Chapter 1

Chapter 1

The construction of Containing Spaces In the present chapter we give the main construction of the Containing Spaces and prove some basic properties of this construction. In Section 1.1 on the class of all marked spaces using algebras of subsets some equivalence relations are defined. These equivalence relations are an important part of the construction of Containing Spaces. In Section 1.2 we consider an arbitrary indexed collection S of Tospaces (of weight _< T) and an arbitrary co-mark M of S. Using relations considered in Section 1 we define an admissible family RM of equivalence relations on S calling M-standard. Furthermore, for an arbitrary admissible family R of equivalence relation on S, which is a final refinement of RM, we construct by a standard manner a Containing Space denoted by T ( M , R). This space is uniquely determined by the collection S, co-mark M, and the family R. For every element X of S there exists a natural embedding of X into T ( M , R). In Section 1.3 we consider a restriction Q of S and show that if RIQ is a final refinement of MIQ-standard family of equivalence relations on Q, then the Containing Space T(MIQ,RIQ ) can be considered by a natural way as a (specific) subset of the Containing Space T ( M , R). In Section 1.4 we give the notions of a closed (open) restriction: a restriction Q is said to be closed (open) if Q ( X ) is closed (open) in X E S. We also give the notion of an (M, R)-complete restriction, which is a generalization of closed and open restrictions. For such restrictions we study commutative properties of specific subsets with respect to some topological

10

1. The construction of Containing Spaces

operators on spaces. As such operators we consider here the boundary, the closure, the interior, and the complement operator (that is, each subset corresponds to its complement). For example, for the b o u n d a r y operator the corresponding commutative property can be formulated as follows. Suppose t h a t Q_

{Ox . x e

s}

is a (M, R)-complete restriction. We put B d ( Q ) - {Bd(QX) 9 X r S}. Then, under some simple conditions concerning the corresponding family of equivalence relations on S, we prove the following property: BdT(TIQ) - TIBd(Q). In Section 1.5 we give a generalization of the main construction. Actually, this generalization concerns the families of equivalence relations. Finally, in Section 1.6 some basic notations which we follow in the next chapters are given.

1.1

Marked spaces and standard equivalence relations

In this section for every s E ~c \ {~} on the class of all marked spaces the so-called s-standard equivalence relation is defined and studied. The given results are used in the next section for the construction of Containing Spaces. The following two lemmas are easily proved. In the sequel, we shall refer to L e m m a 1.1.1 in cases where we want to use some relation between elements of an algebra and where it is known t h a t the same relation between the corresponding elements of an isomorphic algebra is true. 1.1.1 L e m m a . Let X and Y be two sets, A an algebra of X and let i be an isomorphism of A into T)(Y). Suppose that U, V, and W are elements of A. Then, the relations U = X , U -- ~, U C V, and U N V = W are equivalent to the relations i ( g ) = Y , i ( g ) = ~, i(U) C i(V), and i(u) n i(v) - i(w), D

1.1. Marked spaces and standard equivalence relations

11

1.1.2. L e m m a . Let X and Y be two sets, g a subset of T)(X), and let A be the minimal algebra of X containing the set G. The following statements are true: (1) If i is a homomorphism of A into T)(Y), then i(A) is the minimal algebra of Y containing the set i(G). (2) If h~ and h2 are two homomorphisms of A into P ( Y ) such that hi (U) = h2(U) for every U E ~, then hi = h2. [3 D e f i n i t i o n . Let X be a marked space, {U~ 9 5 E ~-} the corresponding m a r k of X, and s E $c \ {0}. By the s-algebra of X we mean the minimal algebra of X containing the sets Uff, 5 E s. This algebra is denoted by A x . The sets U ~ and X \ Uff (which are elements of A X) are denoted also by X(~,0 ) and X(5,~), respectively. By definition, if ~) -r t C s, then A X c A X. In the sequel, it is convenient to consider an isomorphism of the algebra A x of a marked space X onto an algebra of a set, which is independent on the space X. As such a set we take the set 2 s. (We recall t h a t according to our notations 2 - {0, 1}.) Notation. we set

Let X be a marked space and s E De\ {1~}. For every f E 2 s

X(s,f ) --N{X(5,f(5))" 5 E 8}. As a finite intersection of elements of A X, X(s,f ) E A X. We denote by 2~( the subset of 2 ~ consisting of all elements f of 2 ~ such t h a t X(~,f) r O. Since the sets X(5,0) and X(5,1) (for a fixed 5) are disjoint and their union is the space X (true even if X - O) the sets X(s,f ) (for a fixed s) are mutually disjoint and their union is also the space X. Therefore, for every x E X there exists a unique element f of 2 s such t h a t x E X(s,f). We define a mapping d X of X into 2 s setting

f. It is supposed t h a t d X (X) - O if X - O. Obviously, 2~( - d X (X). For every 5 E s we set

u(X, s, 5 ) -

{f E 2~('f(5)-

For every element u of 7)(2~) we set

0}.

12

1. The construction of Containing Spaces X ( s , u ) - U{X(s,f)" f 6 u}.

It is supposed that X(s,u ) - ~ if u - 0. As a finite union of elements of A X, X(~,~) E A X. We denote by i ) the mapping of the set 7)(2)) into A X such that for every u 6 7)(2~),

~k(~) - x(~,~). 1.1.3. L e m m a . Let X be a marked space and s E ~ \ {0}. The mapping i~x is an isomorphism of the algebra 7)(2~) onto the algebra A x such that for every 5 E T,

i k ( ~ ( x , ~, 5)) - x(~,o).

(1.1.1)

P r o o f . Let v c 7 ) ( 2 ~ ) a n d u - 2~( \ v. Then, by properties of the sets X(s,f) we have x(~,~) - u { x ( ~ , ~ ) 9 f e ~ } - u { x ( ~ , ~ ) 9 f c 2 k \ v} -

X \ ( U { X ( ~ , f ) ' f e v } ) - X \ X(~,~). This means that i~c (n) - X \ i~ (v). Let v, w E 7)(2~) and u - v U w. As the above,

x ( ~ , ~ ) - u{x(~,~) 9 f e ~ } -

u{x(~,~) 9 f e v u ~ } -

that is, i3((u) - i~ (v) U i~((w). The proving properties show that i~ is a homomorphism of 7 ) ( 2 ~ ) i n t o A X. Since X(s,f) ~ 0 for every f C 2~ we have that X(s,~) :fi X(s,v) for distinct elements u and v of 7)(2~(), that is i~c is one-to-one; and, therefore, it is an isomorphism. Now, we prove the equality (1.1.1). Let 5 E s and x E X(5,0). For every c C s let k(c) denote the unique element of the set {0, 1} for which x E X(~,k(~)). Let f be the element of 2 s such that f(c) - k(c) for every c C s. By the definition of the set X(s,y), x E X(s,f ). On the other hand, by assumption, k(5) - 0 and therefore f C u(X, s, 5). By the definition of

1.1. Marked spaces and standard equivalence relations

13

the set X(s,u(X,~,5)) , x belongs to this set. Thus, X(5,0) C X(s,u(X,~,5)). Conversely, let x C X(~,~(x,~,5)). Then, there exists an element f of u(X, s, 5) such that x E X(s,f). By the definition of the set u(X, s, 5), f(5) - O. Therefore, x C X(5,0), which means that X(s,u(x,s,5)) c X(5,0). Thus, X(~,~(x,~,~)) = X(~,o). Since Asx is the minimal algebra containing the sets X(6,o), 5 C s, the equality (1.1.1) implies that i ) ( 7 ) ( 2 ) ) ) - A x, that is the mapping i~ is onto, which completes the proof of the lemma. E] D e f i n i t i o n . For every s E 9c \ {0} on the class of all marked spaces an equivalence relation, denoted by ~*m ~ is defined as follows: two marked spaces X and Y are ~sm -equivalent if there exists an isomorphism i of A X onto AsY such that i(x(e,o))

-

for every 5 E s. This equivalence relation is called standard, more precisely, s-standard. The isomorphism i, which by Lemma 1.1.2 is uniquely determined, is called natural. Lemma 1.1.2 implies also that ~ C ~ t m if 0 -/: t C s. Moreover, the restriction ilAx of the above natural isomorphism i to A X is the natural isomorphism of A X onto AtY. 1.1.4. L e m m a . Let X and Y be marked spaces and s e ff \ {0}. The following condition are equivalent: (1) The marked spaces X and Y are ~n-equivalent. (2) (3)

-

25. =

e

P r o o f . (1) => (2). Let i be the natural isomorphism of A x onto AsY. Then, we have

N{Y(5,/(5)))" 5 C s} -- Y(s,f) for every f C 2 ) . Since i is an isomorphism, Y(s,f) # 0 if X(s,f ) # O. This fact means that 2~( C 2~,. Similarly, 2~, C 2 ) . Thus, 2~ - 2~,. (2) => (3). It follows immediately by the definitions of the sets u(X, s, 5) and u(Y, s, 5).

14

1. The construction of Containing Spaces

(3) => (2). By Lemmas 1.1.2 and 1.1.3 the set 7)(2~() is the minimal algebra of 2 s containing the sets u(X, s, 5), 5 E s. The same is true for the space Y. This fact means t h a t 79(2~() - 7)(2~); and, therefore, 2 ) - 2~. (2) => (1). By L e m m a 1.1.3 the mapping i - i ~ o (i~v) -1 is an isomorphism of a x onto Asv such t h a t i(X(a,o)) - IQ5,0) for every 5 E s. This fact means t h a t the marked spaces X and Y are ~ - e q u i v a l e n t . D

Let {U~( 9 6 E r} be the mark of a marked space X, O a subspace of X , and let {U Q 9 6 E r} be the trace on O of the m a r k {U~( ' S E T } . Then, 1.1.5 L e m m a .

d~ (x) - d? (x) for every z E Q and s E 9c \ {1~}. Therefore, dX(Q) = dQs(Q). (We note t h a t the mapping dsQ is constructed with respect to the m a r k { U ~ : 5 E r } of Q).

P r o o f . Let s E ~ \ {0}, x C Q and d X ( x ) f E 2 s. We need to prove t h a t dQ(x) - f. By the definition of the mapping d X we have x E X(s,f ) - A{X(~,I(~)) 9 5 E s}. Therefore, x ~ O n (n{x(5,z(5)) 9 5 < s}) - n { Q n x(5,f(5)) 9 5 c s} -

This means t h a t dsQ ( x ) -

1.2

f. IS

The Containing Spaces

In the present section we give the construction of Containing Spaces. These spaces are used throughout the book. D e f i n i t i o n s . Let S be an indexed collection of spaces and M a co-mark of S. For every s E j r \ {0}, the s-standard equivalence relation ~Srn defines on S an equivalence relation, denoted by ~ 4 , as follows: two elements X and Y of S are ~ - e q u i v a l e n t if and only if the marked spaces X and Y (with the chosen marks M ( X ) and M ( Y ) , respectively) are ~ n - e q u i v a l e n t . We also set ~ S x S if s - 0. The indexed family RM -- { ~ ' s

E ~)

1.2. The Containing Spaces

15

of the above defined equivalence relations is called M-standard. By Lemma 1.4, RM is admissible. An admissible family R of equivalence relations on S is said to be Madmissible if R is a final refinement of the M-standard family RM. A g r e e m e n t . In what follows of this section it is assumed that an arbitrary non-empty indexed collection of spaces (of weight _< T) denoted by S is given. It is also assumed that an arbitrary co-mark of S is given. This co-mark is denoted by M. Moreover, we assume that for every X E S, M(X)-

{ U ~ " 6 E T}.

Whenever an element X of S is considered to be marked, then M ( X ) is considered to be the chosen mark of X. Finally, it is assumed that an M-admissible family of equivalence relations on S is given. This family is denoted by

D e f i n i t i o n . Suppose that S contains a non-empty element. Consider the set of all pairs (x, X), where X E S and x E X. (Two such pairs (x, X) and (y,Y) are considered to be distinct if either X and Y are distinct elements of the indexed set S or X and Y are the same element of the indexed set S and x ~ y.) On this set, an equivalence relation denoted by ~RM is defined as follows" two pairs (x, X) and (y, Y) are ~RM-equivalent if X ~s y and d X (x) - dy (y) for every s E ~ \ {0}. (We note that the condition dX(x) - dY(y) is true simultaneously for every s E ~c \ {0} if and only if for every 5 E 7- either xEU~; andyEU~ orx~U~ (andy~U~.) N o t a t i o n . In what follows, the set of all ~a-equivalence classes is denoted by T ( M , R) - T (that is, according to the Introductory notations, T - C(~RM)). It is assumed that T ( M , R ) - @ if all elements of S are empty. For every element H of CO(R) (about notation see the Introductory Remarks) the set of all a E T, for which there exists an element (x, X) of a such that X C H, is denoted by T(M, R, H) - T(H). (Therefore, T(H)

1. The construction of Containing Spaces

16

coincides with the set of all a E T such t h a t for every (x, X) E a, X E H.) These subsets of T will be called primary. N o t a t i o n . Let s r 0 and t be elements of ~ such t h a t ~ t c ~ - , ~ and let H be an element of C ( ~ t ) . Suppose t h a t an element X of H is chosen. We denote by 2 h the set 2~c and by u ( H , s, 5) the set u ( X , s, 5) for every 5 C s. Since ~ t c ~ a and, therefore, X ~,,s y for every X, Y E H, by L e m m a 1.1.4 the sets 2~ and u ( H , s, 5) do not depend on a particular choise of a representative from H. For every u E P ( 2 ~ ) we denote by T(s,u)(H) the set of all elements a of T for which there exists an element (x, X ) of a such t h a t X E H and x E X(s,u). If u -- u ( H , s, 5) for some 5 E s (and, therefore, X(s,u) - X(a,0) for every X E H), then the set T(~,~)(H) is also denoted by T(a,0)(H). (We note t h a t since the set T(a,0)(H) coincides with the set of all elements a of T such that there exists an element (x, X) of a for which X E H and x E X(a,0), this set is independent of the element s of 9c for which ($ E s and ~ t c ~ . ) Finally we set A H - {T(~,~)(H) 9 u E 7)(2h)}. Since the set T(s,u)(H) is a subset of the set T ( H ) , A N can be considered as a set of subsets of T ( H ) . 1.2.1 L e m m a . Let s,t E ~ , s =/= O, ~ t c ~ 4 , H E C ( ~ t ) , and u E 7~(2~). Then, the set T(s,u)(H) coincides with the set of all a E T such that for every (x, X ) E a we have X E H and x E X(s,u ) . P r o o f . Consider the set T(s,~)(H). It suffices to prove t h a t if a E T(s,u)(n) and (y, Y) e a, then Y e H and y E Y(s,u). Let a E T(s,u)(H). There exists an element (x, X) of a such t h a t X E H and x E X(s,u). Also, there exists an element f of u such t h a t x E X(~,I ), t h a t is, d x (x) - f. Now, let (y, Y) E a. Since ( x , X ) , (y, Y) E a, by definition, X ~ t y and d x (x) - d~ ( y ) - f, which means t h a t Y E H and y ~ ~ , s / c ~ , ~ / - [~ 1.2.2 L e m m a .

Let s, t E 9r, s =/= 0, ~ t c ~ ,

and H E C ( ~ t ) . Let also

1.2. The Containing Spaces

17

T(s,u ) (H) U T(s,v ) (H) - T(s,uuv ) (H) and

(1.2.1)

T ( H ) \ T(~,~)(H) - T(~,w)(H).

(1.2.2)

P r o o f . We prove (1.2.1). Let a C T(s,u)(H) U T(s,v ) (H).

Then, there exists (x, X) E a such that X E H and either x E X(s,u) or x E X(s,v). This means that x E X(s,~ ) U X(~,v) - X(~,uuv). Hence, a E T(~,~u~)(H). Conversely, let a e T(s,uuv)(H). Then, there exists ( x , X ) E a such that X E H and x E X(s,~uv) - X(s,~) U X(s,v), that is either x E X(s,u) or x E X(s,v), which means that a E T(s,u)(H)U T(s,v)(H). This proves relation (1.2.1). Relation (1.2.2) is proved similarly. D 1.2.3 C o r o l l a r y . If s, t E ~ , s # O, ~ t c ~ , set A H is an algebra of the set T(H). V!

and H E C(~t), then the

1.2.4 L e m m a . Let s, t E ~ , s # O, ~ t c ~ and H e C(~t). Then, the m a p p i n g i h of the set 7)(2~i ) onto the set A H, for which i h ( u ) - T(s,u ) (H) for every u E 2~i, is an isomorphism. P r o o f . By Lemma 1.2.2, i~ is a homomorphism. Therefore, it suffices to prove that i~_I is one-to-one. Let u, v E 7)(2{_i) and u :fi v. Let X E H. Then, 2~_I - 2~ and by Lemma 1.1.3, X(s,u ) -r X(s,v ). Hence, there exists an element x of X, which belongs to one of the sets X(s,u ) and X(s,v ) and does not belong to the other. Let a be the element of T containing the pair ( x , X ) . Then, by the definition of the sets T(s,~)(H ) and Lemma 1.2.1, a belongs to one of the sets T(s,u)(H) and T(s,v)(H ) and does not belong to the other, which means that T(s,u ) (H) # T(s,v ) (H), that is the mapping i~_I is one-to-one. [3 1.2.5 C o r o l l a r y . Let s, t E if, s # ~, , , ~ t c ~ , H E C(-,)), and X E H. Then, the m a p p i n g i - i~ o (isx) -1 is an isomorphism of a X onto A H such that i(X(~,o)) - T(&0)(H) for every 5 E s. D

18

1. T h e c o n s t r u c t i o n o f C o n t a i n i n g Spaces

N o t a t i o n . For every (~ E ~- and H E C<>(R) (in particular, for every H E C(R)) we denote by U~r (H) the set of all a E T for which there exists an element (x, X) of a such t h a t X E H and z E U~. Therefore, if for some element s of $- we have (~ E s and H E C ( ~ t ) , where t E ~c and ~ t c ~ - , ~ , then U~r (H) - T(5,0)(H) E A H. It is easy to see that such an element s of /F exists if and only if H E C ( ~ t ) , where t E ~ and ~-,tc~-,{M5}. N o t a t i o n . Let ~ be a subset of T. We denote by B T the set of all sets of the form U~r(H), where 5 E T, H E C(~-,t), and ~tC~-,{M5} and we put B,T,~- { U ~ ( H ) E B T" 5 r t~}. 1.2.6 L e m m a . T h e set B T is a base for a t o p o l o g y on t h e set T. Moreover, i f t~ is a s u b s e t o f T such t h a t for e v e r y X E S t h e set

is a base for X , then the set B,T~ is a base for the s a m e t o p o l o g y on T .

P r o o f . Let ~ be a subset of ~- such t h a t for every X E S the set {U/z 9 5 E ~} is a base for X. We shall prove that" (a) the e m p t y set is an element of the set B,T~, (b) the union of all elements of B,T~ is the set T, and (c) if an element a of T is contained in the intersection of two elements of B T, then there exists an element of B,T~ containing a, which is contained in this intersection. Let X E S. Since {U~c 9 5 C ~} is a b a s e for X, the e m p t y set is an element of this base; and, therefore, there exists (~ E ~ such t h a t U~c - 0. Let s be an element of 5 such t h a t 5 E s and ~sC~{M5} and let H be the ~s_ equivalent class of X. Then, for every Y E H the algebra A X is isomorphic (by the natural way) to the algebra Ay, which means t h a t U~ - 0 and, therefore U~r (H) - 0 Since U~r (H) E B,T~ we have 0 E B T Now, let a E T and (x, X) E a. There exists an element 5 of ~ such 9

~

9

t h a t x E U~c. Let t be an element of 9c such t h a t ~tC~{M5}. Denote by H the ~t-equivalence class of X. By definition, a E U~r (H) E B,T~ C B T, which means t h a t the union of all elements of B,T~ is the set T. Finally, let U~r(H), UT(L) E B T and a E U~r(H)c~UT(L/. Suppose t h a t q, p E f f , H E C ( ~ q ) , a n d L E C ( ~ P ) . Then, X E H n L a n d x E U ~ cAU x. Therefore, there exists an element ~/of n such t h a t x E Ux c Ufi N U x . Let

1.2. The Containing Spaces

19

s - {5, c, r/} and let t be an element of ,T such that ~-,tc~-,~ and q U p c t. Let also E be the ~-,t-equivalence class of X. Then, UT(E) E B T,,~ and E C H N L. By definition, a E UT(E). Since Uhr(E) - T(5,0)(E) E Ap, U T ( E ) - T(c,0)(E) E A E, and UT(E) -- T(v,0)(E ) E A E, by Lemma 1.1.1 and Corollary 1.2.5 we have UT (E) C U~r (E) M UT (E) C U~r (H) N UT (L). The proven properties imply that the set B,T is a base for a topology on T and that the set B,T~ is also a base for this topology. V1 D e f i n i t i o n . The set T equiped with the topology for which the set B T is a base, will be called the Containing Space for the indezed collection S corresponding to the co-mark M and the family R. Since IBTI _< the weight of T is _< T. In what follows, whenever we shall refer to the space T we shall mean just this space. The subspaces of T of the form T(L), where L E C ~(R), will be called primary subspaces of T.

1.2.7 L e m m a . Let 5 E T and H E C+(R) \ {r The following statem e n t s are true: (1) There exists an d e m e n t t of ~ such that H is a union of ~t_ equivalence classes. (2) The set U~ (H) is open in T and coincides with the set of all a E T such that for every (x, X ) E a, X E H and x E U ~ . (3) The set T(H) is simultaneously open and closed in T. (4) The set C1T(U~(H)) coincides with the set of all a E T such that there exists an element ( z , X ) of a for which X C H and z E C1x(U~(), as well as, with the set of all a E T such that for every ( z , X ) E a, X E H

9

clx(uJ).

(5) The set BdT(U~C(H)) coincides with the set of a11 a E T such that there exists an d e m e n t (z, X ) of a for which X E H and z E B d x ( U ~ ) as well as with the set of all a E T such that for every (z, X) E a, X E H and z E Bdx(g~(). (We note that in general the sets U~(H), C1T(U~(H)), and BdT(U~r(H)) do not belong to any algebra AH.) P r o o f . (1). Since C + (R) is the minimal ring of subsets of S containing the set C(R) there exists (see Introductory Remarks) a finite set u consisting of finite subsets of C(R) such that

20

1. The construction of Containing Spaces

H=U{Hv:vEu}, where Hv is the intersection of all elements of v E u. Suppose that v - {H~,..., Hvk~ }. For every i E {1, ..., k~} we denote by s vi an element of the set .T such that H~v E C(~s~) and put sv

- - U{S/v

9 i E {1, ..., k v } } .

Let t = U { s v : v E u} and E be an element of C ( ~ t) such that H V/E # 0. Then, there exists an element v of u such that E A H~ r ~) and, therefore, E M H / v 5r 0 for every i E {1,...,kv}. By the choice o f t , E C H/v and, therefore, E C H r . Thus, E C H, which means that H is a (finite) union of elements of C(~t). (2). By (1) there exists an element t of $- such that H = E o U ... U E n ,

where Ei C C(~t), i - 0, ..., n. Moreover, we can suppose that ~tC~{MS}. Then, U~: (H) - U5r (Eo) U... U U~r (En). For every i - 0, ..., n the set U~r(Ei) belongs to the base B T and to the set n{6} .Ei Therefore, any such set is open in T and by Lemma 1.2.1 coincides with the set of all a E T such that for every (x, X) E a, X E Ei and x E U~c. This fact means that the set U~r (H) is also open in T and coincides with the set of all a E T, such that for every (x, X) E a, X E H and x E U~c. (3). The openness of the set T ( H ) follows by statements (1) and (2) and by relation T(H)-

U { U ~ ( H ) " 8 E T},

which is easily proved. The closeness of this set follows by relation T \ T ( H ) - U{T(E)" E E C ( ~ t ) , E r H}, where t is the element of 5 determined in (1). This relation is also easy to prove.

1.2. The Containing Spaces

21

(4). Let a E T. First we prove that if for some element (x, X) of a, X E H and x E CIx(U~:), then a E C1T(U~r(H)). Indeed, let (x,X) E a, X E H and x E C1x(Uff). Suppose that UT(L)is an arbitrary element of B T containing a. Then, X E L and x E Ux . Therefore, there exists a point y of Uff D Ux . Let b be the point of T containing the pair (y, X). By the definition of the sets U s r ( H ) a n d UT(L), b E U~r(H)D UT(L), which means that a E C1T(U~r(H)). Now, suppose that a E C1T(U~r(H)). We prove that for every (x,X) E a, X E H and x E C1x(Uff). Let (x,X) E a. Denote by L the ~t_ equivalence class of X, where t is the element of )c determined in (1). If L N H r O, then T(L) is an open neighbourhood (see (2)) of a such that T(L) M U~r(H) - O, which is a contradiction. Therefore, X E L C H. On the other hand, if x ~ C1x(Uff), then there exists an element c of ~- such that x E UX and Ux DU~ - 0. Let s be an element of $c such that t C s and ~--~'~M , and let E be the ~S-equivalence class of X. Then, E C L and a E UT(E). Since UT(E) -- T(c,0)(E) E A E and U~r(E) - T(5,0)(E ) E A E, by Lemma 1.1.1 and Corollary 1.2.5, U T ( E ) D U~r(E) - 0. It is easy to verify that UT (E) gl U~ (H) C UT (E) D U~ (E). This fact means that UT ( E ) A U~r (H) - 0, which is a contradiction. The proven properties completes the proof of (4). (5). Let a E BdT(U~r(H)) and ( x , X ) E a. Since a is an element of C1T(UT(H)) \ UT(H) by (2) and ( 4 ) i t follows that X E H, x r UX, and x E C1x(U~), that is x E B d x ( U ~ ) . Now let a E T, (x,X) E a, x E B d x ( U ~ ) and X E H. Since x E C1x(U~ c) \ U~( by (4) and the definition of the set U~r(H) we have a E C1T(U~r(H)) \ u~r(H), which means that a E BdT(U~r(H)). The proven properties completes the proof of (5). [-1 R e m a r k . We note that if H - 0, then the sets U~r (H), 6 E T, and the set T ( H ) are empty; and therefore, properties (2)-(5) of the above lemma are satisfied. Notation.

Let ~ be a subset of r and L an element of C o (R). We set

B T - {u~r(H) 9 5 E ~ and H E C(R)}, B~>,n - {Usr ( H ) " 5 E t~ and H r C <>(R)},

22

1. The construction of Containing Spaces B L - {U~r(H) E B T" H C L}, B L ,~ - { U ~ : ( H ) E B w ~,~ " H c L }

9

If n - 7, then the families B T , B T L and B T, 0,~, B~, (>,s are denoted by B T, B~, BL, and B~, respectively. Remark.

It is easy to verify t h a t for every L E C ~ (R) we have

T B~>,~ - {U N T(L)" U E B<>,~}.

On the other hand the relation B L - {U N T ( L ) " U E B T} in general is not true. 1.2.8 C o r o l l a r y . If for a subset ~ of 7 and for every X

E

S the family

is a base for X , then the families B T and B~,~ are bases for T. Therefore, for every L E C ~ (R) the families B L and B~,~ are bases for T(L). [-7 D e f i n i t i o n . Suppose that ~ is a subset of ~- and L E C(~(R). The families B T and B L (respectively, B~,n and B~),~) are called ~-standard (respectively, (~, ~)-standard) families. If moreover for every X E S the family {U~c 9 ~ E ~} is a base for X, then the above ~-standard and (~, ~)-standard families are called respectively ~-standard and (~, ~)-standard bases for the corresponding spaces T and T(L). In the case where ~ = 7, the considered ~-standard and ((~, s ) - s t a n d a r d families or bases are called standard and O-standard families or bases, respectively. 1.2.9 P r o p o s i t i o n .

The space T is a T0-space (of weight <_ ~-).

P r o o f . Let a and b be two distinct elements of T. We need to find an open subset of T, which contains one of the elements a and b and does not contain the other.

23

1.2. The Containing Spaces

Let ( x , X ) E a and (y, Y) E b. First, we consider the case where for some s E ~ , the spaces X and Y are not ~S-equivalent. In this case, if H is the ~S-equivalence class of X, then the open set T ( H ) (see L e m m a 1.2.7) contains a and does not contain b. Now, let X ~s y for every s E ft. Since a -~ b, f - d x (x) =/= dsY (y) - 9 for some s E 3v \ {~)}. This fact means t h a t there exists (~ E s such that f(6) r g(~), t h a t is either f(/~) = 0 and g(6) = 1 or f(6) = 1 and g(6) = O. Let H be the ~ - e q u i v a l e n c e class of X and Y. Then, the open set U~r (H) (see L e m m a 1.2.7) contains one of the point a and b and does not contain the other. Thus, T is a T0-space. [3 D e f i n i t i o n . Let X E S. Then, for every x E X there exists a uniquely determined element a of T which contains the pair (x, X ) . We define a mapping e X of X into T setting eX(x) - a. The following proposition shows t h a t e x is an embedding. This embedding will be called natural. 1.2.10 P r o p o s i t i o n .

For every X C S the mapping e x is an embedding

of X into T. P r o o f . Let X E S. If x and y are distinct points of X , then there exists an element (~ of ~- such t h a t the set U ~ contains one of these points and does not contain the other. Therefore, if s E $- and 6 E s, then d x (x) =fi d x (y). This fact means t h a t the pairs (x, X ) and (y, X ) belong to distinct elements of the space T, t h a t is the mapping e X is one-to-one. Now, let (~ be an a r b i t r a r y element of ~- and H be an a r b i t r a r y element of C+(R) such t h a t X C H. By the definition of the mapping e X, the relation x E U~: is equivalent to the relation e x (x) - a E U~r(H). Since the non-indexed set { U ~ 9 (~ E ~-} is a base for X and B T is a base for T, this fact implies t h a t the mapping e x and the inverse to e X mapping (e~) -1 of e X (X) onto X are continuous, which means t h a t e f is an embedding. V1 The definitions of the space T and the mappings e X imply the following consequence. 1.2.11 C o r o l l a r y . The following relation is true

T-

X e S}. D

24

1.3

1. The construction of Containing Spaces

Specific subsets of the Containing Spaces

In this section for a given restriction Q of S we define a subset of T ( M , R) denoted by TIQ. These specific subsets of T(M, R) play an important role throught the book. A g r e e m e n t 1. In this section it is supposed that S, M, and R are the same as in Section 1.2. In particular, M-

{M(X)-

{U~:" 5 C T}" X C S} and

R= {~: ~ ~ ?}. Moreover, it is supposed that an arbitrary restriction { Q x . x e s}

Q_

of S is fixed. We note that, in general, the trace on Q of the M-standard family of equivalence relations on S is not the Mlo-standard family of equivalence relations on Q. This fact justify the following definition. Definition. The M-admissible family R of equivalence relations on S is said to be (M, Q )- admissible if RIQ is an MIQ-admissible family of equivalence relations on Q. 1.3.1 L e m m a . The M-admissible family R of equivalence relations on S is (M, Q)-admissible if and only if for every element s of ? \ {0} there exists an element t of 5 \ {0} such that for every X, Y E S the relation

x ~v

i~pli~ ~l~tio~ d~ (Q~) - dy(Q~).

P r o o f . Suppose that the condition of the lemma is satisfied. We need to prove that the family R is (M, Q)-admissible. Since R is M-admissible it suffices to prove that the family

aIQ = {~~lQ : ~ e f } is a final refinement of the MIQ-standard family of equivalence relations on Q. Denote the last family by

RQ

-

{~~.~ e f}.

25

1.3. Specific subsets of the Containing Spaces

Let s E ~-\ {0}. Denote by t an element of ~" satisfying the condition of the lemma. We prove that the trace on Q of the equivalence relation ~t on S, that is the equivalence relation ~tlo on Q is contained in the equivalence relation ~-,~ on Q. Indeed, let QX, QY E Q such that QX ~t]QQY. This means that X ~t y . By the condition of the lemma, the last relation Qx QX O,Y implies that dX(Q X) - dYs (QY). By Lemma 1.1.5, ds ( ) - ds (QY) and by Lemma 1.1.4, Qx ~,,~ QY. Thus, the equivalence relation ~tlQ is contained in the equivalence relation ~ . Conversely, suppose that the family R is (M, Q)-admissible. Let s C \ {0}. Since the family RIQ i~ Mlo-admissible there exists an element t of )r \ {0} such that the equivalence relation ~tlQ on Q is contained in the equivalence relation ~ on Q. Now, let X, Y C S such that X ~t y . Then, QX ~tloQY and by the above Qx ~ QY. By Lemma 1.1.4 this means that dQsx (QX) _ dQsY (QY). This relation and Lemma 1.1.5 imply that dX(Q X) - dYs(QY). E] R e m a r k . Lemma 1.3.1 implies the existence of (M, Q)-admissible faroilies of equivalence relations on S. For example, such a family is the admissible family R0 - { ~ " s E 5 } for which X ~ Y if and only if X ~ 4 Y and dXs (Q X) - dYs (QY). N o t a t i o n . Suppose that R is an (M, Q)-admissible family of equivalence relations on S. Then, besides of the space T we can also consider the Containing Space T(M]Q, R Q) for the indexed collection Q of spaces corresponding to the co-mark M [ Q - { { U ~ x - Q A U~( " S E T} " QX E Q} and the M]o-admissible family RIO of equivalence relations on Q. This Containing Space is denoted briefly by T]Q. Let L be an element of C(>(R) and E - LIQ. Then, the subspace T(MIQ , RiO , LIQ ) of T(MIQ , RiO ) is denoted also by T(E) - T(LIQ ). For the Containing Space T]Q the standard, t~-standard, ~-standard, and (~, t~)-standard families or bases, where n C T, are denoted by BTIQ,

B

tiv ly. If E is

of C (RIQ) th n

by B E , B E , B~ , and B E ~,~ we denote the corresponding families or bases for the subspace T(E) of TIQ. The elements of all the above families or bases are denoted by

U~ q (H),

where d E T and H E C* (RIQ).

1. The construction of Containing Spaces

26

Agreement 2.

In the rest of this section it is assumed that R is

(M, Q)-admissible. 1.3.2 L e m m a . For every dement b of T]Q there exists a unique element a of T such that for every x E Q x the pair (x, Qx) belongs to b if and only if the pair (x, X) belongs to a. P r o o f . Let b 6 TIQ. Consider an element (y, QY) of b and denote by a the element of T containing the pair (y, Y). Now, let (x, QX) be also an element of b. Then,

QY Qx ,.,.,~IQQY and d~ x (x) - d~ (y)

(1.3.1)

for every s E $c \ {fl}. By Lemma 1.1.5,

Qx

y

QY

(1.3.2)

Therefore, X ,..,s y and d x (x) -- dy (y) for every s E ~ - \ (0}. This fact means that the pairs ( x , X ) and (y, Y) belong to the same element of the set T. Thus, (r X) E a. Conversely, let x E Qx and ( x , X ) E a. Then, relation (1.3.3)is true for every s E )v \ {0}. Lemma 1.1.5 implies relation (1.3.2). Relations (1.3.2) and (1.3.3) imply relation (1.3.1) for every s E ~ \ {0}, which means that the pairs (x, QX) and (y, QY) belong to the same element of the set TIQ. Thus, (x, QX) E b. It is clear that the element a of T satisfying the condition of the lemma is uniquely determined. D N o t a t i o n . Let b be an arbitrary element of TIQ and let a be the unique element of T satisfying the condition of Lemma 1.3.2. We denote TQ

by e T Q the mapping of TI0 into T such that e T

(b)-

a.

1.3.3 L e m m a . Let 5 E T, H E C <~(R), and L - H]Q. Then, nn element TIQ b of TiQ belongs to U~r Q (L) if and only if the element a - e T (b) belongs to U~r (H). P r o o f . For every X E S the indexed set

27

1.3. Specific subsets of the Containing Spaces

nu2 is the trace o n Q X of the m a r k M ( X ) .

LetbET[Q

TIQ ande T (b)-aeT.

Suppose that b E U : Q (L) and let (z, Qx) c b. By L e m m a 1.3.2, (x, X) C a. Then, QX E L and x E U~Q~ - Q x N U~ and, therefore, X C H and x E U~:, which means that a E U~r (H). Similarly we prove that if a E UT (H), then b E UTIQ (L). F7 1.3.4. P r o p o s i t i o n . T[Q into the space T.

TQ

The mapping e T

is an embedding of the space q-,

P r o o f . First, we prove that the mapping e T q is one-to-one. Indeed, let b l and b2 be two distinct points of the space TIQ. Since T l o is a T0-space (see Proposition 1.2.9) there exist elements 5 of 7 and L of C(>(R Q) such that one of the points bl and b2 belongs to the open set U~rlQ (L) and the other does not belong to this set. Let H be the element of C (>(R) such that L is the trace on Q of H. By Lemma 1.3.3 one of the points al - e T Q ( b l ) and a2 - e TIQ (b2) belongs to the set U~r (H) and the other does not belong TO to this set. This means that al 7! a2, that is e T is one-to-one. The continuity of the mappings e TO T and (e T q) -1 follows by L e m m a 1.3.3 and by the fact that the elements of the base B T Q of the space TIQ v

has the form

(L)

of

of

T

the form U~:(H). Thus, the mapping e T q is an embedding. D T]Q D e f i n i t i o n . The embedding e T is called the natural embedding of T(MIQ , R]Q) into T ( M , R). In what follows we identify a point b of T(MIQ,RIQ) with the point TQ e T (b) - a of T ( M , R) and consider the space T(MIQ, RIQ) as a subspace TQ of the space T ( M , R) and the embedding c T as the identical embedding of TIO into T. The subsets of this form will be called specific subsets of T ( M , R).

1.3.5 L e m m a . Q x into X . Then,

QX Let X E S and let eX be the identical embedding of

QX QX e T Q o e T Q - e x T oe x .

(1.3.4)

1. The construction of Containing Spaces

28 P r o o f . Let x C QX. Let also

Qx

(e X o e X )(x) -- e X ( x )

-

a.

Then, a is the point of T containing the pair (x, X). On the other hand, Qx QX by the definition of the m a p p i n g e T q, eTi Q(x) -- b is the point of TIo TQ containing the pair (x, QX). By the construction of the embedding e T we TIQ have e T (b) - a. This proves relation (1.3.4). [-1 The definitions of the mappings e~ and e T Q and the definitions of the subsets T ( H ) , U ~ ( H ) , and U~ I~ (HIQ) imply the following two corollaries, which are proved similarly to L e m m a 1.3.5. 1.3.6. C o r o l l a r y . The following relations are true: TI q _ u{ex(Qx).

X

E

S} and

T(HIQ) _ U{eT x(QX). X E H } for every H E C + (R). V1

1.3.7 C o r o l l a r y . For every 5 C "1- and for every H E C o (R) we have T ( M I Q , RIQ , HIQ ) - T ( M I Q , RIQ ) N T ( H ) and

TIQ n Uf(H) - U~ IQ(HIQ). D D e f i n i t i o n s . Let Q be a subset of a space X. A family B of open subset of X is said to be a positional base or briefly a p-base for Q in X if the e m p t y set is an element of B and the set {Q n V : V E B} is a base for the open subsets of the subspace Q. A family B of open subset of X is said to be a pos-base for Q in X if the e m p t y set is an element of B and for every x E Q and an open neighbourhood U of x in X there exists an element V of B such t h a t xEVcU. Any base for the open subsets of X is called also a ps-base for Q in X. The proof of the next l e m m a is similar to the proof of L e m m a 1.2.6.

1.4. Commutative properties of the specific subspaces

29

1.3.8 L e m m a . Let n be a subset o f t such that the family

is ~ p-base (respectively, a pos-base) for QX in X E S. Then, the family B T and, therefore, the family B~>,~ is a p-base (respectively, a pos-base) for TIQ in T. (We note that ~ is independent of X) D

1.3.9 Corollary. Under the conditions of L e m m a 1.3.8, for every L E C + ( R ) t h e family BL; and, therefore, the family B~,~ is ~ p-b~se (respectively, a pos-bnse) for T(MIQ , RIQ , LIQ ) in T(M, I~, L). [-1

1.4

C o m m u t a t i v e properties of the specific subspaces

A g r e e m e n t . It is supposed that S, M, R, and Q are the same as in Section 1.3 (see Agreements 1 and 2). In particular, it is supposed that the family R is (M, Q)-admissible.

1.4.1 L e m m a . Let K { K X 9 X E S} b e a r e s t r i c t i o n of S such that K X c QX. (Therefore, K can be also considered as a restriction of Q). If the family R of equivalence relations on S is (M, Q)-admissible and (M, K)-~dmissible, then the family RIQ of equivalence relations on Q is (MIQ, K)-admissible. P r o o f . Let R be (M, Q)-admissible and (M,K)-admissible. Then, by definition, RIQ is MIQ-admissible. We need to prove that the family (RIQ)IK is (MIQ)lK-admissible. It is easy to verify that (RIQ)IK = RIK and (MIQ)IK = MIK. Therefore, it suffices to prove that the family RIK is MIK-admissible. But this follows immediately by the condition of the lemma that R is (M, K)-admissible. D N o t a t i o n . Let K - { K X 9 X E S} be a restriction of S such that K x c Q x . Suppose that the family R is (M, Q)-admissible and (M, K)admissible. Then, by Lemma 1.4.1, TIK can be considered as a subset of TIQ. The corresponding identical embedding is denoted by e TIK Iq"

The proof of the next lemma is similar to the proof of Lemma 1.3.5.

1. The construction of containing spaces

30 1.4.2

Under the conditions of L e m m a 1.4.1 the following

Lemma.

equality is true: TIQ

eT

T

K O

o eT

--

eT K

. [--1

D e f i n i t i o n . A restriction F - { F X 9 X E S} is said to be an (M, R)complete restriction if the family R is (M, F)-admissible and the subset TIF of T satisfies the following condition: for every point a of TIF and for every element (x, X) of a we have x E F X. D e f i n i t i o n . Suppose that for every element A of a set A, F(a)-

{Fx(a) 9x ~ s}

is a restriction of S. The union (respectively, the intersection) of the restrictions F(A) is the restriction {F x 9 X E S} of S for which

F X - u { F x (~,)-,~ c A} (respectively,

F x - - n { F x ( a ) . a c 1}) for every X E S. The defined restrictions are denoted respectively by V{F(A)" A E A} and A{F(A)" A E A}. 1.4.3 L e m m a . Suppose that for every element A of a set A of cardinality <_ r an (M, R)-complete restriction F(A) orS is given and let F be either the restriction A{F(A)" A E A} or the restriction V{F(A)" A E a}. If the family R is (M, F)-admissible, then F is an (M, R)-complete restriction. P r o o f . Let F(a)-

{Fx(a) 9x e s},

- A { F ( ~ ) - ~ ~ A} - { F x . X ~ S}, a E TIF-

T ( M I F , R I F ) , and ( x , X ) E a.

1.4. Commutative properties of the specific subspaces

31

There exists a pair (y, Y) of a such that y E F Y. Therefore, y E F Y (A) for every A E A. By Corollary 13.6, a E TIF(a). Since by assumption F(A) is an ( M , R ) - c o m p l e t e restriction, x E F X (A) for every A E A, which means that x E F X. Thus, F is also an (M, R)-complete restriction. The case where F = V{F(A): A ~ A} is proved similarly. [-7 Notation.

We set

CI(Q)Bd(Q)-

{C1x(QX) 9 x E S}, { B d x ( Q X ) 9 X E S},

I n t ( Q ) - { I n t x ( Q X ) 9 x E S}, C o ( Q ) - {X \ Q x . X E S}, and u~ - {uJ-x

~ s } , ~ ~ ~.

Obviously, the indexed sets CI(Q), B d ( Q ) , I n t ( Q ) , C o ( Q ) , and U5 are restrictions of S. D e f i n i t i o n . A restriction F - {F X 9 X E S} of S is said to be closed (respectively, open) if for every X E S, F x is a closed (respectively, an open) subset of X. Obviously, the restrictions CI(Q), B d ( Q ) , and C o ( U s ) , for 5 E 7, are closed and the restrictions I n t ( Q ) and Us, for 5 E 7- are open. 1.4.4 L e m m a . For every 5 E T the family R is (M, Us)-admissible and (M, Co(Us))-admissible. P r o o f . We prove t h a t R is (M, Us)-admissible. Let 6 E T. Consider an arbitrary element s of ? and set t = s U {5}. By L e m m a 1.3.1 it suffices to prove t h a t if X, Y E S and X ~ t y , then dx(U~ () - d y ( U ~ ) . Indeed, let f E d x (U~C). This means t h a t

u J n (n{x(~,s(~)) 9 : ~ ~}) r 0. By the choice of t and L e m m a 1.1.1,

1. The construction of containing spaces

32

which means that f E dy(U~). Thus, dx(U~ () C dy(U~). Similarly, dy(U~) C dx(U~) and, therefore, dy(U~) - dx(U~C). Hence, the family R is (M, U5)-admissible. Similarly we prove that R is (M, Co(U~))admissible. [El 1.4.5 L e m m a . For e v e r y (~ E 7" and H E C (>(R) the following relation

is true: TIu~ n T(H) - U~ (H). P r o o f . An element a of T belongs to T[ue N T(H) if and only if there exists an element (x, X) of a such that x E U~c and X E H, that is if and only if a E U5r (H). K1 1.4.6 L e m m a . Suppose that Q is a closed restriction of S. Then, the following statements are true: (1) The subset TIQ of T is dosed. (2) If, moreover, R is (M, Co(Q))-admissible, then (1.4.1)

TICo(Q) - T \ TIQ. (3) The restriction Q is an (M, R)-complete restriction.

P r o o f . Let a be a point of T for which there exists an element (x, X) of a such that x ~ QX. Since QX is closed in X there exists an element 5 of 7 such that x E U~ and U~c A QX _ O. Let s - {8}. Since R is (M, Q)-admissible by Lemma 1.3.1 there exists an element t of .7" such that ~ t c N ~ and dX(Q X) - dy(Q Y) for every Y E S for which

X ~ty. Let H be the ~-,t-equivalence class of J~. neighbourhood of a in T. We prove that U~r (H) C~TIQ - 0.

Then, U~(H) is an open

(1.4.2)

Indeed, in the opposite case there exists a point b belonging to the set U~r(H) N TIQ. Let (y, QY) be an element of b as a point of T]Q. By Lemma 1.3.2, b as a point of the space T contains the pair (y, Y). Since b E U~(H), we have Y E H and y E U~. Therefore, X ~t y . By the choice of t, X ~ 4 Y and dx(QX) _ dYs(QY). Since y E QY there exists a point z E QX such that d x (z) - dy (y). This equality and relation y E U~

1.4. Commutative properties of the specific subspaces

33

imply that z C U~. Therefore, QX n U~( 7~ O, which contradicts the choice of 5. Thus, the relation (1.4.2) is proved. Now we prove the statements of the lemma. (1). By Corollary 1.3.6 as the above point a we can take any point of the set T \ TIQ. In this case, the relation (1.4.2) implies that the set TIQ is closed. (2). As the point a we can take any point of the set TIco(Q). In this case, relation (1.4.2) means that a ~ TIQ , that is TIQ n T I C o ( Q ) - 0. The Corollary 1.3.6 implies that TIQ U TIco(Q) -- T. The last two relations are equivalent to the relation (1.4.1). (3). If Q is not an (M,R)-complete restriction, then there exists a point a of TIQ and an element ( x , X ) of a such that x ~ Qx. Then, relation (1.4.2) is true for some open neighbourhood U6r (H) of a, which is a contradiction. Thus, Q is an (M, R)-complete restriction. [:] 1.4.7 L e m m a . Suppose that the restriction Q of S is open and the family R is (M, Co(Q))-admissible. Then: (1) The following relation is true: TICo(Q) -- T \ TIQ.

(1.4.3)

(2) The subset TIQ of T is open. (3) The restriction Q is an (M, R)-complete restriction. P r o o f . (1). The relation (1.4.3) follows immediately by the statement (2) of Lemma 1.4.1 if instead of the restriction Q we consider the closed restriction Co(Q). (Note that C o ( C o ( Q ) ) = Q). (2). Since the restriction Co(Q) is closed by Lemma 4.6 the subset Tlco(q) of T is closed. Therefore, by relation (1.4.3) the subset TIQ of T is open. (3). Let a C TIQ and (x,X) E a. I f x ~ QX, then x c X \ Q X . By Corollary 1.3.6, a E TIco(Q ), which contradicts relation (1.4.3). D 1.4.8 L e m m a . Then,

Suppose that the f~mily R is (M, Cl(Q))-~dmissible.

TIcI(Q) -- C1T(TIo ). P r o o f . Since the restriction CI(Q) is closed by Lemma 1.4.6 the subset Tlcl(O) of T is closed. Therefore, since TIQ C TIcI(Q ), it suffices to prove that TIcI(Q) C C1T(TIQ ).

1. The construction of containing spaces

34

Let a E Tlcl(q ). Then, by Corollary 1.3.6 there exists an element (z, X) of a such that z E C1x(QX). Suppose that a ~ C1T(TIQ ). Then, there exists a neighbourhood U6r(H) E B T of a such that U6r(H)A TIo - 0. Corollary 1.3.6 implies that U~ N QY - 0 for every Y E H. Since X E H, U~ NQ x - O. This fact means that z ~ Clx(QX), which is a contradiction. Thus, a E C1T(TIQ) and, therefore, TIcI(O) C C1T(T[Q). [3 1.4.9 L e m m a . Suppose that Q is (M, R)-complete and the family R is (M, Int(Q))-admissible and (M, Co(Int(Q)))-admissible. Then, Tlint(q ) -- IntT(TIq ). P r o o f . Obviously, the restriction I n t ( Q ) is open. By assumptions of the lemma and Lemma 1.4.7, the subset Tlint(Q ) is open in T. Therefore, since Tlint(Q ) C TIQ , it suffices to prove that IntT(TIq ) C Tlint(Q ). Let a E IntT(TIq ). There exists an open neighbourhood U6r(H) E B T of a such that U6r(H) c IntT(TIq ). Let ( z , X ) E a. Then, z E Ux and X E H. We prove that U~( C QX. Indeed in the opposite case there exists a point y belonging to the set U~c \ Qx. Let b be the point of T containing the pair (y,X). Then, b E U6r(H). Since Q is an ( M , R ) complete restriction, (y,X) E b, and y ~ QX we have b r TIO , which contradicts the fact that UT(H) C IntT(TIo ) C TIo. Thus, Ux c Qx, which means that z E Intx(Q X) and, therefore, a E Tlint(Q ). !-1 1.4.10 L e m m a . Suppose that Q is (M,R)-complete and the family R is (M, Cl(Q))-admissible, (M, Int(Q))-admissible, (M, C o ( I n t ( Q ) ) ) ~dmissible, and (M, Bd(Q))-admissible. Then, T I B d ( Q ) - BdT(TIo ). P r o o f . Obviously, BdT(T]o) - C1T(TIo) \ IntT(Tlq). Lemmas 1.4.8 and 1.4.9 imply respectively that TIcI(Q) - C1T(T]Q) a n d T l i n t ( Q ) -- IntT(Tlq). Therefore, it is suffices to prove that

(1.4.4)

1.4. Commutative properties os the specific subspaces TIBd(Q) --

WIC,(Q) \

35

TlInt(Q).

Let a E TIBd(Q). There exists an element (x,X) of a such that x E Bdx(QX). Then, x E C1x(Q X) and, therefore, a E TIcI(Q ). On the other hand, since B d ( Q ) is a closed restriction, by Lemma 1.4.6, B d ( Q ) is an (M, R)-complete restriction. This fact means that for every (y, Y) E a we have y E Bdy(QY), that is y ~ Inty(QY). By Corollary 1.3.6, a ~ Tlint(Q), that is a E TICI(Q) \ Wlint(Q ). Conversely, let a E Tlcl(q ) \ TlInt(q ). Since el(Q) and I n t ( Q ) are (M, R)-complete restrictions, for every (x, X) E a we have

x E C1x(Q x) \ Intx(QX), that is x E Bdx(QX). This fact means that a E TIBd(Q). Thus, relation (1.4.4) is proved. [7 1.4.11 Corollary. Suppose that for some (~ C 7 the family R is (M, Cl(U~))-admissible and (M, Bd(U~))-admissible. Then, TIBd(U~) - BdT(WIu~).

Moreover, for every L E C 9 (R) we have TIBd(Ue) C7W(L) - BdT(L)(U~: (L)). Proof. Lemma 1.4.4 implies that the family R is (M, U~)-admissible, (M, Int(U~))-admissible, and (M, Co(Int(U~)))-admissible. By Lemma 1.4.7 the restriction U~ is an (M,R)-complete restriction. (Note that Int(U~) = U~.) Therefore, the first part of the corollary follows by Lemma 1.4.10. The second part of the corollary follows by the first part and Lemma 1.4.5 using the fact that T(L) is an open and closed subset of T. [-7 1.4.12 L e m m a . Suppose that Q is (M, R)-complete and the family R is (M, Co(Q))-admissible. Then, Co(Q) is also (M,R)-complete restric-

tion and TlCo(Q) -- T \ Tin.

(1.4.5)

Proof. Let a E Tlco(0). There exists an element (x, X) of a such that x E X \ Q X . Let (y,Y) E a. I f y ~ Y \ Q Y , t h e n a E TI0 and since Q

1. The construction of Containing Spaces

36

is a (M, R)-complete restriction we have x C QX, which is a contradiction. Therefore, y E Y \ QY, which means t h a t C o ( Q ) i s a ( M , R ) - c o m p l e t e restriction. This property also implies that a ~ TIQ. B y t h e above TICo(Q) C T \ T I Q . On the other hand TICo(Q)OTIQ - T. The last two relations imply (1.4.5). E]

1.5

A generalization of the construction of Containing Spaces

In this section we generalize the construction of Containing Spaces given in Sections 1.1 and 1.2. This generalization concerns the indexed families R and RM of equivalence relations on S. First, instead of the indexing set /~ we consider here the set 7)~(T) consisting of all subsets of ~- of cardinality less t h a n a fixed infinite cardinal y. Furthermore, the elements of the M - s t a n d a r d family is defined here by ~-algebras (see below the definition) instead of algebras. Finally, the property (b) of the definition of an admissible family (see the Introductory Remarks) is replaced by the following" the number of ~S-equivalence classes is less t h a n a fixed infinite cardinal #. For Containing Spaces to be T0-spaces of weight _< 7- the cardinals L, and # must satisfy some additional conditions. The given generalization is carried out in parallel to the construction of Sections 1.1 and 1.2. The definitions, lemmas and propositions considered here have a similar correspondence in mentioned sections. Many of them use the same notations and have the same formulation as the corresponding ones though they are related here to the "infinite case". The proofs of almost all lemmas and propositions are ommited. These proofs are similar to that of the corresponding lemmas and propositions of Sections 1.1 and 1.2.

Agreement

1. In this section we denote by y and # two fixed infinite cardinals. It is supposed that the following conditions are satisfied: (1) The cardinals u and ~- are not 7r-accessible for any cardinal ~r less than ~ (that is, the sum of less t h a n 7c many cardinals, which are less t h a n or ~- is less t h a n u or ~-, respectively). (2) 17)v(T)l < ~-, (therefore, y <_ T). (3) IP(7)(7c))1 < # _< T + for every cardinal 7c less than ~,. It is easy to see that the above conditions are satisfied if ~ - a~ and # is an arbitrary cardinal such that a~ < # < ~-+.

1.5. A generalization of the construction of Containing Spaces

37

R e m a r k . In the case where u = # = a~, the construction of Containing Spaces given here coincides with t h a t of Sections 1.1 and 1.2. The Containing Spaces corresponding to the case L, = ~- = a~ and # = aJ+ will be used in C h a p t e r 9. D e f i n i t i o n s . By a u-algebra of subsets of a set X, or briefly by a ualgebra of X , we mean a set of subsets of X which is closed under the

complements and under the sums of less t h a n u elements of this set. As for algebras we have: (a) the e m p t y set and the set X are elements of any u-algebra of X , (b) the set of all subsets of X is a L,-algebra of X , and (c) the intersection of any number of u-algebras of X is also a u-algebra of X. Therefore, any given set of subsets of X is contained in a u-algebra of X. The intersection of all such u-algebras is called the minimal u-algebra containing the given set of subsets of X . Let A be a u-algebra of a set X and i be a mapping of A into the set 7)(Y) of all subsets of a set Y. The mapping i is said to be a homomorphism if the following conditions are satisfied: (a) i ( X \ U) = Y \ i(U), U e A, and

(b) i(u{Ux :

e A}) = u{i(Ux):

e A}

for every set A of cardinality < u and elements U~ of A. It is easy to verify t h a t if i is an homomorphism, then the set i(A) is a u-algebra of the set Y. An one-to-one h o m o m o r p h i s m is called an isomorphism. It is easy to see t h a t if i is an isomorphism of A onto i(A) C 7)(Y), then the inverse mapping of i(A) onto A C Y)(X) is also an isomorphism. It is also clear t h a t the composition of two homomorphisms is a homomorphism; and, therefore, the composition of two isomorphisms is an isomorphism. The following two lemmas correspond to Lemmas 1.1.1 and 1.1.2 concerning algebras.

Let X and Y be sets, A a ~-algebra of X, and i an isomorphism of A into T)(Y). Suppose that U, V, and U~ are elements of A, where A E A and IAI < ~. Then, the relations 1.5.1 L e m m a .

N{U~:AEA}=V,U=0,

are equivalent to the relations

U=X,

andUcV

38

1. The construction of Containing Spaces n{i(u

) :

A} =

i(u) = v ,

O,

i(u) c i(v),

respectively. V] 1 . 5 . 2 L e m m a . Let X and Y be sets, G a subset of TJ(X), and A the minimal u-algebra os X containing G. The following statements are true: (1) Is is an isomorphism os A into T~(Y), then i(A) is the minimal u-algebra of Y containing the set i(~). (2) If hi and h2 are two homomorphisms of A into 79(Y) such that h i ( U ) = h2(U) for every U E g, then hi = h2. [~ D e f i n i t i o n . Let X be a m a r k e d space w i t h a m a r k {U~v 9 5 E T} a n d let s E 79,(T) \ {0}. T h e m i n i m a l u - a l g e b r a of X c o n t a i n i n g the sets U~v and for 5 E s is called the s-algebra of the m a r k e d space X and it is d e n o t e d by A x . Notation. Let X be a m a r k e d space w i t h a m a r k {U x 9 5 E T} a n d let s E 7),(7) \ {0}. For every 5 E r we set

X(6,0 ) -- Uff and X(5,1 ) -- X \ V ~ . For every f E 2 s we set

X(s,f ) --N{X(5,f(5))" 5 E 8}. We d e n o t e by 2~( the set of all f E 2 s such t h a t X(s,I) r ~. For every x E X there exists a u n i q u e element f of 2 s such t h a t x E X(s,f ). We define a m a p p i n g d X of X into 2 s setting dX(x) - f. It is s u p p o s e d t h a t d x (X) - 0 if X - 0. Obviously, d x (X) - 2~(. For every 5 E s we set

u(X, s, 5 ) = { f E 2 ~ : f ( 5 ) = For every element u of 7)(2~) we set

X(s,u ) --U{X(s,f))" f E u} if u r 0 and X(s,~,) - {Dif u - 0.

0}.

1.5. A generalization of the construction of Containing Spaces

39

Finally, we denote by A(2~) the set of all elements u of 7)(2~<) such t h a t X(s,~ ) E A x and by i~< the mapping of A(2~<) into A x such t h a t i~x(U) X(s,u) for every u E A(2~<). In particular, 0 E A(2~<) and i~<(0) - 0 E A X. Also, for every f E 2 ) , {f} E A(2~() and i~(({f}) - X(~,f) E A X. 1.5.3 L e m m a . Let X be a marked space and s E T),(T) \ {0}. Then, the set A(2~c ) is a .-algebra of 2~x; u ( X , s, 5) E A(2~) for every 5 E s; and the mapping iSx is an isomorphism of A(2~() onto A x such that -

for every 5 E s. [:] The above l e m m a and L e m m a 1.5.2 imply the following consequence. 1.5.4 C o r o l l a r y . Let X be a marked space and s e 7)~(T)\ {0}. Then, the set A(2~) is the minimal .-algebra of 2~x containing the sets u ( X , s, 5), 5 E s. [:] D e f i n i t i o n . For every s E 7).(T) on the class of all marked spaces an equivalence relation, denoted by ~sm, is defined as follows: two marked s p a c e s X and Y is s a i d t o b e ~s equivalent if either s - 0 or s ~ ~ and if there exists an isomorphism i of A X onto Ay, called natural, such that i(X(&o)) - 1~5,0) for every 5 E s. By L e m m a 1.5.2 such an isomorphism is uniquely determined. It is easy to see t h a t if q) :/: t C s E T)~(T) X and Y are two ~ sm_ equivalent marked spaces and if i is the corresponding natural isomorphism of A X onto Ay, then X ~ Y and the restriction to A X of i is the natural isomorphism of A X onto AtY, that is the equivalence relation ~ n is contained in the equivalence relation ~ tYKt" 1.5.5 L e m m a . Let X and Y be marked spaces and s ~ P . ( ~ ) \ {0}. The following condition are equivalent: (1) X and Y are ~Sm-equivalent. (2)

-

(3) u ( X , s, 5) = u(Y, s, 5) for every 5 E s.

(4) A(21 )= 1.5.6 Lemma. Let s be an element of T).(T)\{@}. Then, the cardinality of the set C ( ~ ms ) of all equivalence classes of the relation ~sm considered

40

1. The construction of Containing Spaces

on a given set of marked sp~ces is less than or equal to IP(2~)I . Moreover, there exists a set of marked spaces such that [ C ( ~ S m ) l - 17)(2s)1 . Proof.

By L e m m a 1.5.5, ] C ( ~ S ) l _< 179(2s)1. We for some set of m a r k e d spaces. For same l e m m a it suffices to prove t h a t for every subset a space X and an indexed base B X for X such t h a t corresponding to the m a r k e d space X (with the m a r k Let A be a subset of 2 ~. We set X = A. For every

prove t h a t actually this fact a n d by the A C 2 s there exists the subset 2~c of 2 ~ B X) is equal to A. (~ E s we set

u(A, s, (~) - { f E A " f ((~) - 0}. On the set X we consider a topology for which the finite intersections of elements of the set

{u(A, s, 5)" (~ E s} U { X \ u(A, s, 5)" (~ E s} is a base for the open sets. Since Isl _< ~- the weight of the c o n s t r u c t e d space X is < ~-. Let f and g be two distinct elements of X. Then, there exists an element (~ E s such t h a t f((~) 7~ 9(5). Therefore, either f((~) - 0 and g((~) - 1 or f(~) - 1 and g(~) - 0. Then, either f E u(A, s, 5) and g ~ u(A, s, 5) or f ~ u(A, s, 5) and g E u(A, s, 5). This fact means t h a t X is a T0-space. Let B x - {U~ c"/~ E 7} be the m a r k of X such t h a t U ~ - u(A, s, (~), (~ E s. By definition, an element f E 2 s belongs to 2~c if and only if n{x(~,s(~)). ~ ~ s} r O. It is easy to verify t h a t the i n t e r s e c t i o n A{X(t~,f((~)) 9 (~ E 8} is not e m p t y if and only if f E A. Thus, f E 2~( if and only if f E A, which m e a n s t h a t

2)-A.

D

D e f i n i t i o n . Let S be an indexed collection of spaces and M a c o - m a r k of S. For every s E P,(~-) we define on S an equivalence relation d e n o t e d by ~ as follows: two elements X and Y of S are ~ - e q u i v a l e n t if a n d only if either s - 0 or s -r 0 and the m a r k e d spaces X and Y with the marks M ( X ) and M ( Y ) , respectively, are ~ n - e q u i v a l e n t . T h e family

1.5. A generalization of the construction of Containing Spaces

41

of equivalence relations on S is called (M, y)-standard. Definitions. family R

Let S be an indexed collection of spaces.

An indexed

-

of equivalence relations on S is said to be p-admissible if the following conditions are satisfied: (a) if s c t E 7),(T), then the equivalence relation ~t is contained in the equivalence relation ~s, (b) for every s E P~(7) the number of ~S-equivalence classes is less than #, and (c) ~ s = S x S if s = 0. The set :,

e

is denoted by C(R). The minimal p-algebra of S containing the set C(R) is denoted by C~ (R). By Lemmas 1.5.5 and 1.5.6 and the condition that IP(P( ))I < ~ for every cardinal ~ < L, (see Agreement 1 of this section) the (M, L,)-standard family of equivalence relations on S is p-admissible. Let R0 - { ~ : s C P , ( 7 ) } and R1 - { ~ : s E 7),(~-)} be two #admissible families of equivalence relations on S. It is said that R1 is & final refinement of R0 if for every s E 7)~(7) there exists an element t of 7)~(7) such that ~ c ~ . A #-admissible family R of equivalence relations on S is said to be (M, y, #)-admissible if R is a final refinement of the (M, ~)-standard family

R~. A g r e e m e n t 2. In what follows of this section it is assumed that an arbitrary indexed non-empty collection of spaces (of weight _< 7) denoted by S is given. It is also assumed that an arbitrary co-mark of S is given. This co-mark is denoted by M. Moreover, we assume that for every X E S, M(X)-

{ U ~ " # r ;}.

Finally, it is assumed that an (M, L,, #)-admissible family of equivalence relations on S is given. This family is denoted by

1. The construction of Containing Spaces

42

D e f i n i t i o n . Suppose t h a t S contains a non-empty element. On the set of all pairs ( x , X ) , where X E S and x E X, we consider an equivalence relation denoted by ~ and defined as follows: two pairs (x, X ) and (y, Y) are ~ - e q u i v a l e n t if X ~s y and d X (x) -- dy (y) for every s C 7)u(T) \ {0}. Notation. The set of all equivalence classes of the relation ~ 4 is denoted by T ( M , R) - T. It is supposed t h a t T ( M , R) - 0 if all elements of S are empty. For every element H of C~ (R) the set of all a E T for which there exists an element (x, X) of a such t h a t X E H is denoted by T ( M , R, H) - T ( H ) . It is easy to see that if a E T ( H ) , then for every (x, X ) E a, X E H. N o t a t i o n . Let s =/= 0 and t be elements of P , ( T ) such t h a t ~ t c ~ and let H be an element of C ( ~ t ) . Suppose t h a t an element X of H is chosen. We denote by 2~ and a(2 ) the sets 2 ) and A ( 2 ) ) , respectively, and for every ~ E s by u(H, s, 5) we denote the set u ( X , s, 5). Since X ~ Y for every X, Y E H, the sets 2~, a ( 2 ~ ) , and u ( H , s, ~) are independent of the element of H we choose to define them. (See L e m m a 1.5.5). For every u E A(2~) we denote by T(s,~)(H) the set of all elements a of T for which there exists an element (x, X) of a such t h a t X E H and x E X(s,u). Finally we set A n - {T(~,u)(H) 9 u E A(2{_I) }. Since T(s,u ) (H) is a subset of T ( H ) , A N can be considered as a set of subsets ofT(H). 1.5.7 L e m m a . Let s , t E T),(T), s r O, ~ t c ~ , H E C ( ~ t ) , and u E A(2{_I). Then, the set T(s,u ) (H) coincides with the set of all a E T such that for every (x, X ) E a we have X E H and x E X(s,u ) . 73

1.5.8 Lemma. Let s, t E T),(T), s ~= O, ~ t c ~ ,

and H E C(~t). Let

also A be a set of cardinality < u and u, v)~ E A(2~_I), A E A. Then, setting w - 2 h \ u and v - U{v), 9 A E A},

1.5. A generalization of the construction of Containing Spaces

43

we have T ( H ) \ T(s,u)(H) - T(s,w ) (H) and T(s,v) (H) - U{T(s,v~)(H)" :k E A}. E] 1.5.9 C o r o l l a r y . I f s, t E 7J,(r), s # O, ~ t C ~ 4 , the set A M is a ~-algebra of the set T ( H ) . [-]

and H E C ( ~ t ) , then

1.5.10 Lemma. Let s , t E P~,(r), s --/= 0, ~ t C ~ h and H E C ( ~ t ) . Then, the m a p p i n g i ~ of the set a ( 2 ~ ) onto the set AHs , for which i ~ ( u ) T(s,u)(H) for every u E A ( 2 h ) , is an isomorphism. K] 1.5.11 C o r o l l a r y . Let s, t E 7).(r), s ~: 0, ~ t c ~ , H E C ( ~ t ) , and X E H. Then, the m a p p i n g i i~io(i~) -1 is an isomorphism of the algebra A X onto the algebra A H such t h a t for every 5 E s, i(X(5,o)) - T(s,u)(H), where u = u ( H , s, 5). [2] -

-

N o t a t i o n . For every 5 E r and n E C~ (R) we denote by U~: ( n ) the set of all a E T for which there exists an element ( a , X ) of a such t h a t X E H and a E U~c. Therefore, if for some elements s and t of 7)~(r) we have 5 E s, H E C ( ~ t ) , and ~ t c ~ 4 , then

U~ (n)

-

T(s,~)(n) E A~,

where u - u ( H , s, 5).

We denote B T the set of all subsets of T of the form U~r (H), Notation. where 5 C s and H E C ( ~ s ) for some t, s E 7)~(~-) such t h a t ~ t c ~ } . We set B T - {U~r(H) 9 5 r 7 and H E C(R)}, B~> - {U~r ( H ) " 5 E r and H E C~ (R)}. Also for every subset ~ of r we set B,T~ -- { U ~ ( H ) E B T" 6 r n}, - {Uf(H)

B

a

44

1. T h e construction of Containing Spaces

B~>,~ - {U~(H) E B~>" 5 E n}. Since I79u(~-)l _< T and for every s E 7)u(w), IC(~S)[ < # _< w+ (see Agreement 1 and Lemma 1.5.6) we have IBTI _< IBTI _< lEVI _< w. 1.5.12 L e m m a .

T h e set B T is a base for a topology on the set T. Moreover, if ~ is a subset of T such that for every X E S the set

is

a

base for X , then the set B,T~ is

a

base for the same topology on T. R

D e f i n i t i o n . The set T equipped with the topology for which the set B T is a base is called the Containing Space f o r the indexed collection S corresponding to the co-mark M and the f a m i l y R. In what follows whenever we refer to the space T, we mean just this space. 1.5.13 L e m m a .

L e t 5 E T and H E C~(R).

Then, the following

s t a t e m e n t s are true; (1) There exists an element t of T~,(T) such that H is a union o f ~ tequivalence classes. (2) T h e set UT(H) is open in T and coincides with the set of all a E T such t h a t for every (x, X) E a, X E H and x E U ~ . (3) T h e set T ( H ) is s i m u l t a n e o u s l y open and closed in T. (4) T h e set C1T(U~r(H)) coincides with the set of all a E T such t h a t there exists an element ( x , X ) of a for which X E H and x E Clx(U~ c) as well as with the set of all a E T such t h a t for every (x, X ) E a, X E H and

BdT(U~(H))

(5) The set coincides with the set of all a E T such t h a t there exists an element ( x , X ) of a for which X E H and x E Bdx(U~x) as well as with the set of all a E T such that for every (x, X) E a, X E H and x E B d x ( U f i ) . I-I 1.5.14 C o r o l l a r y . T h e sets B T and B ; are bases for the space T. Moreover, if for a subset ~ of T and for every X E S the set { U ~ 9 5 E ~} is a base for X , then the sets B T and B~>,~ are also bases for T. I-7

1.5.15 Proposition. weight ~ T. F3

T h e Containing Space T is a To-space o f the

1.6. A g r e e m e n t concerning notations of the n e x t chapters

45

D e f i n i t i o n . Let X C S. For every z E X there exists a uniquely determined element a of T which contains the pair (z, X). We define a mapping e x of X into T setting eX (x) - a.

The next proposition shows that e x is an embedding. This embedding is called natural. 1.5.16 P r o p o s i t i o n . For every X E S the m a p p i n g e X is an e m b e d d i n g of X into T. [-1

1.6

A g r e e m e n t concerning notations of the next chapters

In the next chapters we shall construct many Containing Spaces corresponding to different indexed collections of spaces, co-marks, and families of equivalence relations. For the corresponding notations we shall follow the notations introduced in the present chapter. Thus, in general outline we shall follow the given below rules. (1) In the notations of the Containing Spaces, co-marks, families of equivalence relations, standard bases, and their elements we do not indicate the corresponding indexed collections of spaces. These collections will be clear by the context or will be explicitly indicated. The letter "S" is used only for the notations of indexed collections of spaces. However, we use other letters for such notations since many restrictions of some given indexed collection of spaces, denoted by different letters, will also be considered as an indexed collection of spaces. (2) The co-marks and families of equivalence relations on indexed collections of spaces used for the construction of Containing Spaces are always denoted by the letters "M" and "R", respectively, and equipped if neecessary with other letters or symbols as indices. Some of the considered indexed co-bases (that is, co-marks) are denoted without use of the letter "M". However, the Containing Spaces corresponding to such co-marks are not needed for our considerations. (These co-marks are used in order to define standard bases in Containing Spaces corresponding to extensions of these co-marks).

46

1. The construction of Containing Spaces

(3) For the Containing Spaces we shall use two kind of notations. The first kind is the "whole" notation consisting of the letter "T" and in brackets the corresponding co-mark and the family of equivalence relations. For example, if the co-mark is denoted by M ~ and the family of equivalence relations by I{~, then the corresponding Containing Space is denoted by T ( M ~ , R ~ ) . The second kind is the "brief" notation consisting of the letter "T" equipped, if neccessary, with some indices. For example, T ~ can be considered as the brief notation of the space T ( M ~ , R~). The brief notation of a Containing Space is always defined throughout the book with the symbol " - " using the whole notation of this space. The next rule is the only exception for such a definition of the brief notation. (4) There is an "ordinary situation" which is the most frequent case. In these instances, the considered indexed collection of spaces is denoted by the letter "S", the co-mark is denoted by the letter "M", and the Madmissible family of equivalence relations is denoted by the letter "R". All these letters are considered without indices. In this case and only in this case, the brief notation of the corresponding Containing Space is the letter "T" without indices. Because of this fact in the ordinary situation we sometimes use the brief notation without defining it. Moreover, the whole notation of the corresponding Contained Space is not always indicated. We note that in the present chapter we considered exactly this situation. (5) For the notations of the restrictions of indexed collections of spaces we use different letters equipped if neccessary with some indices. All such letters are capital and bold-faced. For the corresponding specific subsets of Containing Spaces we also use two kinds of the notations. The first kind is the "whole" notation obtained by the whole notation of the Containing Space replacing the corresponding co-mark and the family of equivalence relations by its traces on the restriction. For example, if the Containing Space is denoted by T ( M ~ , R~) and the restriction is denoted by Ue, then the corresponding specific subset is denoted by T(M~Iu~, R~Iu~). The second kind is the "brief" notation. This notation is obtained as "the trace" of the brief notation of the Containing Space on the restriction. For example, if the brief notation of the Containing Space T ( M ~ , R ~ ) is T~, then the brief notation of the specific subset T(M~Iu~, R~Iu~) is In a non-ordinary situation the brief notation of a specific subset is always defined through the symbol "=" using the whole notation of this

1.6. Agreement concerning notations of the next chapters

47

subset. In the ordinary situation we sometimes use the brief notation of a specific subset without to define it. In this case we do not indicate the whole notation of this subset. This action means, for example, that TIQ is the brief notation of the specific subset T(MIQ , RIO ) of the Containing Space T ( M , R). We note that the last situation is the most frequent case. (6) For the primary subspaces we also use the "whole" and "brief" notations. In the ordinary situation the whole notation is T ( M , R, H) and the brief is T(H), where H is the corresponding element of C0(R). In the non-ordinary situation the whole and brief notations are obtained by replacing in T ( M , R, H) and T(H) the letters "M", "R"; and "H" by the notations of the corresponding co-mark, the family of equivalence relations, and the corresponding element of the minimal ring containing equivalence classes of all elements of the last family, respectively. We note that in the brief notations of primary subspaces in the nonordinary situation the letter "T" is always used without indices, which means that "T" is not the brief notation of the corresponding Containing Space (that is, it is not the same as for the specific subsets). Whenever we use the brief notation of a primary subspace (in a non-ordinary situation), we define this notation by using the symbol " - " . For example, if the Containing Space is denoted by T ( M ~ , R ~ ) and n is an element of C ( R ~ ) , then the whole notation of the corresponding primary subspace is T ( M ~ , R ~ , H) and the brief is T(H). In particular, if Ue is a restriction of the considered indexed collection of spaces, then T(HIu~) is the brief notation of the primary subspace T ( M ~ I u ~ , R Iu ,HIu ) of the Containing Space T ( M ~ I u ~ , R~lu~ ). (7) In the next chapters in the ordinary situation the ~-standard and (~, ~)-standard families of (or bases for the) open subsets of the Containing Space T will be denoted (as in Section 1.2 of the present chapter) by B T and B T respectively, where ~ is a subset of w Also, for every element L of C(R) or C~(R) the n-standard and (+, ~)-standard families of open subsets of the primary subspace T(L) of T will be denoted by B L and B L respectively. Moreover, the elements of all these families will be denoted by U~r (H), where ~ E ~ and H E C(R) or H E C + (R), respectively. In the case where ~ = w, the letter "~" is omitted. Of course, instead of the above letters "~", "L", and "H" we can use other letters or group of symbols. Usually, ~ is the image of the set w under an indicial mapping 0, that is, =

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1. The construction of Containing Spaces

In a non-ordinary situation the notations of all these families and their elements are obtained by the above, replacing the letter "T" by the brief notation of the corresponding Containing Space and the letter "L" by the notation of the corresponding element of the minimal ring containing the equivalence classes of all elements of the considered family of equivalence relations. We note that in the notations of the elements of the above families there are a few cases where instead of the letter "U", we use the letter "V". In these cases the notations will be clearly explained.