 ðX; dÞ is a metric space. Its spherical neighborhoods have the following properties. (1) cx ˛ X, there is one neighborhood at least. cBðx; εÞ, have x ˛ Bðx; εÞ. (2) x ˛ X, for any two spherical neighborhoods Bðx; ε1 Þ and Bðx; ε2 Þ, there exists Bðx; ε3 Þ such that Bðx; ε3 Þ3Bðx; ε1 ÞXBðx; ε2 Þ. (3) If y ˛ Bðx; εÞ, then there exists Bðy; ε1 Þ3Bðx; εÞ.
A.2.2 Topological Space Definition 2.2.1 X is a non-empty set. T is a family of subsets of X. If T satisfies the following conditions (1) X; B ˛ T (2) A; B ˛ T , AXB ˛ T (3) T 1 3T , W A ˛ T A˛T 1
then T is a topology of X. ðX; T Þ is a topologic space. Each member of T is called an open set on ðX; T Þ. ðX; dÞ is a metric space. For A3X and cx ˛ A, if there exists Bðx; εÞ3A, then A is an open set on X. Let T d be a family of all open sets on X. It can be proved that T d is a topology on X. ðX; T d Þ is called a topologic space induced from d. Definition 2.2.2 ðX; T Þ is a topologic space (ðX; T Þ always indicates a topologic space below). For x ˛ X and U ˛ T , if x ˛ U, then U is called a neighborhood of x denoted by UðxÞ. For x ˛ X, the set of all neighborhoods of x is called a system of neighborhoods of x, denoted by Ux. Proposition 2.2.1 ðX; dÞ is a topologic space. For x ˛ X, Ux is a neighborhood system of x. We have
Some Concepts and Properties of Point Set Topology 337 (1) cx ˛ X; Ux 6¼ B and cu ˛ Ux , then x ˛ u. (2) If u; v ˛ Ux , then uXv ˛ Ux . (3) If u ˛ Ux , there exist v3u; v ˛ Ux such that for cy ˛ v have v ˛ Uy .
A.2.3 Induced Set, Close Set and Closure Definition 2.3.1 For A3ðX; T Þ; x ˛ X, if cu ˛ Ux , uXðA=fxgÞ 6¼ B, then x is called an accumulation (limit) point of A. Set A0 of all accumulation points of A is called an induced set of A. Proposition 2.3.1 For cA; B3ðX; T Þ, we have (1) (2) (3) (4)
B0 ¼ B A3B A0 3B0 ðAWBÞ0 ¼ A0 WB0 ðA0 Þ0 3AWA0
Definition 2.3.2 For A3ðX; T Þ, if all accumulation points of A belong to A, then A is a close set. Proposition 2.3.2 A is close 5Ac is open. Proposition 2.3.3 Assume that F is a family of all close sets on ðX; T Þ. We have (1) X; B ˛ F (2) If A; B ˛ F , then AWB ˛ F . (3) If F 13 F , then X A ˛ F . A ˛ F1
Definition 2.3.3 For A3ðX; T Þ, letting A ¼ AWA0 , A is called a closure of A. Proposition 2.3.4 For cA; B3ðx; T Þ, we have (1) B ¼ B (2) A3 A
338 Addenda A (3) AWB ¼ AWB (4) A ¼ A Definition 2.3.4 For A3ðX; dÞ; x ˛ X, define dðx; AÞ ¼ inffdðy; xÞjy ˛ Ag. Proposition 2.3.5 For A3ðX; dÞ, we have (1) x ˛ A0 5dðx; ðA fxgÞÞ ¼ 0 (2) x ˛ A5dðx; AÞ ¼ 0
A.2.4 Interior and Boundary Definition 2.4.1 For A3ðX; T Þ, letting A0 ¼ fxjdu ˛ U; x ˛ u3Ag, A0 is called the interior (core) of A. Proposition 2.4.1 For A3ðX; T Þ, we have (1) (2) (3) (4) (5) (6)
A is open 5A0 ¼ A ðA0 Þc ¼ ðAc Þ; ðAc Þ0 ¼ ðAÞc X0 ¼ X A0 3A ðAXBÞ0 ¼ A0 XB0 ðA0 Þ0 ¼ A0
Definition 2.4.2 For A3ðX; T Þ; x ˛ X, if cu ˛ Ux , uXA 6¼ B and uXAc 6¼ B, x is called a boundary point of A. The set of all boundary points of A is called boundary of A, denoted by vA. Proposition 2.4.2 For A3ðX; T Þ, we have (1) (2) (3) (4) (5)
A ¼ AWvA A0 ¼ A vA vA ¼ AXðAc Þ vA0 3vA; vðAÞ3vA vðAWBÞ3vAWvB; vðvBÞ3vB
Some Concepts and Properties of Point Set Topology 339
A.2.5 Topological Base and Subbase Definition 2.5.1 ðX; T Þ is a topologic space. For B3T and cu ˛ T , if there exists B1 3B such that u ¼ W v, then B is a base of T . v ˛ B1
Proposition 2.5.1 ðX; dÞ is a space. ðX; T d Þ is a topologic space induced from d. Then, T 1 ¼{all spherical neighborhoods of x, cx ˛ X } is a base of T d . Proposition 2.5.2 B is a family of open sets on ðX; T Þ. Then, B is a base 5cu ˛ T and x ˛ u, there is v ˛ B such that x ˛ v3u. Proposition 2.5.3 B is a family of subsets of X and satisfies (1) X ¼ W u u˛B
(2) If B1 ; B2 ˛ B, for cx ˛ B1 XB2, there exists BðxÞ ˛ B such that x ˛ BðxÞ3B1 XB2 . Then, let T ¼ fAjA ¼ W u; cB1 3Bg be a topology of X and B be a base of T . u ˛ B1
Definition 2.5.2 ðX; T Þ is a space. 4 is a sub-family of T . If si ˛ 4; i ¼ 1; 2; .; n; n ˛ N, letting s1 Xs2 X:::Xsi ˛ B, i.e., B is a family of sets composed by the intersections of any finite number of elements in 4, then B is a base of T , and 4 is a subbase of T .
A.2.6 Continuous Mapping and Homeomorphism Definition 2.6.1 f : ðX; T 1 Þ/ðY; T 2 Þ is a mapping. If cu ˛ T 2 ; f 1 ðuÞ ˛ T 1 , then f is a continuous mapping. If x ˛ X, cw ˛ T 2 and f ðxÞ ˛ w, have f 1 ðwÞ ˛ T 1 , then f is continuous at x. Proposition 2.6.1 For f : ðX; T 1 Þ/ðY; T 2 Þ, the following statements are equivalent. (1) f is a continuous mapping (2) If B is a base of Y, then cu ˛ B, f 1 ðuÞ ˛ T 1 . (3) f 1 ðFÞ is a preimage of any close set F in Y; f 1 ðFÞ is close in X.
340 Addenda A (4) 4 is a subbase of Y; cu ˛ 4, have f 1 ðuÞ ˛ T 1 . (5) cA3X, have f ðAÞ3ðf ðAÞÞ. (6) cB3Y, have f 1 ðBÞIðf 1 ðBÞÞ. Proposition 2.6.2 For f : ðX; T 1 Þ/ðY; T 2 Þ; x ˛ X, the following statements are equivalent. (1) f is continuous at x. (2) For all neighborhoods uðf ðxÞÞ of f ðxÞ, there exists uðxÞ ˛ U such that f ðuðxÞÞ3uðf ðxÞÞ. Proposition 2.6.3 If f : ðX; T 1 Þ/ðY; T 2 Þ and g : ðY; T 2 Þ/ðZ; T 3 Þ are continuous, then g+f : ðX; T 1 Þ/ðZ; T 3 Þ is continuous. Definition 2.6.3 ðX; T 1 Þ and ðY; T 2 Þ are two spaces. If there exists f : ðX; T 1 Þ/ðY; T 2 Þ, where f is a 1-1 surjective and bicontinuous mapping, i.e., both f and f 1 are continuous, then f is called a homeomorphous mapping from X to Y, or X and Y are homeomorphism.
A.2.7 Product Space and Quotient Space Definition 2.7.1 T 1 and T 2 are two topologies on X. If T 1 3T 2 , T 1 is called smaller (coarser) than T 2 . fT a ; a ˛ Ig is a family of topologies on X. If there exists T a0 such that cT a , T a0 3T a , then T a0 is called the smallest (coarsest) topology in fT a g. Similarly, we may define the concept of the largest (finest) topology. Proposition 2.7.1 Assume that ca ˛ I, fa : X/ðYa ; T a Þ. There exists the smallest (coarsest) topology among topologies on X that make each fa continuous. Proposition 2.7.2 Assume that ca ˛ I, fa : X/ðYa ; T a Þ. There exists the largest (finest) topology among topologies on X that make each fa continuous. Corollary 2.7.2 Assume that f : ðX; T Þ/Y. There exists the largest (finest) topology among topologies on Y that make f continuous. The topology is called the quotient topology with respect to T and f .
Some Concepts and Properties of Point Set Topology 341 Definition 2.7.2 For A3ðX; T Þ, letting T A ¼ fuju ¼ AXv; v ˛ Tg, ðA; T A Þ is called the subspace of ðX; T Þ. Definition 2.7.3 Assume that X ¼
Q a˛I
Xa ; I 6¼ B, where
Q a˛I
Xa indicates the product set. fðXa ; T a Þ; a ˛ Ig
is a family of topologic spaces. Let pa : X/Xa be a projection. T is the smallest topology among topologies on X that make pa ðca ˛ IÞ continuous. ðX; T Þ is called the product Q topologic space of fðXa ; T a Þg, denoted by ðX; T Þ ¼ ðXa ; T a Þ. a˛I
Proposition 2.7.3 Assume that ðX; T Þ is a product topologic space of fðXa ; T a Þ; a ˛ Ig. Letting 4 ¼ fp1 a ðua Þjua ˛ T a ca ˛ Ig, 4 is a subbase of T . Proposition 2.7.4 Assume that ðX; T Þ is a product topologic space of fðXa ; T a Þ; a ˛ Ig. f : ðY; T 0 Þ/ðX; TÞ is continuous 5ca ˛ I, pa +f : ðY; T 0 Þ/ðXa ; T a Þ is continuous. Proposition 2.7.5 Assume that ðX; T Þ is a product topologic space of fðXa ; T a Þ; a ˛ Ig. Then, series fxi g on X converges to x0 ˛ X5ca ˛ I, series pa ðxi Þ on Xa converges to pa ðx0 Þ. Where, the definition of convergence is that for fxi g3ðX; T Þ; x ˛ X, if cu ˛ Ux , there exists n0 such that when n > n0 , xn ˛ u. Then fxi g is called to be converging to x, denoted by lim xn ¼ x. n/N
Definition 2.7.4 R is an equivalence relation on ðX; T Þ. Let p be a nature projection X/X=R ðpðxÞ ¼ ½xÞ, and ½T be the finest topology that makes p continuous. ðX=R; ½T Þ is called the quotient space of ðX; T Þ with respect to R. Where, X=R may be indicated by ½XR, or ½X. Proposition 2.7.6 Assume that ð½X; ½T Þ is a quotient topologic space of ðX; T Þ with respect to R. Then, ½T ¼ fuju3½X; p1 ðuÞ ˛ T; p : X/½Xg. Definition 2.7.5 For f : ðX; T Þ/Y, letting T =f ¼ fuju3Y; f 1 ðuÞ ˛ T g, T =f is called the quotient topology of ðX; T Þ with respect to f. We have a topologic space ðY; T =f Þ and ðY; T =f Þ is a congruence space of T and f.
342 Addenda A Proposition 2.7.7 f : ðX; T 1 Þ/ðY; T 2 Þ is an open (close) surjective mapping. Then, T 2 ¼ T 1 =f . Proposition 2.7.8 ðY; T 2 Þ is an congruence space of T and f. Assume that f : ðX; T 1 Þ/ðY; T 2 Þ and g : ðY; T 2 Þ/ðZ; T 3 Þ. Then, g is continuous 5g+f is continuous.
A.3 Separability Axiom A.3.1 T0 , T1 , T2 Spaces Definition 3.1.1 ðX; T Þ is a space. For cx; y ˛ X; x 6¼ y, there is u ˛ Ux such that y;u, or there is u ˛ Uy such that x;u, X is called T0 space. Definition 3.1.2 ðX; T Þ is a space. For cx; y ˛ X; x 6¼ y, there must be u ˛ Ux ; v ˛ Uy such that y;u; x;v, X is called T1 space. Definition 3.1.3 ðX; T Þ is a space. For cx; y ˛ X; x 6¼ y, there must be u ˛ Ux ; v ˛ Uy such that uXv ¼ B, X is called T2 space, or Hausdorff space. Proposition 3.1.1 X is a T0 space 5cx; y ˛ X; x 6¼ y, fxg 6¼ fyg, where fxg is the closure of singleton fxg. It means that the closures of any two different singletons are different. Proposition 3.1.2 ðX; T Þ is a topologic space. The following statements are equivalent. (1) X is a T1 space. (2) Each singleton on X is a close set. (3) Each finite set on X is a close set. Proposition 3.1.3 ðX; T Þ is a T1 space 5cx ˛ X, the intersection of all neighborhoods containing x is just fxg. Proposition 3.1.4 A3ðX; T Þ, X is a T1 space. Then, cx ˛ A0 5cu ˛ Ux , uXðA fxgÞ is an infinite set.
Some Concepts and Properties of Point Set Topology 343 Proposition 3.1.5 ðX; T Þ is a T2 space. Then, the convergent series on X has only one limit point. Proposition 3.1.6 ðX; T Þ is a T2 space 5 the diagonal D ¼ fðx; xÞjx ˛ Xg of product topologic space on X X is a close set.
3.2 T3 , T4 , Regular and Normal Space Definition 3.2.1 In space ðX; T Þ,cA3X, A is close. For x;A, if there exist open sets u and v, uXv ¼ B, such that x ˛ v; A3u, then X is called a T3 space. Definition 3.2.2 In space ðX; T Þ, for cA; B3X, if there exist open sets u and v such that A3u; B3v; uXv ¼ B, then X is called a T4 space. Proposition 3.2.1 ðX; T Þ is a T3 space 5cx ˛ X and u ˛ Ux , there exists v ˛ Ux such that v3u. Proposition 3.2.2 ðX; T Þ is a T4 space 5 for any close set A in X and any open set u that contains A, i.e., A3u, there exists open set v such that A3v3v3u. Proposition 3.2.3 ðX; T Þ is a T4 space 5 For close sets A; B3X; AXB ¼ B, there exists a continuous mapping f such that ðX; T Þ/½0; 1 and f ðAÞ ¼ 0; f ðBÞ ¼ 1. Proposition 3.2.4 (Tietz Theorem) ðX; T Þ is a T4 space 5 For any close set cA3X and any continuous function f0 : A/½0; 1 on A, there exists a continuous expansion f : X/½0; 1 of f0 on X. Definition 3.2.3 If ðX; T Þ is a T1 and T3 space, then X is called a regular space. Definition 3.2.4 If ðX; T Þ is a T1 and T4 spaces, then X is called a normal space. Proposition 3.2.5 ðX; T Þ is a normal space 0X is a regular space 0X is a T2 space 0X is a T1 space 0X is a T0 space.
344 Addenda A
A.4 Countability Axiom A.4.1 The First and Second Countability Axioms Definition 4.1.1 If ðX; T Þ has countable base, then X is said to satisfy the second countability axiom. Definition 4.1.2 If in ðX; T Þ, for cx ˛ X, there exists countable local base, then X is said to satisfy the first countability axiom. Proposition 4.1.1 Real space R satisfies the second countability axiom. Proposition 4.1.2 If ðX; dÞ is a metric space, then X satisfies the first countability axiom. Proposition 4.1.3 If ðX; T Þ satisfies the second countability axiom, then X satisfies the first countability axiom. Proposition 4.1.4 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuously open and surjective mapping. If X satisfies the second (or first) countability axiom, then Y will satisfy the second (or first) countability axiom. Definition 4.1.3 If ðX; T Þ has property P and any sub-space of X also has the property P, property P is called having heredity. Q If for cXi has property P and their product space X ¼ Xi also has property P, then P is i ˛ I called having integrability. The relation among separation axiom, countability axiom, heredity and integrability is shown in Table 4.1.1. Table 4.1.1
heredity integrability
T0
T1
T2
T3
T4
A1
A2
Separable
O O
O O
O O
O O
O O
O O
O
Where, A1 and A2 are the first and second countability axioms, respectively. O (countable) means that the product space of the countable number of metric spaces is metrizable.
Distance O O(countable)
Some Concepts and Properties of Point Set Topology 345 Proposition 4.1.5 For f : ðX; T 1 Þ/ðY; T 2 Þ, if X is countable, then f is continuous at x ˛ X 5cxi /x, have f ðxi Þ/f ðxÞ.
A.4.2 Separable Space Definition 4.2.1 If D3ðX; T Þ and D ¼ X, then D is called dense in X, or D is a dense subset of X. Proposition 4.2.1 Assume that D is a dense subset in ðX; T Þ. f : X/R and g : X/R are two continuous mappings. Then, f ¼ g5 f ¼ g on D. Definition 4.2.2 If ðX; T Þ has dense countable subsets, X is called a separable space. Proposition 4.2.1 If ðX; T Þ satisfies A2 , then X is separable. Proposition 4.2.2 If a separable metric space satisfies A2 , then it must be A1 . The relation among A1 , A2 and metric spaces is shown below. Separable A2
metric space
A1 Separable
Where, A / C indicates that property A with the addition of property B infers property C. B
A.4.3 Lindelof Space Definition 4.3.1 A is a family of sets and B is a set. If B3 W A, then A is called a cover of set B. When A˛A
A is countable or finite, A is called a countable or finite cover. If a family A of sets covers B and sub-family A1 of A also covers B, then A1 is called a sub-cover of A. If each set of cover A is open (closed), then A is called an open (closed) cover.
346 Addenda A Definition 4.3.2 In ðX; T Þ, for any open cover of X, there exists countable sub-cover, X is called a Lindelof space. Proposition 4.3.1 If ðX; T Þ satisfies A2 , then X is a Lindelof space. Corollary 4.3.1 An n-dimensional Euclidean space Rn is a Lindelof space. Proposition 4.3.2 If ðX; dÞ is a Lindelof space, then X satisfies A2 . Proposition 4.3.3 If any sub-space in ðX; T Þ is a Lindelof space, then each uncountable subset A of X must have accumulation points of A.
A.5 Compactness A.5.1 Compact Space Definition 5.1.1 In ðX; T Þ, if each open cover of X has its finite sub-covers, then X is called a compact space. Definition 5.1.2 Assume that A is a family of sets. If each finite sub-family in A has non-empty intersection, then A is said to have the finite intersection property. Proposition 5.1.1 ðX; T Þ is compact 5 each family of close sets that has the finite intersection property in X has non-empty intersection. Proposition 5.1.2 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. If X is compact, then f ðXÞ is also compact. Proposition 5.1.3 Each close subset of a compact set is compact.
Some Concepts and Properties of Point Set Topology 347 Proposition 5.1.4 If Xi ; i ˛ I, is compact, then their product space is compact as well.
A.5.2 Relation between Compactness and Separability Axiom Proposition 5.2.1 A compact subset in T2 is close. Proposition 5.2.2 A compact T2 space is a normal space. Proposition 5.2.3 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. If X is compact and Y is T2 , then f is a close mapping, i.e., mapping a close set to a close set. Proposition 5.3.4 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous and 1-1 surjective mapping. If X is compact and Y is T2 , then f is homeomorphous. Proposition 5.2.5 If A3Rn and Rn is an n-dimensional Euclidean space, then A is compact 5 A is a bounded close set. Proposition 5.2.6 f ðX; T Þ/R is a continuous mapping. If X is compact, there exist x0 ; y0 ˛ X such that cx ˛ X, f ðx0 Þ f ðxÞ f ðy0 Þ.
A.5.3 Some Relations in Compactness Definition 5.3.1 A topological space ðX; T Þ is countably compact if every countable open cover has a finite subcover. Definition 5.3.2 A topological space ðX; T Þ is said to be limit point compact if every infinite subset has a limit point. Definition 5.3.3 A topological space ðX; T Þ is sequentially compact if every infinite sequence has a convergent subsequence.
348 Addenda A
Compactness
Countable compactness Lindelof A2
Limit point compactness T1
Sequential compactness
In metric space, especially in n-dimensional Euclidean space, the four concepts of compactness, limit point compactness, countable compactness, and sequential compactness are equivalent.
A.5.4 Local Compact and Paracompact Definition 5.4.1 In ðX; T Þ, for each point on X there exists a compact neighborhood, and X is called a local compact space. Definition 5.4.2 Assume that A1 and A2 are two covers of X. If each member of A1 is contained by some member of A2 , then A1 is called the refinement of A2 . Definition 5.4.3 In ðX; T Þ, A is a cover of subset A. If cx ˛ A, there exists uðxÞ ˛ U such that uðxÞ only intersects with the finite number of members in A, then A is called a the local finite cover of A. Definition 5.4.4 In ðX; T Þ, for each open cover of A on X, there exists local finite cover A1, where A1 is the refinement of A, then X is called a paracompact space. Proposition 5.4.1 Each locally compact and T2 space are normal spaces. Proposition 5.4.2 Each paracompact normal space is a regular space. The relation among compactness, paracompactness and local compactness is shown below. Paracompact
Compact A2+T2
Locally Compact
Some Concepts and Properties of Point Set Topology 349
A.6 Connectedness A.6.1 Connected Space Definition 6.1.1 Assume that A; B3ðX; T Þ. If ðAXBÞWðAXBÞ ¼ B, then A and B are separate subsets. Definition 6.1.2 In ðX; T Þ, if there exist non-empty separate subsets A and B on X such that X ¼ AWB, then X is said to be disconnected. Non-disconnected spaces are called connected spaces. Proposition 6.1.1 In ðX; T Þ, the following conditions are equivalent. (1) X is disconnected (2) X can be represented by the union of two non-empty and mutually disjoint close sets, i.e., X ¼ AWB, AXB ¼ B, where A and B are non-empty close sets (3) X can be represented by the union of two non-empty and mutually disjoint open sets. (4) There exists non-empty both open and close proper subset on X. Definition 6.1.3 For A3ðX; T Þ, if A is regarded as a sub-space of X, then it’s connected; A is called a connected subset of X. Proposition 6.1.2 Y3ðX; T Þ is disconnected 5 there exist non-empty separate subsets A and B on X and Y ¼ AWB. Proposition 6.1.3 Assume that Y is a connected subset on ðX; T Þ. If A and B are separate subsets on X,Y ¼ AWB, then Y3A or Y3B. Proposition 6.1.4 Assume that A3ðX; T Þ is a connected subset. Let A3B3A. Then, B is a connected subset, especially A is connected. Proposition 6.1.5 Assume that fAa;a ˛ I g is a family of connected sets on ðX; T Þ and X Aa 6¼ B. a˛I Then, W Aa is connected. a˛I
350 Addenda A Proposition 6.1.6 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. If X is connected, then f ðXÞ is connected on Y. Proposition 6.1.7 If X1 ; X2 ; /Xn are connected spaces, then their product space X ¼ connected.
n Y
Xi is also
1
From R1 is connected, Rn is connected. Proposition 6.1.8 If f : ðX; T Þ/R is continuous, X is connected, and there exist a; b ˛ X such that f ðaÞ < f ðbÞ, then for cf ðaÞ < r < f ðbÞ, there must have c ˛ X such that f ðcÞ ¼ r. Proposition 6.1.9 If f : S1 /R is a continuous mapping, where S1 is a unit circle, then there exists z ˛ S1 such that f ðzÞ ¼ f ðz0 Þ, where z0 ¼ z.
A.6.2 Connected Component and Local Connectedness Definition 6.2.1 Assume that x and y are two points on topologic space ðX; T Þ. If there exists a connected set A3X such that x; y ˛ A, then x and y are called connected. The connected relation among points on ðX; T Þ is an equivalence relation. Definition 6.2.2 Each equivalent class with respect to connected relations on ðX; T Þ is called a connected component of X. Definition 6.2.3 For A3ðX; T Þ, if A is regarded as a sub-space, its connected component is called a connected component of subset A of X. Definition 6.2.4 In ðX; T Þ, for each neighborhood u of x ˛ X, there exist connected neighborhood v such that x ˛ v3u, then X is called local connected at point x. If for cx ˛ X X is local connected at x, then X is called a local connected space. Proposition 6.2.1 In ðX; T Þ, C is a connected component of X, then
Some Concepts and Properties of Point Set Topology 351 (1) If Y is a connected subset on X and YXC 6¼ B, then Y3C. (2) C is a connected subset. (3) C is a close set on X. Proposition 6.2.2 In ðX; T Þ, the following statements are equivalent. (1) X is a local connected space. (2) Any connected component of any open set of X is open. (3) There exists a base on X such that its each member is connected. Proposition 6.2.3 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. X is local connected. Then, f ðXÞ is also local connected. Proposition 6.2.4 If X1 ; X2 ; /Xn are local connected spaces, then their product space is also local connected. Proposition 6.2.5 If A3ðX; T Þ is a connected open set, then A must be a connected component of ðvAÞc .
A.6.3 Arcwise Connected Space Definition 6.3.1 f : ½0; 1/ðX; T Þ is a continuous mapping and is called an arc (or path) that connects points f ð0Þ and f ð1Þ on ðX; T Þ, where f ð0Þ and f ð1Þ are called start and end points of arc f , respectively. If f ð0Þ ¼ f ð1Þ, then f is called a circuit. If f is an arc on X, then f ð½0; 1Þ is called a curve on X. For cx; y ˛ X, if there exists an arc f : ½0; 1/ðX; T Þ such that f ð0Þ ¼ x and f ð1Þ ¼ y, then X is an arcwise connected space. For A3ðX; T Þ, regarding A as a sub-space, if A is arcwise connected, then A is an arcwise connected subset of X. Definition 6.3.2 For x; y ˛ ðX; T Þ, if there is an arc on X that connects x and y, then x and y are arcwise connected. All points on X are an equivalent relation with respect to arcwise connected relations.
352 Addenda A Definition 6.3.3 The points on ðX; T Þ that belong to an equivalent class with respect to arcwise connected relations are called an arcwise connected component of X. Proposition 6.3.1 If ðX; T Þ is arcwise connected, then X is connected. Proposition 6.3.2 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. If X is an arcwise connected space, then f ðXÞ is also an arcwise connected space. Proposition 6.3.3 If X1 ; X2 ; /Xn are arcwise connected spaces, then their product space is also an arcwise connected space. Corollary 6.3.3 Rn is an arcwise connected space. Proposition 6.3.4 (Bond Lemma) Assume that A; B3ðX; T Þ are close sets and X ¼ AWB. f1 : A/ðY; T 2 Þ and f2 : B/ðY; T 2 Þ are continuous mappings. f1 jAXB ¼ f2 jAXB, i.e., f1 and f2 are the same on f ðxÞ; x ˛ A AXB. Let f ðxÞ ¼ 1 . Then, f : ðX; T 1 Þ/ðY; T 2 Þ is continuous. f2 ðxÞ; x ˛ B Proposition 6.3.5 For A3Rn, if A is an open connected set, then A is arcwise connected. Definition 6.3.4 In ðX; T Þ, cx ˛ X for any neighborhood uðxÞ of x, if there exists a connected neighborhood vðxÞ such that x ˛ vðxÞ3uðxÞ, then X is called local arcwise connected. Proposition 6.3.6 If A3ðX; T Þ is local arcwise connected and A is connected, then A is arcwise connected. Proposition 6.3.7 If ðX; T Þ is local arcwise connected and A3X is an open connected subset, then A is arcwise connected. Proposition 6.3.8 The continuous image of a local arcwise connected space is also local arcwise connected.
Some Concepts and Properties of Point Set Topology 353 Proposition 6.3.9 If X1 ; X2 ; /Xn are local arcwise connected, then their product space is also local arcwise connected. Definition 6.3.5 ðX; dÞ is a metric space. For x; y ˛ X and cε > 0, if there exist a set of points x0 ¼ x; x1 ; x2 ; /; xn ¼ y; xi ˛ X such that dðxi ; xiþ1 Þ < ε; i ¼ 0; 1; 2; /; n 1, then ðx0 ; x1 ; /; xn Þ is called a εchain that connects points x and y. The above materials are from [Xio81]. The interested readers can also refer to [Eis74].
A.7 Order-Relation, Galois Connected and Closure Space A.7.1 Order-Relation and Galois Connected Definition 7.1.1 Assume that ‘ ’ is a binary relation on U and satisfies reflexivity and transitivity properties, i.e.,cx ˛ U, x x and cx; y; z ˛ U, if x y and y z, then x z, ‘ ’ is called a pre-order or quasi-order on U. Especially, if ‘ ’ satisfies transitivity and anti-reflexivity, i.e., for cx ˛ U, x x does not hold, then ‘ ’ is a strict pre-order on U denoted by ‘ < ’ generally. Definition 7.1.2 If a pre-order relation satisfies anti-symmetry, i.e., cx; y ˛ U, x y; y x0x ¼ y, then is called a partial order relation on U. ðU; Þ is called a partial ordered set. If a pre-order relation satisfies symmetry, i.e.,cx; y ˛ U, x y0y x, then is called an equivalence relation on U. Symbol is not used to denote equivalence relations generally. Definition 7.1.3 Assume that is a semi-order (partial-order) relation on U. For any two elements x; y ˛ U, if their supremum supfx; yg and infimum inf fx; yg exist, then ðU; Þ is a lattice. For a lattice ðU; Þ, x n y and x o y are used to represent the supremum and infimum of two elements x and y generally. Especially, if for any V4U, supfxjx ˛ Vg and inf fxjx ˛ Vg exist, then ðU; Þ is called a complete lattice. Definition 7.1.4 ðU; Þ is a semi-order set. cx; y ˛ U, if self-mapping 4 : U/U satisfies the following conditions
354 Addenda A (1) x 4ðxÞ (increasing property) (2) x y04ðxÞ 4ðyÞ (order-preserving) (3) 4ðxÞ ¼ 4ð4ðxÞÞ (idempotent) Then, 4 is a closure operator on ðU; Þ. Correspondingly, if a self-mapping f : U/U on U satisfies order-preserving, idempotent and decreasing property, i.e., fðxÞ x, then f is called an interior operator on ðU; Þ. Note 7.1.1 4 is a closure operator on ðU; Þ. A set 4ðUÞbfx ˛ Ujdy ˛ U s:t: 4ðyÞ ¼ xg of images is just a set composed by all fixed points of 4, i.e., 4ðUÞ ¼ fx ˛ Uj4ðxÞ ¼ xg. Elements of 4ðUÞ are called to be closed under the mapping 4. Especially, if ðU; Þ is a complete lattice, then 4ðUÞ is also a complete lattice. Note 7.1.2 Assume that 4 is a closure operator on a complete lattice ð2U ; 4Þ, where U is any given set and 2U is a power set of U. Then, 4 uniquely corresponds to a family U42U of subsets of U and U satisfies (1) U ˛ U, (2) cU 4U, XU ˛ U, U is called a Moore family on U, and two-tuple ðU; 4Þ is a closure system. Please refer to Davey and Priestley (1992) for more details. Galois Connection (Davey and Priestley, 1992)
Definition 7.1.5 Assume that ðU; U Þ and ðV; V Þ are a pair of semi-order structures. f : U/V and g : V/U are a pair of mappings. The domains of f and g are U and V, respectively. If f and g satisfy For cx ˛ U and cy ˛ V, xU gðyÞ5f ðxÞV y. Then, ðf ; gÞ is called a Galois connection between ðU; U Þ and ðV; V Þ as shown below, ðU; U Þ
ðV; V Þ
Proposition 7.1.1 ðf ; gÞ is a Galois connection between ðU; U Þ and ðV; V Þ, where f : U/V and g : V/U. If x; x1 ; x2 ˛ U and y; y1 ; y2 ˛ V, then we have the following conclusions. (1) xm gð f ðxÞÞ; f ðgðyÞÞn y (2) x1 U x2 0f ðx1 ÞV f ðx2 Þ; y1 V y2 0gðy1 ÞU gðy2 Þ
Some Concepts and Properties of Point Set Topology 355 (3) f ðgðf ðxÞÞÞ ¼ f ðxÞ; gðf ðgðyÞÞÞ ¼ gðyÞ Conversely, assume that f and g are a pair of mappings between ðU; U Þ and ðV; V Þ. For cx; x1 ; x2 ˛ U and cy; y1 ; y2 ˛ V, the above two conditions (1) and (2) hold. Then, f and g are a Galois connection between ðU; U Þ and ðV; V Þ. Proposition 7.1.2 Assume that ðf ; gÞ is a Galois connection between ðU; U Þ and ðV; V Þ, where f : U/V and g : V/U. Then the combination mapping g+f is a closure operator on ðU; U Þ, and f +g is an interior operator on ðV; V Þ.
A.7.2 Closure Operation and Closure Space The concept of closure operation that we previously introduced is under the order theory sense. The terms of closure operation, closure space and related properties that we will introduce below have the topologic sense, especially under E. Cech sense, i.e., based on set theory and always assuming that there does not appear paradox (Cech, 1966). Definition 7.2.1 U is a domain. If mapping cl : 2U /2U satisfies the following three axioms, where 2U is a power set of U, (cl1) clðBÞ ¼ B (cl2) cX4U, X4clðXÞ (cl3) cX4U and cY4U, clðXWYÞ ¼ clðXÞWclðYÞ then, cl is called a closure operation on U. Correspondingly, two-tuples ðU; clÞ is a closure space, and clðXÞ is a cl closure of subset X. If not causing confusion, the closure clðXÞ of subset X is denoted by X. Proposition 7.2.1 If ðU; clÞ is a closure space, then (1) clðUÞ ¼ U (2) For cX4U and cY4U, if X4Y, then clðXÞ4clðYÞ (3) For any family Xi ði ˛ I Þ of subsets of U, have clð X Xi Þ4 X clðXi Þ i˛I
i˛I
Definition 7.2.2 CðUÞ is a set composed by all closure operations on U, i.e., CðUÞ ¼fmjm is a closure operation on U}. Define a binary relation on CðUÞ as cm; n ˛ CðUÞ; n m5cX4U; nðXÞ4mðXÞ
356 Addenda A If n m holds, then closure operation m is said to be coarser than n. Equivalently, n is said to be finer than m. Theorem 7.2.1 Binary relation is a semi-order relation on CðUÞ. ðCðUÞ; Þ has a greatest element m1 and a least element m0 . That is, for cX4U, if X 6¼ B, then m1 ðXÞ ¼ U, otherwise m1 ðBÞ ¼ B; and m0 ðXÞ ¼ X. Furthermore, for any subset fmi ji ˛ Ig of CðUÞ and cX4U, we have ðsupfmi ji ˛ I gÞðXÞ ¼ Wfmi ðXÞji ˛ I g, i.e., CðUÞ is order complete with respect to . Definition 7.2.2 ðU; clÞ is a closure space. Mapping intcl : 2U /2U , induced by closure operation cl, is called an interior operation, denoted by int. Its definition is as follows cX4U; intðXÞ ¼ U clðU XÞ Correspondingly, intðXÞ is called clinterior of X, or simply interior. Proposition 7.2.3 ðU; clÞ is a closure space. If int is defined by Definition 7.2.2, then (int1) intðUÞ ¼ U (int2) cX4U, intðXÞ4X (int3) cX4U and cY4U, intðXXYÞ ¼ intðXÞXintðYÞ Assume that int 2U /2U satisfies axioms int1w int3. Define an operation cl as follows cX4U; clðXÞ ¼ U intðU XÞ It can be proved that cl is a closure operation on U and intcl ¼ int. If IðUÞ is a set of mappings int on U that satisfy axioms int1w int3, then there exists one-one correspondence between CðUÞ and IðUÞ. Or a closure operation and an interior operation are dual. Definition 7.2.3 ðU; clÞ is a closure space. int is a dual interior operation of cl. For cX4U, if clðXÞ ¼ X, then X is called a close set. If clðU XÞ ¼ U X, or equivalently, intðXÞ ¼ X, then X is called an open set. Proposition 7.2.4 ðU; clÞ is a closure space. int is a dual interior operation of cl. We have (1) intðBÞ ¼ B
Some Concepts and Properties of Point Set Topology 357 (2) For cX4U and cY4U, if X4Y, then intðXÞ4intðYÞ. (3) For any family Xi ði ˛ I Þ of subsets of U, have W intðXi Þ4intð W Xi Þ i˛I
i˛I
Definition 7.2.4 A topological closure operation on U is a closure operation cl that satisfies the following condition ðcl4ÞcX4U; clðclðXÞÞ ¼ clðXÞ If cl is a topological closure operation, then closure space ðU; clÞ is a topological space. Proposition 7.2.5 If ðU; clÞ is a closure space, then each condition shown below is the necessary and sufficient condition that ðU; clÞ is a topological space. (1) The closure of each subset is a close set (2) The interior of each subset is an open set (3) The closure of each subset equals to the intersection of all close sets that include the subset (4) The interior of each subset equals to the union of all open sets that include the subset. Theorem 7.2.2 Assume that O~ is a family of subsets of set U that satisfies the following conditions ~ U ˛ O~ (o1) B ˛ O, ~ i.e., O~ is closed for any union operation ~ WfAjA ˛ O~1 g ˛ O, ~ (o2) cO1 4O, ~ AXB ˛ O, ~ i.e., O~ is closed for finite intersection operation. (o3) cA; B ˛ O, Let CO~ ¼ fcljcl is a closure operation on U and the set composed by all open sets of ~ ðU; clÞ is just Og. Then, there just exists a topological closure operation clT on CO~ such that clT is the roughest element on CO~ . Theorem 7.2.3 Assume that C~ is a family of subsets of set U that satisfies the following conditions ~ U ˛ C~ (c1) B ˛ C, ~ i.e., C~ is closed for any union operation ~ ~ XfAjA ˛ C~1 g ˛ C, (c2) cC 1 4C, ~ ~ (c3) cA; B ˛ C, AWB ˛ C, i.e., C~ is closed for finite intersection operation. Then, there just exists a topological closure operation clT on U such that C~ is just a set that composed by all close sets on ðU; clT Þ.
358 Addenda A Using open set as a language to describe topology, axioms (o1) w (o3) are used. However, conditions (cl1) w (cl4) are called axioms of Kuratowski closure operator. Kuratowski closure operator, interior operator that satisfies axioms (int1) w (int3) and (int4):cX4U intðintðXÞÞ ¼ intðXÞ, open set and neighborhood system are equivalent tools for describing topology. For describing non-topologic closure spaces, only closure operations, interior operations and neighborhood systems can be used, but open set or close set cannot be used as a language directly. In some sense, closure spaces are more common than topologic spaces. We will discuss continuity, connectivity and how to construct a new closure space from a known one below. A closure operation cl on a domain set U is defined as a mapping from 2U to itself, where domain DomðclÞ ¼ 2U and codomain RanðclÞ42U . Closure operation cl is completely defined by binary relation R4U 2U , i.e., cx ˛ U and cX4U, xRX5x ˛ clðXÞ. Obviously, we have clðXÞ ¼ R1 ðXÞbfy ˛ UjyRXg. Compared to cl, relation R more clearly embodies the intuitive meaning of closure operation, i.e., what points are proximal to what sets. Naturally, the intuitive meaning of continuous mappings is the mapping that remains the ‘x is proximal to subset X’ relation. Definition 7.2.5 f is a mapping from closure space U to closure space V. For x ˛ U and cX4U, if x ˛ X, have f ðxÞ ˛ f ½X holds, then f is called continuous at x. If f is continuous at any x, then f is called continuous. Theorem 7.2.4 f is a mapping from closure space U to closure space V. The following statements are equivalent. (1) f is a continuous mapping (2) For cX4U, f ½X4f ½X holds. (3) For cY4V, f 1 ½Y4f 1 ½Y holds. Definition 7.2.6 f is an 1-1 correspondence (bijective mapping) from closure space U to closure space V. Both f and f 1 are continuous mappings. Then, f is called a homeomorphous mapping from U to V, or V is a homeomorph of U. Definition 7.2.7 If there exists a homeomorphous mapping from closure space U to V, then U and V are called homeomorphous closure spaces.
Some Concepts and Properties of Point Set Topology 359 Definition 7.2.8 If a closure space U has property P such that all spaces that homeomorphous to U have the property, then P is called the topological property. Obviously, the homeomorphous relation is an equivalent relation on the set composed by all closure spaces. Definition 7.2.9 ðU; mÞ is a closure space. For X4U, if there exist subsets X1 and X2 on U such that X ¼ X1 WX2 , and if ðmðX1 ÞXX2 ÞWðX1 WmðX2 ÞÞ ¼ B, then X1 ¼ B or X2 ¼ B, then X is called a connected subset of ðU; mÞ. Definition 7.2.10 f is a continuous mapping from closure space ðU; mÞ to closure space ðV; nÞ. If X4U is a connected subset, then f ðXÞ is a connected subset on ðV; nÞ. Below we will discuss how to generate a new closure operation from a known closure operation, or a set of closure operations. Two generated approaches are discussed, the generated projectively and generated inductively. The product topology and quotient topology discussed in point topology are special cases of the above two generated approaches in closure operation. Definition 7.2.11 fðU i ; mi Þji ˛ I g is a set of closure spaces. For any i ˛ I, the closure operation on V generated inductively by mapping fi : U i /V is defined as follows cX4V; ni ðXÞ ¼ XWfi mi fi1 ðXÞ The above closure operation is the finest one among all closure operations that make fi continuous. The closure operation on V generated inductively by a set ffi ji ˛ I g of mappings is defined as follows ðsupfvi ji ˛ I gÞðXÞ ¼ XW fi mi fi1 ðXÞ i ˛ I The above closure operation is the finest one among all closure operations that make each fi , i ˛ I continuous. Proposition 7.2.6 ðU; mÞ is a closure space. R is an equivalence relation on U, and its corresponding quotient set is ½U, where p : U/½U, pðxÞ ¼ ½x. The closure operation n generated
360 Addenda A inductively by p is defined as a quotient closure operation on ½U. And for cX4½U, nðXÞ ¼ XWp m p1 ðXÞ ¼ p m p1 ðXÞ Definition 7.2.12 fðU i ; mi Þji ˛ I g is a set of closure spaces. For any i ˛ I, the closure operation on V generated projectively by fi : V/U i is defined as follows cX4V; ni ðXÞ ¼ fi1 ðmi ðfi ðXÞÞÞ The above closure operation is the coarsest one among all closure operations that make fi continuous. The closure operation on V generated projectively by a set ffi ji ˛ I g of mappings is defined by inf fni ji ˛ I g. It is the coarsest one among all closure operations that make each fi , i ˛ I continuous. Note that ðinf fni ji ˛ I gÞðXÞ is not necessarily the Xffi1 ðmi ðfi ðXÞÞÞji ˛ I g. And the latter is not necessarily a closure operation, unless a set fðU i ; mi Þji ˛ I g of closure spaces satisfies a certain condition (Cech, 1966).
A.7.3 Closure Operations Defined by Different Axioms Two forms of closure that we mentioned previously are denotes by closure operator and closure operation, respectively. The former is under order theory sense and the latter is under topologic sense. In fact, the term of closure does not have a uniform definition. In different documents it might have different meanings. We introduce different definitions of closure, quasi-discrete closure space, Allexandroff topology, etc. below. U is a domain. Assume that Cl : 2U /2U is a given mapping. For cX4U, ClðXÞ is called the closure of subset X. ðU; ClÞ is called the most general closure space. Int : 2U /2U is a dual mapping of Cl, i.e., IntðXÞbU ClðU XÞ. IntðXÞ is called the interior of subset X. For convenience, for cX; Y ˛ 2U , the following axioms are introduced (Table 7.3.1). Table 7.3.1
Neighborhood space Closure space Smith space Cech closure space Topological space Alexandroff space Alexandroff topology
(CL0)
(CL1)
(CL2)
A A A A A A A
A A A A > > >
A A A A A A
A: the axiom satisfied by definition >: the property induced by definition.
(CL3)
(CL4)
(CL5)
A A A A > >
A A
A A
Some Concepts and Properties of Point Set Topology 361 (CL0) (CL1) (CL2) (CL3) (CL4) (CL5)
ClðBÞ ¼ B X4Y 0 ClðXÞ4ClðYÞ X4ClðXÞ ClðXWYÞ4ClðXÞWClðYÞ ClðXÞ ¼ ClðClðXÞÞ for any family fXi ji ˛ I g of subsets on U, W ClðXi Þ ¼ Clð W Xi Þ.
where, (CL1)þ(CL3) are equivalent to axiom
i˛I (CL3)0 : ClðXWYÞ
i˛I
¼ ClðXÞWClðYÞ.
Using the dual interior operation Int of Cl, we have the following equivalent axioms (CL0)w(CL5). For cX; Y ˛ 2U , we have (INT0) (INT1) (INT2) (INT3) (INT4) (INT5)
IntðUÞ ¼ U X4Y 0 IntðXÞ4IntðYÞ IntðXÞ4X IntðXÞXIntðYÞ4IntðXXYÞ IntðXÞ ¼ IntðIntðXÞÞ for any family fXi ji ˛ I g of subsets on U, X IntðXi Þ ¼ Intð X Xi Þ. i˛I
i˛I
where, (INT1)þ(INT3) are equivalent to INT30 : IntðXXYÞ ¼ IntðXÞXIntðYÞ. Note 7.3.1 Under the general order theory sense, the closure space is defined by axioms (CL1), (CL2) and (CL4). For example, the closure operation defined by Definition 7.1.4 is called closure operator. When considering the inclusion relation between a power set and a subset, the axiom (CL0) may be or may not be satisfied. Note 7.3.2 Under the Cech’s sense, the closure space is called pre-topology and is defined by axioms (CL0)w(CL3). In Definition 7.1.4, axioms (CL0) and (CL3) are replaced by (CL3)’. The topology described by the Kuratowski closure operator that satisfies axioms (CL0), (CL2), (CL3)’ and (CL4) is equivalent to the above description, since axiom (CL3)’ may induce axiom (CL3), and (CL4)þ(CL3) may induce (CL3)’. The distinction between the closure space in the Cech’s sense and the topologic space in general sense is the satisfaction of the idempotent axiom or not. So the former is the extension of the latter. Note 7.3.3 Axiom (CL5) is called Alexandroff property. The topologic space that satisfies the Alexandroff property is called Alexandroff topology. In Cech (1966) and Galton (2003), axiom (CL5) is called quasi-discrete property. The Cech closure space that satisfies quasidiscrete property is called quasi-discrete closure space.
362 Addenda A Note 7.3.4 To describe the closure space, except closure and interior operations, the neighborhood and the filter convergent sequence can be used equivalently. In Table 7.3.1, the neighborhood and Smith spaces (Kelly, 1955; Smith, 1995) originally are described by neighborhood language; we use the equivalent closure axioms.
Some Concepts and Properties of Point Set Topology A.1 Relation and Mapping A.1.1 Relation Definition 1.1.1 X and Y are any two sets. fðx; yÞjx ˛ X; y ˛ Yg is called a Cartesian product of X and Y, denoted by X Y, where ðx; yÞ is a pair of ordered elements. x is the first coordinate of ðx; yÞ, and y is the second coordinate of ðx; yÞ. X is a set of the first coordinates of X Y, and Y is a set of the second coordinates of X Y. Definition 1.1.2 X and Y are two sets. For any R3 X Y, R is called a relation from X to Y. Assume that R is a relation from X to Y. If ðx; yÞ ˛ R, then x and y are Rrelevant, denoted by xRy. Set fxjdy; ðx; yÞ ˛ Rg is called the domain of R, denoted by DðRÞ. Set fyjðx; yÞ ˛ R; x ˛ DðRÞg is called the range of R, denoted by RðRÞ. For A3X, letting fyjdx ˛ A; ðx; yÞ ˛ Rg ¼ RðAÞ, RðAÞ is called a set of images (or image) of A. For B3Y, letting fxjdy ˛ B; ðx; yÞ ˛ Rg ¼ R1 ðBÞ, R1 ðBÞ is called the preimage of B. Definition 1.1.3 For R3X Y; S3Y Z, letting T ¼ fðx; zÞjdy ˛ Y; ðx; yÞ ˛ R; ðy; zÞ ˛ Sg, T is called the composition of R and S, denoted by T ¼ S+R. For R3X Y, letting R1 ¼ fðy; xÞjðx; yÞ ˛ Rg3Y X, R1 is called the inverse of R. Proposition 1.1.1 For R3X Y; S3Y Z and T3Z U, we have
333
334 Addenda A (1) ðR1 Þ1 ¼ R (2) ðS+RÞ1 ¼ R1 +S1 (3) T+ðS+RÞ ¼ ðT+SÞ+R (4) cA; B3X, RðAWBÞ ¼ RðAÞWRðBÞ and RðAXBÞ3RðAÞXRðBÞ (5) ðS+RÞðAÞ ¼ SðRðAÞÞ Note that in (4) RðAXBÞ3RðAÞXRðBÞ rather than RðAXBÞ ¼ RðAÞXRðBÞ.
A.1.2 Equivalence Relation Definition 1.2.1 Assume that R is a relation from X to X (or a relation on X) and satisfies (1) DðXÞ ¼ fðx; xÞjx ˛ Xg3R (Reflexivity) (2) R ¼ R1 (Symmetry) (3) R+R3R (Transitivity) R is called an equivalence relation on X. Assume that R is an equivalence relation on X. For cx ˛ X, letting ½xR ¼ fyjyRx; y ˛ Xg, ½xR is an Requivalent set of x. Definition 1.2.2 For ca; Aa 3X, if Aa XAb ¼ B; a 6¼ b and W Aa ¼ X, then fAa g is a partition of x. a
Proposition 1.2.1 R is an equivalence relation on X. Then, f½xR jx ˛ Xg is a partition of X.
A.1.3 Mapping and OneeOne Mapping Definition 1.3.1 F is a relation from X to Y. For cx ˛ X, if there exists a unique y ˛ Y such that ðx; yÞ ˛ F, then F is called a mapping from X to Y, denoted by F : X/Y. If RðFÞ ¼ Y, F is called surjective, where RðFÞ is the range of F. For x1 ; x2 ˛ X, if x1 6¼ x2 0Fðx1 Þ 6¼ Fðx2 Þ, F is called 1-1 mapping. Proposition 1.3.1 f : X/Y is a mapping. For cA; B3X, we have f ðAWBÞ ¼ f ðAÞWf ðBÞ f ðAXBÞ ¼ f ðAÞXf ðBÞ A3f f 1 ðAÞ
Some Concepts and Properties of Point Set Topology 335 If A3B, then f ðAÞ3f ðBÞ. For cA; B3Y, we have f 1 ðAWBÞ ¼ f 1 ðAÞWf 1 ðBÞ f 1 ðAXBÞ ¼ f 1 ðAÞXf 1 ðBÞ f f 1 ðAÞ 3A; f 1 AC ¼ f 1 ðAÞC
If A3B, then f 1 ðAÞ3f 1 ðBÞ. If f is surjective, then cA3X, f ðAC ÞIf ðAÞC . If f is a 1-1 mapping, then f ðAC Þ3f ðAÞC . Where, AC is the complement of A. f 1 is the inverse of f. If f is surjective and 1-1 mapping, then A ¼ f 1 ðf ðAÞÞ and f ðAC Þ ¼ f ðAÞC . Definition 1.3.2 Assume that X is a Cartesian product of X1 ; X2 ; /; Xn . Let x ¼ ðx1 ; x2 ; /; xn Þ ˛ X. Define pi : X/Xi ; pi ðxÞ ¼ xi . pi is the projection of X on Xi , or a set of the i-th coordinates.
A.1.4 Finite Set, Countable Set and Uncountable Set Definition 1.4.1 A and B are two sets. If there exists a 1-1 surjective mapping from A to B, A and B are called equinumerous. Any set that is not equinumerous to its proper subsets is a finite set. A set that is equinumerous to the set N of all natural numbers is a countable set. An infinite set that is not equinumerous to the set N of all natural numbers is an uncountable set. Theorem 1.4.1 (Bernstein) If A and the subset of B are equinumerous, and B and the subset of A are also equinumerous, A and B are equinumerous.
A.2 Topology Space A.2.1 Metric Space X is a non-empty set. d : X X/R is a mapping, where R is a real set. cx; y; z ˛ X, d satisfies: (1) dðx; yÞ 0 and dðx; yÞ ¼ 05x ¼ y
336 Addenda A (2) dðx; yÞ ¼ dðy; xÞ (3) dðx; zÞ dðx; yÞ þ dðy; zÞ Then, d is a distance function on X and ðX; dÞ is a metric space. Definition 2.1.2 ðX; dÞ is a metric space. For x ˛ X; cε > 0, fyjdðx; yÞ < ε; y ˛ Xg ¼ Bðx; εÞ. Bðx; εÞ is called a spherical neighborhood with x as its center and ε as its radius, or simply ε neighborhood. Proposition 2.1.1 ðX; dÞ is a metric space. Its spherical neighborhoods have the following properties. (1) cx ˛ X, there is one neighborhood at least. cBðx; εÞ, have x ˛ Bðx; εÞ. (2) x ˛ X, for any two spherical neighborhoods Bðx; ε1 Þ and Bðx; ε2 Þ, there exists Bðx; ε3 Þ such that Bðx; ε3 Þ3Bðx; ε1 ÞXBðx; ε2 Þ. (3) If y ˛ Bðx; εÞ, then there exists Bðy; ε1 Þ3Bðx; εÞ.
A.2.2 Topological Space Definition 2.2.1 X is a non-empty set. T is a family of subsets of X. If T satisfies the following conditions (1) X; B ˛ T (2) A; B ˛ T , AXB ˛ T (3) T 1 3T , W A ˛ T A˛T 1
then T is a topology of X. ðX; T Þ is a topologic space. Each member of T is called an open set on ðX; T Þ. ðX; dÞ is a metric space. For A3X and cx ˛ A, if there exists Bðx; εÞ3A, then A is an open set on X. Let T d be a family of all open sets on X. It can be proved that T d is a topology on X. ðX; T d Þ is called a topologic space induced from d. Definition 2.2.2 ðX; T Þ is a topologic space (ðX; T Þ always indicates a topologic space below). For x ˛ X and U ˛ T , if x ˛ U, then U is called a neighborhood of x denoted by UðxÞ. For x ˛ X, the set of all neighborhoods of x is called a system of neighborhoods of x, denoted by Ux. Proposition 2.2.1 ðX; dÞ is a topologic space. For x ˛ X, Ux is a neighborhood system of x. We have
Some Concepts and Properties of Point Set Topology 337 (1) cx ˛ X; Ux 6¼ B and cu ˛ Ux , then x ˛ u. (2) If u; v ˛ Ux , then uXv ˛ Ux . (3) If u ˛ Ux , there exist v3u; v ˛ Ux such that for cy ˛ v have v ˛ Uy .
A.2.3 Induced Set, Close Set and Closure Definition 2.3.1 For A3ðX; T Þ; x ˛ X, if cu ˛ Ux , uXðA=fxgÞ 6¼ B, then x is called an accumulation (limit) point of A. Set A0 of all accumulation points of A is called an induced set of A. Proposition 2.3.1 For cA; B3ðX; T Þ, we have (1) (2) (3) (4)
B0 ¼ B A3B A0 3B0 ðAWBÞ0 ¼ A0 WB0 ðA0 Þ0 3AWA0
Definition 2.3.2 For A3ðX; T Þ, if all accumulation points of A belong to A, then A is a close set. Proposition 2.3.2 A is close 5Ac is open. Proposition 2.3.3 Assume that F is a family of all close sets on ðX; T Þ. We have (1) X; B ˛ F (2) If A; B ˛ F , then AWB ˛ F . (3) If F 13 F , then X A ˛ F . A ˛ F1
Definition 2.3.3 For A3ðX; T Þ, letting A ¼ AWA0 , A is called a closure of A. Proposition 2.3.4 For cA; B3ðx; T Þ, we have (1) B ¼ B (2) A3 A
338 Addenda A (3) AWB ¼ AWB (4) A ¼ A Definition 2.3.4 For A3ðX; dÞ; x ˛ X, define dðx; AÞ ¼ inffdðy; xÞjy ˛ Ag. Proposition 2.3.5 For A3ðX; dÞ, we have (1) x ˛ A0 5dðx; ðA fxgÞÞ ¼ 0 (2) x ˛ A5dðx; AÞ ¼ 0
A.2.4 Interior and Boundary Definition 2.4.1 For A3ðX; T Þ, letting A0 ¼ fxjdu ˛ U; x ˛ u3Ag, A0 is called the interior (core) of A. Proposition 2.4.1 For A3ðX; T Þ, we have (1) (2) (3) (4) (5) (6)
A is open 5A0 ¼ A ðA0 Þc ¼ ðAc Þ; ðAc Þ0 ¼ ðAÞc X0 ¼ X A0 3A ðAXBÞ0 ¼ A0 XB0 ðA0 Þ0 ¼ A0
Definition 2.4.2 For A3ðX; T Þ; x ˛ X, if cu ˛ Ux , uXA 6¼ B and uXAc 6¼ B, x is called a boundary point of A. The set of all boundary points of A is called boundary of A, denoted by vA. Proposition 2.4.2 For A3ðX; T Þ, we have (1) (2) (3) (4) (5)
A ¼ AWvA A0 ¼ A vA vA ¼ AXðAc Þ vA0 3vA; vðAÞ3vA vðAWBÞ3vAWvB; vðvBÞ3vB
Some Concepts and Properties of Point Set Topology 339
A.2.5 Topological Base and Subbase Definition 2.5.1 ðX; T Þ is a topologic space. For B3T and cu ˛ T , if there exists B1 3B such that u ¼ W v, then B is a base of T . v ˛ B1
Proposition 2.5.1 ðX; dÞ is a space. ðX; T d Þ is a topologic space induced from d. Then, T 1 ¼{all spherical neighborhoods of x, cx ˛ X } is a base of T d . Proposition 2.5.2 B is a family of open sets on ðX; T Þ. Then, B is a base 5cu ˛ T and x ˛ u, there is v ˛ B such that x ˛ v3u. Proposition 2.5.3 B is a family of subsets of X and satisfies (1) X ¼ W u u˛B
(2) If B1 ; B2 ˛ B, for cx ˛ B1 XB2, there exists BðxÞ ˛ B such that x ˛ BðxÞ3B1 XB2 . Then, let T ¼ fAjA ¼ W u; cB1 3Bg be a topology of X and B be a base of T . u ˛ B1
Definition 2.5.2 ðX; T Þ is a space. 4 is a sub-family of T . If si ˛ 4; i ¼ 1; 2; .; n; n ˛ N, letting s1 Xs2 X:::Xsi ˛ B, i.e., B is a family of sets composed by the intersections of any finite number of elements in 4, then B is a base of T , and 4 is a subbase of T .
A.2.6 Continuous Mapping and Homeomorphism Definition 2.6.1 f : ðX; T 1 Þ/ðY; T 2 Þ is a mapping. If cu ˛ T 2 ; f 1 ðuÞ ˛ T 1 , then f is a continuous mapping. If x ˛ X, cw ˛ T 2 and f ðxÞ ˛ w, have f 1 ðwÞ ˛ T 1 , then f is continuous at x. Proposition 2.6.1 For f : ðX; T 1 Þ/ðY; T 2 Þ, the following statements are equivalent. (1) f is a continuous mapping (2) If B is a base of Y, then cu ˛ B, f 1 ðuÞ ˛ T 1 . (3) f 1 ðFÞ is a preimage of any close set F in Y; f 1 ðFÞ is close in X.
340 Addenda A (4) 4 is a subbase of Y; cu ˛ 4, have f 1 ðuÞ ˛ T 1 . (5) cA3X, have f ðAÞ3ðf ðAÞÞ. (6) cB3Y, have f 1 ðBÞIðf 1 ðBÞÞ. Proposition 2.6.2 For f : ðX; T 1 Þ/ðY; T 2 Þ; x ˛ X, the following statements are equivalent. (1) f is continuous at x. (2) For all neighborhoods uðf ðxÞÞ of f ðxÞ, there exists uðxÞ ˛ U such that f ðuðxÞÞ3uðf ðxÞÞ. Proposition 2.6.3 If f : ðX; T 1 Þ/ðY; T 2 Þ and g : ðY; T 2 Þ/ðZ; T 3 Þ are continuous, then g+f : ðX; T 1 Þ/ðZ; T 3 Þ is continuous. Definition 2.6.3 ðX; T 1 Þ and ðY; T 2 Þ are two spaces. If there exists f : ðX; T 1 Þ/ðY; T 2 Þ, where f is a 1-1 surjective and bicontinuous mapping, i.e., both f and f 1 are continuous, then f is called a homeomorphous mapping from X to Y, or X and Y are homeomorphism.
A.2.7 Product Space and Quotient Space Definition 2.7.1 T 1 and T 2 are two topologies on X. If T 1 3T 2 , T 1 is called smaller (coarser) than T 2 . fT a ; a ˛ Ig is a family of topologies on X. If there exists T a0 such that cT a , T a0 3T a , then T a0 is called the smallest (coarsest) topology in fT a g. Similarly, we may define the concept of the largest (finest) topology. Proposition 2.7.1 Assume that ca ˛ I, fa : X/ðYa ; T a Þ. There exists the smallest (coarsest) topology among topologies on X that make each fa continuous. Proposition 2.7.2 Assume that ca ˛ I, fa : X/ðYa ; T a Þ. There exists the largest (finest) topology among topologies on X that make each fa continuous. Corollary 2.7.2 Assume that f : ðX; T Þ/Y. There exists the largest (finest) topology among topologies on Y that make f continuous. The topology is called the quotient topology with respect to T and f .
Some Concepts and Properties of Point Set Topology 341 Definition 2.7.2 For A3ðX; T Þ, letting T A ¼ fuju ¼ AXv; v ˛ Tg, ðA; T A Þ is called the subspace of ðX; T Þ. Definition 2.7.3 Assume that X ¼
Q a˛I
Xa ; I 6¼ B, where
Q a˛I
Xa indicates the product set. fðXa ; T a Þ; a ˛ Ig
is a family of topologic spaces. Let pa : X/Xa be a projection. T is the smallest topology among topologies on X that make pa ðca ˛ IÞ continuous. ðX; T Þ is called the product Q topologic space of fðXa ; T a Þg, denoted by ðX; T Þ ¼ ðXa ; T a Þ. a˛I
Proposition 2.7.3 Assume that ðX; T Þ is a product topologic space of fðXa ; T a Þ; a ˛ Ig. Letting 4 ¼ fp1 a ðua Þjua ˛ T a ca ˛ Ig, 4 is a subbase of T . Proposition 2.7.4 Assume that ðX; T Þ is a product topologic space of fðXa ; T a Þ; a ˛ Ig. f : ðY; T 0 Þ/ðX; TÞ is continuous 5ca ˛ I, pa +f : ðY; T 0 Þ/ðXa ; T a Þ is continuous. Proposition 2.7.5 Assume that ðX; T Þ is a product topologic space of fðXa ; T a Þ; a ˛ Ig. Then, series fxi g on X converges to x0 ˛ X5ca ˛ I, series pa ðxi Þ on Xa converges to pa ðx0 Þ. Where, the definition of convergence is that for fxi g3ðX; T Þ; x ˛ X, if cu ˛ Ux , there exists n0 such that when n > n0 , xn ˛ u. Then fxi g is called to be converging to x, denoted by lim xn ¼ x. n/N
Definition 2.7.4 R is an equivalence relation on ðX; T Þ. Let p be a nature projection X/X=R ðpðxÞ ¼ ½xÞ, and ½T be the finest topology that makes p continuous. ðX=R; ½T Þ is called the quotient space of ðX; T Þ with respect to R. Where, X=R may be indicated by ½XR, or ½X. Proposition 2.7.6 Assume that ð½X; ½T Þ is a quotient topologic space of ðX; T Þ with respect to R. Then, ½T ¼ fuju3½X; p1 ðuÞ ˛ T; p : X/½Xg. Definition 2.7.5 For f : ðX; T Þ/Y, letting T =f ¼ fuju3Y; f 1 ðuÞ ˛ T g, T =f is called the quotient topology of ðX; T Þ with respect to f. We have a topologic space ðY; T =f Þ and ðY; T =f Þ is a congruence space of T and f.
342 Addenda A Proposition 2.7.7 f : ðX; T 1 Þ/ðY; T 2 Þ is an open (close) surjective mapping. Then, T 2 ¼ T 1 =f . Proposition 2.7.8 ðY; T 2 Þ is an congruence space of T and f. Assume that f : ðX; T 1 Þ/ðY; T 2 Þ and g : ðY; T 2 Þ/ðZ; T 3 Þ. Then, g is continuous 5g+f is continuous.
A.3 Separability Axiom A.3.1 T0 , T1 , T2 Spaces Definition 3.1.1 ðX; T Þ is a space. For cx; y ˛ X; x 6¼ y, there is u ˛ Ux such that y;u, or there is u ˛ Uy such that x;u, X is called T0 space. Definition 3.1.2 ðX; T Þ is a space. For cx; y ˛ X; x 6¼ y, there must be u ˛ Ux ; v ˛ Uy such that y;u; x;v, X is called T1 space. Definition 3.1.3 ðX; T Þ is a space. For cx; y ˛ X; x 6¼ y, there must be u ˛ Ux ; v ˛ Uy such that uXv ¼ B, X is called T2 space, or Hausdorff space. Proposition 3.1.1 X is a T0 space 5cx; y ˛ X; x 6¼ y, fxg 6¼ fyg, where fxg is the closure of singleton fxg. It means that the closures of any two different singletons are different. Proposition 3.1.2 ðX; T Þ is a topologic space. The following statements are equivalent. (1) X is a T1 space. (2) Each singleton on X is a close set. (3) Each finite set on X is a close set. Proposition 3.1.3 ðX; T Þ is a T1 space 5cx ˛ X, the intersection of all neighborhoods containing x is just fxg. Proposition 3.1.4 A3ðX; T Þ, X is a T1 space. Then, cx ˛ A0 5cu ˛ Ux , uXðA fxgÞ is an infinite set.
Some Concepts and Properties of Point Set Topology 343 Proposition 3.1.5 ðX; T Þ is a T2 space. Then, the convergent series on X has only one limit point. Proposition 3.1.6 ðX; T Þ is a T2 space 5 the diagonal D ¼ fðx; xÞjx ˛ Xg of product topologic space on X X is a close set.
3.2 T3 , T4 , Regular and Normal Space Definition 3.2.1 In space ðX; T Þ,cA3X, A is close. For x;A, if there exist open sets u and v, uXv ¼ B, such that x ˛ v; A3u, then X is called a T3 space. Definition 3.2.2 In space ðX; T Þ, for cA; B3X, if there exist open sets u and v such that A3u; B3v; uXv ¼ B, then X is called a T4 space. Proposition 3.2.1 ðX; T Þ is a T3 space 5cx ˛ X and u ˛ Ux , there exists v ˛ Ux such that v3u. Proposition 3.2.2 ðX; T Þ is a T4 space 5 for any close set A in X and any open set u that contains A, i.e., A3u, there exists open set v such that A3v3v3u. Proposition 3.2.3 ðX; T Þ is a T4 space 5 For close sets A; B3X; AXB ¼ B, there exists a continuous mapping f such that ðX; T Þ/½0; 1 and f ðAÞ ¼ 0; f ðBÞ ¼ 1. Proposition 3.2.4 (Tietz Theorem) ðX; T Þ is a T4 space 5 For any close set cA3X and any continuous function f0 : A/½0; 1 on A, there exists a continuous expansion f : X/½0; 1 of f0 on X. Definition 3.2.3 If ðX; T Þ is a T1 and T3 space, then X is called a regular space. Definition 3.2.4 If ðX; T Þ is a T1 and T4 spaces, then X is called a normal space. Proposition 3.2.5 ðX; T Þ is a normal space 0X is a regular space 0X is a T2 space 0X is a T1 space 0X is a T0 space.
344 Addenda A
A.4 Countability Axiom A.4.1 The First and Second Countability Axioms Definition 4.1.1 If ðX; T Þ has countable base, then X is said to satisfy the second countability axiom. Definition 4.1.2 If in ðX; T Þ, for cx ˛ X, there exists countable local base, then X is said to satisfy the first countability axiom. Proposition 4.1.1 Real space R satisfies the second countability axiom. Proposition 4.1.2 If ðX; dÞ is a metric space, then X satisfies the first countability axiom. Proposition 4.1.3 If ðX; T Þ satisfies the second countability axiom, then X satisfies the first countability axiom. Proposition 4.1.4 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuously open and surjective mapping. If X satisfies the second (or first) countability axiom, then Y will satisfy the second (or first) countability axiom. Definition 4.1.3 If ðX; T Þ has property P and any sub-space of X also has the property P, property P is called having heredity. Q If for cXi has property P and their product space X ¼ Xi also has property P, then P is i ˛ I called having integrability. The relation among separation axiom, countability axiom, heredity and integrability is shown in Table 4.1.1. Table 4.1.1
heredity integrability
T0
T1
T2
T3
T4
A1
A2
Separable
O O
O O
O O
O O
O O
O O
O
Where, A1 and A2 are the first and second countability axioms, respectively. O (countable) means that the product space of the countable number of metric spaces is metrizable.
Distance O O(countable)
Some Concepts and Properties of Point Set Topology 345 Proposition 4.1.5 For f : ðX; T 1 Þ/ðY; T 2 Þ, if X is countable, then f is continuous at x ˛ X 5cxi /x, have f ðxi Þ/f ðxÞ.
A.4.2 Separable Space Definition 4.2.1 If D3ðX; T Þ and D ¼ X, then D is called dense in X, or D is a dense subset of X. Proposition 4.2.1 Assume that D is a dense subset in ðX; T Þ. f : X/R and g : X/R are two continuous mappings. Then, f ¼ g5 f ¼ g on D. Definition 4.2.2 If ðX; T Þ has dense countable subsets, X is called a separable space. Proposition 4.2.1 If ðX; T Þ satisfies A2 , then X is separable. Proposition 4.2.2 If a separable metric space satisfies A2 , then it must be A1 . The relation among A1 , A2 and metric spaces is shown below. Separable A2
metric space
A1 Separable
Where, A / C indicates that property A with the addition of property B infers property C. B
A.4.3 Lindelof Space Definition 4.3.1 A is a family of sets and B is a set. If B3 W A, then A is called a cover of set B. When A˛A
A is countable or finite, A is called a countable or finite cover. If a family A of sets covers B and sub-family A1 of A also covers B, then A1 is called a sub-cover of A. If each set of cover A is open (closed), then A is called an open (closed) cover.
346 Addenda A Definition 4.3.2 In ðX; T Þ, for any open cover of X, there exists countable sub-cover, X is called a Lindelof space. Proposition 4.3.1 If ðX; T Þ satisfies A2 , then X is a Lindelof space. Corollary 4.3.1 An n-dimensional Euclidean space Rn is a Lindelof space. Proposition 4.3.2 If ðX; dÞ is a Lindelof space, then X satisfies A2 . Proposition 4.3.3 If any sub-space in ðX; T Þ is a Lindelof space, then each uncountable subset A of X must have accumulation points of A.
A.5 Compactness A.5.1 Compact Space Definition 5.1.1 In ðX; T Þ, if each open cover of X has its finite sub-covers, then X is called a compact space. Definition 5.1.2 Assume that A is a family of sets. If each finite sub-family in A has non-empty intersection, then A is said to have the finite intersection property. Proposition 5.1.1 ðX; T Þ is compact 5 each family of close sets that has the finite intersection property in X has non-empty intersection. Proposition 5.1.2 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. If X is compact, then f ðXÞ is also compact. Proposition 5.1.3 Each close subset of a compact set is compact.
Some Concepts and Properties of Point Set Topology 347 Proposition 5.1.4 If Xi ; i ˛ I, is compact, then their product space is compact as well.
A.5.2 Relation between Compactness and Separability Axiom Proposition 5.2.1 A compact subset in T2 is close. Proposition 5.2.2 A compact T2 space is a normal space. Proposition 5.2.3 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. If X is compact and Y is T2 , then f is a close mapping, i.e., mapping a close set to a close set. Proposition 5.3.4 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous and 1-1 surjective mapping. If X is compact and Y is T2 , then f is homeomorphous. Proposition 5.2.5 If A3Rn and Rn is an n-dimensional Euclidean space, then A is compact 5 A is a bounded close set. Proposition 5.2.6 f ðX; T Þ/R is a continuous mapping. If X is compact, there exist x0 ; y0 ˛ X such that cx ˛ X, f ðx0 Þ f ðxÞ f ðy0 Þ.
A.5.3 Some Relations in Compactness Definition 5.3.1 A topological space ðX; T Þ is countably compact if every countable open cover has a finite subcover. Definition 5.3.2 A topological space ðX; T Þ is said to be limit point compact if every infinite subset has a limit point. Definition 5.3.3 A topological space ðX; T Þ is sequentially compact if every infinite sequence has a convergent subsequence.
348 Addenda A
Compactness
Countable compactness Lindelof A2
Limit point compactness T1
Sequential compactness
In metric space, especially in n-dimensional Euclidean space, the four concepts of compactness, limit point compactness, countable compactness, and sequential compactness are equivalent.
A.5.4 Local Compact and Paracompact Definition 5.4.1 In ðX; T Þ, for each point on X there exists a compact neighborhood, and X is called a local compact space. Definition 5.4.2 Assume that A1 and A2 are two covers of X. If each member of A1 is contained by some member of A2 , then A1 is called the refinement of A2 . Definition 5.4.3 In ðX; T Þ, A is a cover of subset A. If cx ˛ A, there exists uðxÞ ˛ U such that uðxÞ only intersects with the finite number of members in A, then A is called a the local finite cover of A. Definition 5.4.4 In ðX; T Þ, for each open cover of A on X, there exists local finite cover A1, where A1 is the refinement of A, then X is called a paracompact space. Proposition 5.4.1 Each locally compact and T2 space are normal spaces. Proposition 5.4.2 Each paracompact normal space is a regular space. The relation among compactness, paracompactness and local compactness is shown below. Paracompact
Compact A2+T2
Locally Compact
Some Concepts and Properties of Point Set Topology 349
A.6 Connectedness A.6.1 Connected Space Definition 6.1.1 Assume that A; B3ðX; T Þ. If ðAXBÞWðAXBÞ ¼ B, then A and B are separate subsets. Definition 6.1.2 In ðX; T Þ, if there exist non-empty separate subsets A and B on X such that X ¼ AWB, then X is said to be disconnected. Non-disconnected spaces are called connected spaces. Proposition 6.1.1 In ðX; T Þ, the following conditions are equivalent. (1) X is disconnected (2) X can be represented by the union of two non-empty and mutually disjoint close sets, i.e., X ¼ AWB, AXB ¼ B, where A and B are non-empty close sets (3) X can be represented by the union of two non-empty and mutually disjoint open sets. (4) There exists non-empty both open and close proper subset on X. Definition 6.1.3 For A3ðX; T Þ, if A is regarded as a sub-space of X, then it’s connected; A is called a connected subset of X. Proposition 6.1.2 Y3ðX; T Þ is disconnected 5 there exist non-empty separate subsets A and B on X and Y ¼ AWB. Proposition 6.1.3 Assume that Y is a connected subset on ðX; T Þ. If A and B are separate subsets on X,Y ¼ AWB, then Y3A or Y3B. Proposition 6.1.4 Assume that A3ðX; T Þ is a connected subset. Let A3B3A. Then, B is a connected subset, especially A is connected. Proposition 6.1.5 Assume that fAa;a ˛ I g is a family of connected sets on ðX; T Þ and X Aa 6¼ B. a˛I Then, W Aa is connected. a˛I
350 Addenda A Proposition 6.1.6 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. If X is connected, then f ðXÞ is connected on Y. Proposition 6.1.7 If X1 ; X2 ; /Xn are connected spaces, then their product space X ¼ connected.
n Y
Xi is also
1
From R1 is connected, Rn is connected. Proposition 6.1.8 If f : ðX; T Þ/R is continuous, X is connected, and there exist a; b ˛ X such that f ðaÞ < f ðbÞ, then for cf ðaÞ < r < f ðbÞ, there must have c ˛ X such that f ðcÞ ¼ r. Proposition 6.1.9 If f : S1 /R is a continuous mapping, where S1 is a unit circle, then there exists z ˛ S1 such that f ðzÞ ¼ f ðz0 Þ, where z0 ¼ z.
A.6.2 Connected Component and Local Connectedness Definition 6.2.1 Assume that x and y are two points on topologic space ðX; T Þ. If there exists a connected set A3X such that x; y ˛ A, then x and y are called connected. The connected relation among points on ðX; T Þ is an equivalence relation. Definition 6.2.2 Each equivalent class with respect to connected relations on ðX; T Þ is called a connected component of X. Definition 6.2.3 For A3ðX; T Þ, if A is regarded as a sub-space, its connected component is called a connected component of subset A of X. Definition 6.2.4 In ðX; T Þ, for each neighborhood u of x ˛ X, there exist connected neighborhood v such that x ˛ v3u, then X is called local connected at point x. If for cx ˛ X X is local connected at x, then X is called a local connected space. Proposition 6.2.1 In ðX; T Þ, C is a connected component of X, then
Some Concepts and Properties of Point Set Topology 351 (1) If Y is a connected subset on X and YXC 6¼ B, then Y3C. (2) C is a connected subset. (3) C is a close set on X. Proposition 6.2.2 In ðX; T Þ, the following statements are equivalent. (1) X is a local connected space. (2) Any connected component of any open set of X is open. (3) There exists a base on X such that its each member is connected. Proposition 6.2.3 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. X is local connected. Then, f ðXÞ is also local connected. Proposition 6.2.4 If X1 ; X2 ; /Xn are local connected spaces, then their product space is also local connected. Proposition 6.2.5 If A3ðX; T Þ is a connected open set, then A must be a connected component of ðvAÞc .
A.6.3 Arcwise Connected Space Definition 6.3.1 f : ½0; 1/ðX; T Þ is a continuous mapping and is called an arc (or path) that connects points f ð0Þ and f ð1Þ on ðX; T Þ, where f ð0Þ and f ð1Þ are called start and end points of arc f , respectively. If f ð0Þ ¼ f ð1Þ, then f is called a circuit. If f is an arc on X, then f ð½0; 1Þ is called a curve on X. For cx; y ˛ X, if there exists an arc f : ½0; 1/ðX; T Þ such that f ð0Þ ¼ x and f ð1Þ ¼ y, then X is an arcwise connected space. For A3ðX; T Þ, regarding A as a sub-space, if A is arcwise connected, then A is an arcwise connected subset of X. Definition 6.3.2 For x; y ˛ ðX; T Þ, if there is an arc on X that connects x and y, then x and y are arcwise connected. All points on X are an equivalent relation with respect to arcwise connected relations.
352 Addenda A Definition 6.3.3 The points on ðX; T Þ that belong to an equivalent class with respect to arcwise connected relations are called an arcwise connected component of X. Proposition 6.3.1 If ðX; T Þ is arcwise connected, then X is connected. Proposition 6.3.2 f : ðX; T 1 Þ/ðY; T 2 Þ is a continuous mapping. If X is an arcwise connected space, then f ðXÞ is also an arcwise connected space. Proposition 6.3.3 If X1 ; X2 ; /Xn are arcwise connected spaces, then their product space is also an arcwise connected space. Corollary 6.3.3 Rn is an arcwise connected space. Proposition 6.3.4 (Bond Lemma) Assume that A; B3ðX; T Þ are close sets and X ¼ AWB. f1 : A/ðY; T 2 Þ and f2 : B/ðY; T 2 Þ are continuous mappings. f1 jAXB ¼ f2 jAXB, i.e., f1 and f2 are the same on f ðxÞ; x ˛ A AXB. Let f ðxÞ ¼ 1 . Then, f : ðX; T 1 Þ/ðY; T 2 Þ is continuous. f2 ðxÞ; x ˛ B Proposition 6.3.5 For A3Rn, if A is an open connected set, then A is arcwise connected. Definition 6.3.4 In ðX; T Þ, cx ˛ X for any neighborhood uðxÞ of x, if there exists a connected neighborhood vðxÞ such that x ˛ vðxÞ3uðxÞ, then X is called local arcwise connected. Proposition 6.3.6 If A3ðX; T Þ is local arcwise connected and A is connected, then A is arcwise connected. Proposition 6.3.7 If ðX; T Þ is local arcwise connected and A3X is an open connected subset, then A is arcwise connected. Proposition 6.3.8 The continuous image of a local arcwise connected space is also local arcwise connected.
Some Concepts and Properties of Point Set Topology 353 Proposition 6.3.9 If X1 ; X2 ; /Xn are local arcwise connected, then their product space is also local arcwise connected. Definition 6.3.5 ðX; dÞ is a metric space. For x; y ˛ X and cε > 0, if there exist a set of points x0 ¼ x; x1 ; x2 ; /; xn ¼ y; xi ˛ X such that dðxi ; xiþ1 Þ < ε; i ¼ 0; 1; 2; /; n 1, then ðx0 ; x1 ; /; xn Þ is called a εchain that connects points x and y. The above materials are from [Xio81]. The interested readers can also refer to [Eis74].
A.7 Order-Relation, Galois Connected and Closure Space A.7.1 Order-Relation and Galois Connected Definition 7.1.1 Assume that ‘ ’ is a binary relation on U and satisfies reflexivity and transitivity properties, i.e.,cx ˛ U, x x and cx; y; z ˛ U, if x y and y z, then x z, ‘ ’ is called a pre-order or quasi-order on U. Especially, if ‘ ’ satisfies transitivity and anti-reflexivity, i.e., for cx ˛ U, x x does not hold, then ‘ ’ is a strict pre-order on U denoted by ‘ < ’ generally. Definition 7.1.2 If a pre-order relation satisfies anti-symmetry, i.e., cx; y ˛ U, x y; y x0x ¼ y, then is called a partial order relation on U. ðU; Þ is called a partial ordered set. If a pre-order relation satisfies symmetry, i.e.,cx; y ˛ U, x y0y x, then is called an equivalence relation on U. Symbol is not used to denote equivalence relations generally. Definition 7.1.3 Assume that is a semi-order (partial-order) relation on U. For any two elements x; y ˛ U, if their supremum supfx; yg and infimum inf fx; yg exist, then ðU; Þ is a lattice. For a lattice ðU; Þ, x n y and x o y are used to represent the supremum and infimum of two elements x and y generally. Especially, if for any V4U, supfxjx ˛ Vg and inf fxjx ˛ Vg exist, then ðU; Þ is called a complete lattice. Definition 7.1.4 ðU; Þ is a semi-order set. cx; y ˛ U, if self-mapping 4 : U/U satisfies the following conditions
354 Addenda A (1) x 4ðxÞ (increasing property) (2) x y04ðxÞ 4ðyÞ (order-preserving) (3) 4ðxÞ ¼ 4ð4ðxÞÞ (idempotent) Then, 4 is a closure operator on ðU; Þ. Correspondingly, if a self-mapping f : U/U on U satisfies order-preserving, idempotent and decreasing property, i.e., fðxÞ x, then f is called an interior operator on ðU; Þ. Note 7.1.1 4 is a closure operator on ðU; Þ. A set 4ðUÞbfx ˛ Ujdy ˛ U s:t: 4ðyÞ ¼ xg of images is just a set composed by all fixed points of 4, i.e., 4ðUÞ ¼ fx ˛ Uj4ðxÞ ¼ xg. Elements of 4ðUÞ are called to be closed under the mapping 4. Especially, if ðU; Þ is a complete lattice, then 4ðUÞ is also a complete lattice. Note 7.1.2 Assume that 4 is a closure operator on a complete lattice ð2U ; 4Þ, where U is any given set and 2U is a power set of U. Then, 4 uniquely corresponds to a family U42U of subsets of U and U satisfies (1) U ˛ U, (2) cU 4U, XU ˛ U, U is called a Moore family on U, and two-tuple ðU; 4Þ is a closure system. Please refer to Davey and Priestley (1992) for more details. Galois Connection (Davey and Priestley, 1992)
Definition 7.1.5 Assume that ðU; U Þ and ðV; V Þ are a pair of semi-order structures. f : U/V and g : V/U are a pair of mappings. The domains of f and g are U and V, respectively. If f and g satisfy For cx ˛ U and cy ˛ V, xU gðyÞ5f ðxÞV y. Then, ðf ; gÞ is called a Galois connection between ðU; U Þ and ðV; V Þ as shown below, ðU; U Þ
ðV; V Þ
Proposition 7.1.1 ðf ; gÞ is a Galois connection between ðU; U Þ and ðV; V Þ, where f : U/V and g : V/U. If x; x1 ; x2 ˛ U and y; y1 ; y2 ˛ V, then we have the following conclusions. (1) xm gð f ðxÞÞ; f ðgðyÞÞn y (2) x1 U x2 0f ðx1 ÞV f ðx2 Þ; y1 V y2 0gðy1 ÞU gðy2 Þ
Some Concepts and Properties of Point Set Topology 355 (3) f ðgðf ðxÞÞÞ ¼ f ðxÞ; gðf ðgðyÞÞÞ ¼ gðyÞ Conversely, assume that f and g are a pair of mappings between ðU; U Þ and ðV; V Þ. For cx; x1 ; x2 ˛ U and cy; y1 ; y2 ˛ V, the above two conditions (1) and (2) hold. Then, f and g are a Galois connection between ðU; U Þ and ðV; V Þ. Proposition 7.1.2 Assume that ðf ; gÞ is a Galois connection between ðU; U Þ and ðV; V Þ, where f : U/V and g : V/U. Then the combination mapping g+f is a closure operator on ðU; U Þ, and f +g is an interior operator on ðV; V Þ.
A.7.2 Closure Operation and Closure Space The concept of closure operation that we previously introduced is under the order theory sense. The terms of closure operation, closure space and related properties that we will introduce below have the topologic sense, especially under E. Cech sense, i.e., based on set theory and always assuming that there does not appear paradox (Cech, 1966). Definition 7.2.1 U is a domain. If mapping cl : 2U /2U satisfies the following three axioms, where 2U is a power set of U, (cl1) clðBÞ ¼ B (cl2) cX4U, X4clðXÞ (cl3) cX4U and cY4U, clðXWYÞ ¼ clðXÞWclðYÞ then, cl is called a closure operation on U. Correspondingly, two-tuples ðU; clÞ is a closure space, and clðXÞ is a cl closure of subset X. If not causing confusion, the closure clðXÞ of subset X is denoted by X. Proposition 7.2.1 If ðU; clÞ is a closure space, then (1) clðUÞ ¼ U (2) For cX4U and cY4U, if X4Y, then clðXÞ4clðYÞ (3) For any family Xi ði ˛ I Þ of subsets of U, have clð X Xi Þ4 X clðXi Þ i˛I
i˛I
Definition 7.2.2 CðUÞ is a set composed by all closure operations on U, i.e., CðUÞ ¼fmjm is a closure operation on U}. Define a binary relation on CðUÞ as cm; n ˛ CðUÞ; n m5cX4U; nðXÞ4mðXÞ
356 Addenda A If n m holds, then closure operation m is said to be coarser than n. Equivalently, n is said to be finer than m. Theorem 7.2.1 Binary relation is a semi-order relation on CðUÞ. ðCðUÞ; Þ has a greatest element m1 and a least element m0 . That is, for cX4U, if X 6¼ B, then m1 ðXÞ ¼ U, otherwise m1 ðBÞ ¼ B; and m0 ðXÞ ¼ X. Furthermore, for any subset fmi ji ˛ Ig of CðUÞ and cX4U, we have ðsupfmi ji ˛ I gÞðXÞ ¼ Wfmi ðXÞji ˛ I g, i.e., CðUÞ is order complete with respect to . Definition 7.2.2 ðU; clÞ is a closure space. Mapping intcl : 2U /2U , induced by closure operation cl, is called an interior operation, denoted by int. Its definition is as follows cX4U; intðXÞ ¼ U clðU XÞ Correspondingly, intðXÞ is called clinterior of X, or simply interior. Proposition 7.2.3 ðU; clÞ is a closure space. If int is defined by Definition 7.2.2, then (int1) intðUÞ ¼ U (int2) cX4U, intðXÞ4X (int3) cX4U and cY4U, intðXXYÞ ¼ intðXÞXintðYÞ Assume that int 2U /2U satisfies axioms int1w int3. Define an operation cl as follows cX4U; clðXÞ ¼ U intðU XÞ It can be proved that cl is a closure operation on U and intcl ¼ int. If IðUÞ is a set of mappings int on U that satisfy axioms int1w int3, then there exists one-one correspondence between CðUÞ and IðUÞ. Or a closure operation and an interior operation are dual. Definition 7.2.3 ðU; clÞ is a closure space. int is a dual interior operation of cl. For cX4U, if clðXÞ ¼ X, then X is called a close set. If clðU XÞ ¼ U X, or equivalently, intðXÞ ¼ X, then X is called an open set. Proposition 7.2.4 ðU; clÞ is a closure space. int is a dual interior operation of cl. We have (1) intðBÞ ¼ B
Some Concepts and Properties of Point Set Topology 357 (2) For cX4U and cY4U, if X4Y, then intðXÞ4intðYÞ. (3) For any family Xi ði ˛ I Þ of subsets of U, have W intðXi Þ4intð W Xi Þ i˛I
i˛I
Definition 7.2.4 A topological closure operation on U is a closure operation cl that satisfies the following condition ðcl4ÞcX4U; clðclðXÞÞ ¼ clðXÞ If cl is a topological closure operation, then closure space ðU; clÞ is a topological space. Proposition 7.2.5 If ðU; clÞ is a closure space, then each condition shown below is the necessary and sufficient condition that ðU; clÞ is a topological space. (1) The closure of each subset is a close set (2) The interior of each subset is an open set (3) The closure of each subset equals to the intersection of all close sets that include the subset (4) The interior of each subset equals to the union of all open sets that include the subset. Theorem 7.2.2 Assume that O~ is a family of subsets of set U that satisfies the following conditions ~ U ˛ O~ (o1) B ˛ O, ~ i.e., O~ is closed for any union operation ~ WfAjA ˛ O~1 g ˛ O, ~ (o2) cO1 4O, ~ AXB ˛ O, ~ i.e., O~ is closed for finite intersection operation. (o3) cA; B ˛ O, Let CO~ ¼ fcljcl is a closure operation on U and the set composed by all open sets of ~ ðU; clÞ is just Og. Then, there just exists a topological closure operation clT on CO~ such that clT is the roughest element on CO~ . Theorem 7.2.3 Assume that C~ is a family of subsets of set U that satisfies the following conditions ~ U ˛ C~ (c1) B ˛ C, ~ i.e., C~ is closed for any union operation ~ ~ XfAjA ˛ C~1 g ˛ C, (c2) cC 1 4C, ~ ~ (c3) cA; B ˛ C, AWB ˛ C, i.e., C~ is closed for finite intersection operation. Then, there just exists a topological closure operation clT on U such that C~ is just a set that composed by all close sets on ðU; clT Þ.
358 Addenda A Using open set as a language to describe topology, axioms (o1) w (o3) are used. However, conditions (cl1) w (cl4) are called axioms of Kuratowski closure operator. Kuratowski closure operator, interior operator that satisfies axioms (int1) w (int3) and (int4):cX4U intðintðXÞÞ ¼ intðXÞ, open set and neighborhood system are equivalent tools for describing topology. For describing non-topologic closure spaces, only closure operations, interior operations and neighborhood systems can be used, but open set or close set cannot be used as a language directly. In some sense, closure spaces are more common than topologic spaces. We will discuss continuity, connectivity and how to construct a new closure space from a known one below. A closure operation cl on a domain set U is defined as a mapping from 2U to itself, where domain DomðclÞ ¼ 2U and codomain RanðclÞ42U . Closure operation cl is completely defined by binary relation R4U 2U , i.e., cx ˛ U and cX4U, xRX5x ˛ clðXÞ. Obviously, we have clðXÞ ¼ R1 ðXÞbfy ˛ UjyRXg. Compared to cl, relation R more clearly embodies the intuitive meaning of closure operation, i.e., what points are proximal to what sets. Naturally, the intuitive meaning of continuous mappings is the mapping that remains the ‘x is proximal to subset X’ relation. Definition 7.2.5 f is a mapping from closure space U to closure space V. For x ˛ U and cX4U, if x ˛ X, have f ðxÞ ˛ f ½X holds, then f is called continuous at x. If f is continuous at any x, then f is called continuous. Theorem 7.2.4 f is a mapping from closure space U to closure space V. The following statements are equivalent. (1) f is a continuous mapping (2) For cX4U, f ½X4f ½X holds. (3) For cY4V, f 1 ½Y4f 1 ½Y holds. Definition 7.2.6 f is an 1-1 correspondence (bijective mapping) from closure space U to closure space V. Both f and f 1 are continuous mappings. Then, f is called a homeomorphous mapping from U to V, or V is a homeomorph of U. Definition 7.2.7 If there exists a homeomorphous mapping from closure space U to V, then U and V are called homeomorphous closure spaces.
Some Concepts and Properties of Point Set Topology 359 Definition 7.2.8 If a closure space U has property P such that all spaces that homeomorphous to U have the property, then P is called the topological property. Obviously, the homeomorphous relation is an equivalent relation on the set composed by all closure spaces. Definition 7.2.9 ðU; mÞ is a closure space. For X4U, if there exist subsets X1 and X2 on U such that X ¼ X1 WX2 , and if ðmðX1 ÞXX2 ÞWðX1 WmðX2 ÞÞ ¼ B, then X1 ¼ B or X2 ¼ B, then X is called a connected subset of ðU; mÞ. Definition 7.2.10 f is a continuous mapping from closure space ðU; mÞ to closure space ðV; nÞ. If X4U is a connected subset, then f ðXÞ is a connected subset on ðV; nÞ. Below we will discuss how to generate a new closure operation from a known closure operation, or a set of closure operations. Two generated approaches are discussed, the generated projectively and generated inductively. The product topology and quotient topology discussed in point topology are special cases of the above two generated approaches in closure operation. Definition 7.2.11 fðU i ; mi Þji ˛ I g is a set of closure spaces. For any i ˛ I, the closure operation on V generated inductively by mapping fi : U i /V is defined as follows cX4V; ni ðXÞ ¼ XWfi mi fi1 ðXÞ The above closure operation is the finest one among all closure operations that make fi continuous. The closure operation on V generated inductively by a set ffi ji ˛ I g of mappings is defined as follows ðsupfvi ji ˛ I gÞðXÞ ¼ XW fi mi fi1 ðXÞ i ˛ I The above closure operation is the finest one among all closure operations that make each fi , i ˛ I continuous. Proposition 7.2.6 ðU; mÞ is a closure space. R is an equivalence relation on U, and its corresponding quotient set is ½U, where p : U/½U, pðxÞ ¼ ½x. The closure operation n generated
360 Addenda A inductively by p is defined as a quotient closure operation on ½U. And for cX4½U, nðXÞ ¼ XWp m p1 ðXÞ ¼ p m p1 ðXÞ Definition 7.2.12 fðU i ; mi Þji ˛ I g is a set of closure spaces. For any i ˛ I, the closure operation on V generated projectively by fi : V/U i is defined as follows cX4V; ni ðXÞ ¼ fi1 ðmi ðfi ðXÞÞÞ The above closure operation is the coarsest one among all closure operations that make fi continuous. The closure operation on V generated projectively by a set ffi ji ˛ I g of mappings is defined by inf fni ji ˛ I g. It is the coarsest one among all closure operations that make each fi , i ˛ I continuous. Note that ðinf fni ji ˛ I gÞðXÞ is not necessarily the Xffi1 ðmi ðfi ðXÞÞÞji ˛ I g. And the latter is not necessarily a closure operation, unless a set fðU i ; mi Þji ˛ I g of closure spaces satisfies a certain condition (Cech, 1966).
A.7.3 Closure Operations Defined by Different Axioms Two forms of closure that we mentioned previously are denotes by closure operator and closure operation, respectively. The former is under order theory sense and the latter is under topologic sense. In fact, the term of closure does not have a uniform definition. In different documents it might have different meanings. We introduce different definitions of closure, quasi-discrete closure space, Allexandroff topology, etc. below. U is a domain. Assume that Cl : 2U /2U is a given mapping. For cX4U, ClðXÞ is called the closure of subset X. ðU; ClÞ is called the most general closure space. Int : 2U /2U is a dual mapping of Cl, i.e., IntðXÞbU ClðU XÞ. IntðXÞ is called the interior of subset X. For convenience, for cX; Y ˛ 2U , the following axioms are introduced (Table 7.3.1). Table 7.3.1
Neighborhood space Closure space Smith space Cech closure space Topological space Alexandroff space Alexandroff topology
(CL0)
(CL1)
(CL2)
A A A A A A A
A A A A > > >
A A A A A A
A: the axiom satisfied by definition >: the property induced by definition.
(CL3)
(CL4)
(CL5)
A A A A > >
A A
A A
Some Concepts and Properties of Point Set Topology 361 (CL0) (CL1) (CL2) (CL3) (CL4) (CL5)
ClðBÞ ¼ B X4Y 0 ClðXÞ4ClðYÞ X4ClðXÞ ClðXWYÞ4ClðXÞWClðYÞ ClðXÞ ¼ ClðClðXÞÞ for any family fXi ji ˛ I g of subsets on U, W ClðXi Þ ¼ Clð W Xi Þ.
where, (CL1)þ(CL3) are equivalent to axiom
i˛I (CL3)0 : ClðXWYÞ
i˛I
¼ ClðXÞWClðYÞ.
Using the dual interior operation Int of Cl, we have the following equivalent axioms (CL0)w(CL5). For cX; Y ˛ 2U , we have (INT0) (INT1) (INT2) (INT3) (INT4) (INT5)
IntðUÞ ¼ U X4Y 0 IntðXÞ4IntðYÞ IntðXÞ4X IntðXÞXIntðYÞ4IntðXXYÞ IntðXÞ ¼ IntðIntðXÞÞ for any family fXi ji ˛ I g of subsets on U, X IntðXi Þ ¼ Intð X Xi Þ. i˛I
i˛I
where, (INT1)þ(INT3) are equivalent to INT30 : IntðXXYÞ ¼ IntðXÞXIntðYÞ. Note 7.3.1 Under the general order theory sense, the closure space is defined by axioms (CL1), (CL2) and (CL4). For example, the closure operation defined by Definition 7.1.4 is called closure operator. When considering the inclusion relation between a power set and a subset, the axiom (CL0) may be or may not be satisfied. Note 7.3.2 Under the Cech’s sense, the closure space is called pre-topology and is defined by axioms (CL0)w(CL3). In Definition 7.1.4, axioms (CL0) and (CL3) are replaced by (CL3)’. The topology described by the Kuratowski closure operator that satisfies axioms (CL0), (CL2), (CL3)’ and (CL4) is equivalent to the above description, since axiom (CL3)’ may induce axiom (CL3), and (CL4)þ(CL3) may induce (CL3)’. The distinction between the closure space in the Cech’s sense and the topologic space in general sense is the satisfaction of the idempotent axiom or not. So the former is the extension of the latter. Note 7.3.3 Axiom (CL5) is called Alexandroff property. The topologic space that satisfies the Alexandroff property is called Alexandroff topology. In Cech (1966) and Galton (2003), axiom (CL5) is called quasi-discrete property. The Cech closure space that satisfies quasidiscrete property is called quasi-discrete closure space.
362 Addenda A Note 7.3.4 To describe the closure space, except closure and interior operations, the neighborhood and the filter convergent sequence can be used equivalently. In Table 7.3.1, the neighborhood and Smith spaces (Kelly, 1955; Smith, 1995) originally are described by neighborhood language; we use the equivalent closure axioms.