- Email: [email protected]
A note on disconnectedness
Chaos, Solitons and Fractals 42 (2009) 3242–3246
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
A note on disconnectedness C.K. Basu a,*, S.S. Mandal b a b
Department of Mathematics, West Bengal State University, Barasat 700126, North 24 Pgs., West Bengal, India Department of Mathematics, Krishnagar Women’s College, Krishnagar 741101, Nadia, West Bengal, India
a r t i c l e
i n f o
Article history: Accepted 28 April 2009
a b s t r a c t The purpose of this note is to characterize various disconnectedness in terms of (star) refinement of covers. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Relation of topology and physics is gaining momentum and is acting as a focus of attention by many mathematicians as well as physicists. El Naschie et al. [1–7] have pointed out that topology plays a significant role in quantum physics, high energy physics and super string theory; in addition, in [3], by the help of set theory, El Naschie derived quantum gravity. Hermann [9], Khalimsky et al. [10], Kong and Koppermann [11] have applied topology in computer science and digital topology. Moor and Peters [13] have investigated computational topology for geometric design. In topology, disconnectedness, as considered as a fundamental concept has been researched by many. M.H. Stone’s contribution [14] towards the topological representation of Boolean algebra in terms of disconnectedness is quite remarkable. Various types of disconnectedness have the potential to play important roles specially, in studying fractals and dimension theory. S. Halayka, in his recent paper [8] used disconnectedness to simplify an analysis of several well-known physical phenomena related to electromagnetism. Because of enormous use and potentiality of various types of disconnectedness, our task in this paper is to offer new characterizations of various types of disconnectedness (e.g. total disconnectedness, zero-dimensionality, extremal disconnectedness, strong zero-dimensionality) in terms of (star) refinement of covers with the belief that such investigations may have possible applications in a broader sense in fractals and quantum physics. By X, we shall mean a topological space without any separation axioms. For a cover U of X and for a subset A of X, the star of A with respect to U is the set StðA; UÞ ¼ [fU 2 U : U \ A – ;g; and for two covers U and V of X, we call U star refines V or H U is a star refinement of V, written U < V, if for each U 2 U; 9V 2 V such that StðU; UÞ V. U is called a refinement of V written as U < V if for each U 2 U, there exists V 2 V such that U V. H A normal sequence of covers is a sequence of open covers U1 ; U2 ; . . . of X such that Unþ1 < Un , for n ¼ 1; 2; . . .; and a normal cover is cover which is U1 in some normal sequence of covers. A family V of covers of X is a normal family iff every cover in V has a star refinement in V [Section 36.10, p. 247 [15]]. A collection l0 of covers of a space X is a base for some uniformity on X iff it satisfies the condition that for U1 ; U2 2 l0 H H there is some U3 2 l0 such that U3 < U1 and U3 < U2 [Section 36.3, p. 245 [15]]. It is well known that if l0 is a base for a covering uniformity l on X, then fStðx; UÞ : U 2 l0 g is a nbd. base at x 2 X in the uniform topology [Section 36.6, p. 246 [15]]. Also if X is any uniformizable topological space then there is a finest uniformity on X, compatible with the topology of X, called the fine uniformity on X, denoted by lF , having a base of all normally open covers of X; such a space is known as the fine space. Further, a T 1 space is paracompact iff every open cover has an open star refinement [Section 20.14, p. 149
* Corresponding author. E-mail address: [email protected] (C.K. Basu). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.056
C.K. Basu, S.S. Mandal / Chaos, Solitons and Fractals 42 (2009) 3242–3246
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[15]], where a space X is called paracompact iff every open cover of X has an open locally finite refinement, which is also a cover of X. A topology r on X is called a subtopology of a space ðX; sÞ if r s. 2. Characterizations of various disconnectedness H
Theorem 2.1. A topological space ðX; sÞ is disconnected iff it has an open cover U consisting of proper subsets of X such that U < U. Proof. Let ðX; sÞ be disconnected. Then there exist U; V 2 s such that U – ;; V – /; U [ V ¼ X and U \ V ¼ ;. So, U ¼ fU; Vg is H UÞ ¼ V V. an open cover of X consisting of proper subsets of X. Here U < U, because StðU; UÞ ¼ U U and StðV; H Conversely, let U ¼ fU b : b 2 Ig be an open cover of X consisting of proper subsets of X, i.e. U b – X; 8b 2 I and let U < U. Because of the result that a cover V of a space Y star refines itself iff for some partition Q of Y; V < Q < V [exercise 36 A (3), p. 249 [15]], we have a partition P of X such that U < P < U. As P < U and U is consisting of proper subsets, P is not a trivial partition, i.e. P – fXg. Let P ¼ fAa : a 2 Kg, where Aa – /; Aa – X; 8a 2 K. We shall show that each Aa is open. Let Aa 2 P and x 2 Aa . Then as U is a cover of X there exists a U bx 2 U such that x 2 U bx . Now as U < P; U bx Aa0 , for some Aa0 2 P. We shall show that a ¼ a0 . If not, then x 2 U bx Aa0 and x 2 Aa . So, Aa ; Aa0 are distinct members of the partition P but Aa0 \ Aa – / – a contradiction. Hence a ¼ a0 . Thus Aa ¼ Aa0 and hence x 2 U bx Aa , and it is true for all x 2 Aa . Hence Aa ¼ [x2Aa U bx is open. Now as P is a non trivial partition, so P ¼ fAa : a 2 Kg contains at least two non empty members say Aa1 ; Aa2 . Now let K1 K be such that a1 2 K1 and a2 2 K K1 ; then U ¼ [a2K1 Aa ; V ¼ [a2KK1 Aa are open sets with U – /; V – /, U [ V ¼ X and U \ V ¼ /. So, ðX; sÞ is disconnected. h Theorem 2.2. If ðY; sÞ is a metrizable Lindeloff space then the following are equivalent: (i) ðY; sÞ is zero-dimensional. (ii) Every open cover U of Y is a normally open cover with a normal sequence of open covers U ¼ U1 ; U2 , . . . such that Ui consists of clopen subsets of ðY; sÞ, for at least one
ði P 2Þ
ðÞ
(iii) Every open cover U of ðY; sÞ has a refinement V consisting of clopen sets only. Proof. We shall first prove that ðiÞ () ðiiÞ. Let ðiÞ holds, i.e. ðY; sÞ is zero-dimensional. Let U ¼ fU a : a 2 Kg be an open cover of Y. As ðY; sÞ is Lindeloff, U has a countable subcover U0 ¼ fU i : i ¼ 1; 2; . . .g, finite or infinite. So,
U0 < U
ðIÞ 1
Now by the reduction theorem [Theorem 1 , p. 279, [12]], there exists a sequence of disjoint clopen sets H1 ; H2 ; . . . . . ., which is finite (when U0 is finite) or infinite (when U0 is infinite) such that [Hi ¼ Y and Hi U i ; 8i. Let V ¼ fHi : i ¼ 1; 2; . . . . . .g. Clearly V is a partition of Y and so H
V
ðIIÞ
Now for each Hi 2 V; StðHi ; VÞ ¼ Hi U i 2 U0 and so H
V < U0
ðIIIÞ
Then (I) and (III) imply H
V
ðIVÞ
Now we construct the sequence U1 ; U2 ,. . . of open covers of Y by taking U1 ¼ U; Ui ¼ V; 8i P 2. (II) and (IV) imply that this is a normal sequence, where Ui ¼ V consists of disjoint clopen sets for all i P 2. So ðiÞ ) ðiiÞ. Conversely, let ðiiÞ holds. Let l0 be the collection of all normally open covers in ðY; sÞ. Now as ðY; sÞ is completely regular, so is uniformizable. Let l be the fine uniformity (having a base l0 ) on Y compatible with s. So, bx ¼ fStðx; UÞ : U 2 l0 g is a nbd. base at each x 2 ðY; sÞ. Now, let G 2 s. Then for x 2 G; Stðx; UÞ G for some U 2 l0 . So, by ðiiÞ there exists a normal sequence of open covers U1 ; U2 ,. . .. . .. such that U ¼ U1 and for at least one iðP 2Þ; Ui consists of clopen sets of ðY; sÞ. If the value of i be n (say), then H x 2 U for some U 2 Un . Now as Un < U1 ¼ U, so x 2 StðU; Un Þ V for some V 2 U and also V Stðx; UÞ. So, x 2 U StðU; Un Þ V Stðx; UÞ G. Therefore x 2 U G and U is a clopen subset of ðY; sÞ. Hence ðY; sÞ is zero-dimensional, i.e. ðiiÞ ) ðiÞ is established. Next, we shall prove that ðiiiÞ () ðiÞ.
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First let ðiÞ holds and U ¼ fU a : a 2 Kg be an open cover of Y. Then by the result that in a zero-dimensional metric space every open set is the union of a sequence of disjoint clopen sets (Corollary 1a, p. 277, [12]), every open set U a 2 U, can be expressed as the disjoint union of clopen sets. So ðiiiÞ follows trivially by taking an cover U0 consisting of such disjoint clopen sets for each U a whose union is U a but not taking U a (which clearly shows U0 < U). So, ðiÞ ) ðiiiÞ holds. Conversely, let ðiiiÞ holds. Let l be the fine uniformity having base l0 of all normally open covers of ðY; sÞ compatible with the topology s of Y. Consequently, bx ¼ fStðx; UÞ : U 2 l0 g forms a local base at x in ðY; sÞ, for each x 2 Y. Let G 2 s and x 2 G. Then x 2 Stðx; UÞ G for some U 2 l0 . Now by (iii), U has a refinement U0 , consisting of clopen sets of ðY; sÞ and hence we have x 2 Stðx; U0 Þ Stðx; UÞ G. As U0 is a cover of X, so x 2 U for some U 2 U0 and hence x 2 U Stðx; U0 Þ G, where U is clopen. Therefore ðY; sÞ is zero-dimensional, i.e. ðiiiÞ ) ðiÞ. h Definition 2.3. A space X is called strongly zero-dimensional iff for every pair of disjoint zero sets of the space, there is a clopen set containing one zero set and missing the other. Theorem 2.4. For a T 2 space X, if bX has no proper non trivial pseudometrizable subtopology then the following are equivalent: (i) X is totally disconnected, (ii) X is strongly zero-dimensional, (iii) bX satisfies property (*). Proof. ðbX; sÞ is obviously Lindeloff.Also ðbX; sÞ is metrizable.Indeed, if possible, let ðbX; sÞ be non-metrizable.Since in a paracompact T 2 space, every open cover has an open star refinement, so every open cover is a normally open cover.As ðbX; sÞ is T 2 , for all x – y 2 bX, there exist disjoint open sets Ox and Oy containing x and y, respectively.Clearly Ox and Oy are proper open subsets of X, for all x – y 2 bX.Fix y0 ; x0 2 bX and take U ¼ fOx ; xð – x0 Þ 2 bX such that Ox0 \ Ox ¼ ;; x0 2 Ox0 ; x 2 Ox ; Ox0 ; Ox 2 sg [ fOx0 2 s : Ox0 \ Oy0 ¼ ;; x0 2 Ox0 ; y0 2 Oy0 ; Oy0 2 sg; then U is an s-open cover of bX.So, U is a normally open cover.Let U ¼ U1 ; U2 ,. . ..be the corresponding normal sequence of open covers.Then l0 ¼ fU1 ; U2 ; . . .g, is a base for some uniformity l on bX.Let sl be the corresponding uniform topology induced by the uniformity l on bX.Then sl is pseudometrizable as l has a countable base l0 .The family bx ¼ fStðx; Ui Þ : i ¼ 1; 2; 3; . . . :g is a nbd.base H at x 2 bX in ðbX; sl Þ.Here it is to be noted that all members of all U;i s are proper open subsets of bX [as Uk < Uk1 and U1 conH sists of proper open subsets of bX, so Uk < U1 , 8k > 1 and hence U 2 Uk ) 9V 2 U1 such thatStðU; Uk Þ V – bX]. Now for Stðx; Ui Þ 2 bx ; x 2 X; Stðx; Ui Þ # StðU; Ui Þ for some U 2 Ui such that x 2 U. Also StðU; Ui Þ V for some V 2 Ui1 and as V – bX, then Stðx; Ui Þ V – bX. So, bx contains many proper subsets of bX. Therefore sl contains proper nonempty open sets, and hence sl is a non-trivial topology on bX. Now if l01 be the collection of all open covers of bX then l01 is the base for the fine uniformity lF on bX [as ðbX; sÞ is paracompact T 2 , so every open cover of ðbX; sÞ is normally open cover, and as the family of all normally open covers of the uniformizable space ðbX; sÞ, forms a base for the fine uniformity lF on bX which induces the topology s ] which gives the s as the uniform topology. Now l0 l01 ) sl # s ¼ slF . As s is not pseudometrizable and sl is pseudometrizable, topology so sl – s – a contradiction. Since X is strongly zero-dimensional iff bX is zero-dimensional, then theorem follows by the direct application of Theorem 2.2. h Theorem 2.5. If every open cover of a T 2 space ðX; sÞ has a refinement which star refines itself then ðX; sÞ is totally disconnected. Proof. Let Y be a subset of X consisting at least two points. Let x 2 Y and keep it fixed. Then for each y 2 Y with x – y, there exist open sets U x (y) and U y ðxÞ containing x and y, respectively, such that U x ðyÞ \ U y ðxÞ ¼ /. So, U x ðyÞ; U y ðxÞ are proper open sets in X and hence U x ðyÞ \ Y and U y ðxÞ \ Y are proper open subsets of Y for all yð – xÞ 2 Y. Now we shall consider the points of X Y. Case I: Suppose cardðX YÞ > 1. For the fixed point x 2 Y and for each y0 2 X Y, there exist open sets U x ðy0 Þ and U y0 ðxÞ containing x and y0 , respectively, such that U x ðy0 Þ \ U y0 ðxÞ ¼ ;. So, fU y ðxÞ : yð – xÞ 2 Yg [ fU x ðyÞ: for some yð – xÞ 2 Yg [ fU y0 ðxÞ : y0 2 X Yg constitutes an open cover of X consisting of proper open subsets of X and intersection of members of this cover with Y are also proper open subsets of Y. Case II: Suppose cardðX YÞ ¼ 1, i.e. X Y ¼ fy0 g. Since ðX; sÞ is T 2 , then as before the cover fU y ðxÞ : yð – xÞ 2 Yg [ fU x ðyÞ: for some yð – xÞ 2 Yg [ fU y0 ðxÞg is consisting of proper open subsets of X; and the intersection of members of this cover with Y are also proper open subsets of Y. So, for each case we obtain an open cover U ¼ fU a : a 2 Kg of X such that / – U a – X and U a \ Y are proper subsets of Y; 8a 2 K. Now, by the given condition U has a open refinement V ¼ fHb : b 2 K0 g such that V < V. Then 0 0 VY ¼ fHb \ Y : b 2 K g is an open cover of Y, where Hb \ Y are proper open subsets in Y for all b 2 K [as Hb U a , for some U a 2 U and U a \ Y – Y hence so is Hb \ Y]. H Now we shall show that VY < VY . Indeed, we always have StðHb \ Y; VY Þ StðHb ; VÞ. But StðHb ; VÞ Hb0 for some Hb0 2 V as V < V. So, StðHb \ Y; VY Þ Hb0 . Again, StðHb \ Y; VY Þ Y. Therefore, StðHb \ Y; VY Þ Hb0 \ Y, where Hb0 \ Y 2 VY . Hence VY < VY . So, Y has an open cover consisting of proper open subsets of Y which star refines itself. Therefore, by Theorem 2.1, Y is disconnected. Hence X is totally disconnected. h
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Problem 2.6. Whether the sufficient condition of Theorem 2.5 is actually a characterization of totally disconnectedness is still unknown to the authors. Theorem 2.7. A space X is extremally disconnected if and only if for every open set U, the cover fU; X Ug of X has an open refinement. Proof. Let for every open set U, the cover U ¼ fU; X Ug of X has an open refinement, say V. Now for each x 2 U, there exists a V x 2 V such that x 2 V x (as V is a cover X). Since V < U, there exists a W x 2 U such that x 2 V x W x . Hence W x is either U or X U. But W x cannot be X U as x R X U. So, W x ¼ U. Thus, x 2 V x U and this holds for each x 2 U. Therefore U is open and hence X is extremally disconnected. Conversely, let X be extremally disconnected. Then for every open set U, the open cover fU; X Ug itself satisfies the condition that required. h Theorem 2.8. Let ðX; sÞ be a separable zero-dimensional space in which every F r set is closed, then ðX; sÞ is extremally disconnected. Proof. Let U be any open set. Now as open subspace of a separable space is separable, X U is separable. Let A ¼ fx1 ; x2 ; . . . ; xi ; . . .g be the countable dense subset of X U. So, A ¼ X U in ðX U; sXU Þ. Now as, X is zero-dimensional so for xi 2 X U, there exists clopen set in ðX; sÞ say, U xi such that xi 2 U xi X U. Now 1 1 [xi 2A fxi g [1 i¼1 U xi X U, i.e. A [i¼1 U xi X U. Now, sXU – closure ðAÞ sXU – closure ð[i¼1 U xi Þ. Therefore 1 1 1 U Þ s – closure ð[ U Þ [ s – closure ðU Þ ¼ [ U X U. Hence X U X U sXU – closure ([1 x x x x i i i i i¼1 i¼1 i¼1 i¼1 1 [1 i¼1 U xi X U. So X U ¼ [i¼1 U xi . As every F r set is closed, so X U is a closed set. Therefore, U is open. Hence X is extremally disconnected. h Corollary 2.9. If in a separable zero-dimensional space every F r set is closed then, every open set is closed. Corollary 2.10. Let ðX; sÞ be a regular separable space in which every F r set is closed, then X is zero dimensional if and only if X is extremally disconnected. Proof. As an extremally disconnected regular space is zero dimensional, hence the result follows.
h
3. Conclusion As far as application of topology in various other branches of mathematics and physics is concerned, the study of disconnectedness in topology plays a predominant role. For example, M.H. Stone [14] asserts that every Boolean algebra has the topological representation as the Boolean algebra COðXÞ of clopen sets for some totally disconnected space X (while the Boolean algebra COðXÞ of clopen sets for a connected space X is not so interesting as it consists only of ; and X). It has been observed that there is a close resemblance of fractals and dimension theory with various types of disconnectedness like total disconnectedness, zero-dimensionality and extremal disconnectedness. Further, analysis of several well-known physical phenomena related to electromagnetism have also been simplified in recent times (2008) by the help of the concept of disconnectedness [8]. We believe, more and more investigations in the arena of disconnectedness, specially, offering new characterizations of such concept, may have possible applications in physics like fractals and quantum mechanics as well as in dimension theory. Acknowledgement The authors are grateful to the learned referee for his valuable comments which improved the paper a great extent. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
El Naschie MS. On the uncertainty of Cantorian geometry and the two-slit experiment. Chaos, Solitons and Fractals 1988;9:517–29. El Naschie MS. On the unification of heterotic strings, M theory and e1 theory. Chaos, Solitons and Fractals 2000:2397–408. El Naschie MS. Quantum gravity from descriptive set theory. Chaos, Solitons and Fractals 2004;19:1339–44. El Naschie MS. Quantum gravity, Clifford algebras, fuzzy set theory and the fundamental constants of nature. Chaos, Solitons and Fractals 2004;20:437–50. El Naschie MS. Two slit experiment as the foundation of E-infinity of high energy physics. Chaos, Solitons and Fractals 2005;25:514–9. El Naschie, Rössler OE. Quantum mechanics and chaotic fractals. Chaos, Solitons and Fractals 1994;4(3):307–9. El Naschie MS, Rössler OE, Oed G. Information and diffusion in quantum physics. Chaos, Solitons and Fractals 1996;7(5) [special issue]. Halayka S. Associating disconnectedness with electromagnetism phenomena. Chaos, Solitons and Fractals 2009;42:469–71. Hermann GT. On topology as applied to image analysis. Comp Vision Graph Image Process 1990;52:409–15. Khalimsky ED, Kopperman R, Meyer PR. Computer graphics and connected topologies an finite ordered sets. Topol Appl 1990;36:1–17. Kong TY, Koppernan R, Meyer PR. A topological approach to digital topology. Am Math Monthly 1991;98:901–17. Kuratowski K. Topology, Vol-I. New York: Academic Press; 1966.
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[13] Moore ELF, Peters TJ, Computational topology for geometric design and molecular design. In: Ferguson DR, Peters TJ, editors. Mathematics in industrychallenges and frontiers, 2003, SIAM; 2005. [14] Stone MH. Application of the theory of Boolean rings to general topology. TAMS 1937;41:375–81. [15] Willard S. General topology. Reading, Mass: Addision-Wesley; 1970.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
A note on disconnectedness C.K. Basu a,*, S.S. Mandal b a b
Department of Mathematics, West Bengal State University, Barasat 700126, North 24 Pgs., West Bengal, India Department of Mathematics, Krishnagar Women’s College, Krishnagar 741101, Nadia, West Bengal, India
a r t i c l e
i n f o
Article history: Accepted 28 April 2009
a b s t r a c t The purpose of this note is to characterize various disconnectedness in terms of (star) refinement of covers. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Relation of topology and physics is gaining momentum and is acting as a focus of attention by many mathematicians as well as physicists. El Naschie et al. [1–7] have pointed out that topology plays a significant role in quantum physics, high energy physics and super string theory; in addition, in [3], by the help of set theory, El Naschie derived quantum gravity. Hermann [9], Khalimsky et al. [10], Kong and Koppermann [11] have applied topology in computer science and digital topology. Moor and Peters [13] have investigated computational topology for geometric design. In topology, disconnectedness, as considered as a fundamental concept has been researched by many. M.H. Stone’s contribution [14] towards the topological representation of Boolean algebra in terms of disconnectedness is quite remarkable. Various types of disconnectedness have the potential to play important roles specially, in studying fractals and dimension theory. S. Halayka, in his recent paper [8] used disconnectedness to simplify an analysis of several well-known physical phenomena related to electromagnetism. Because of enormous use and potentiality of various types of disconnectedness, our task in this paper is to offer new characterizations of various types of disconnectedness (e.g. total disconnectedness, zero-dimensionality, extremal disconnectedness, strong zero-dimensionality) in terms of (star) refinement of covers with the belief that such investigations may have possible applications in a broader sense in fractals and quantum physics. By X, we shall mean a topological space without any separation axioms. For a cover U of X and for a subset A of X, the star of A with respect to U is the set StðA; UÞ ¼ [fU 2 U : U \ A – ;g; and for two covers U and V of X, we call U star refines V or H U is a star refinement of V, written U < V, if for each U 2 U; 9V 2 V such that StðU; UÞ V. U is called a refinement of V written as U < V if for each U 2 U, there exists V 2 V such that U V. H A normal sequence of covers is a sequence of open covers U1 ; U2 ; . . . of X such that Unþ1 < Un , for n ¼ 1; 2; . . .; and a normal cover is cover which is U1 in some normal sequence of covers. A family V of covers of X is a normal family iff every cover in V has a star refinement in V [Section 36.10, p. 247 [15]]. A collection l0 of covers of a space X is a base for some uniformity on X iff it satisfies the condition that for U1 ; U2 2 l0 H H there is some U3 2 l0 such that U3 < U1 and U3 < U2 [Section 36.3, p. 245 [15]]. It is well known that if l0 is a base for a covering uniformity l on X, then fStðx; UÞ : U 2 l0 g is a nbd. base at x 2 X in the uniform topology [Section 36.6, p. 246 [15]]. Also if X is any uniformizable topological space then there is a finest uniformity on X, compatible with the topology of X, called the fine uniformity on X, denoted by lF , having a base of all normally open covers of X; such a space is known as the fine space. Further, a T 1 space is paracompact iff every open cover has an open star refinement [Section 20.14, p. 149
* Corresponding author. E-mail address: [email protected] (C.K. Basu). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.04.056
C.K. Basu, S.S. Mandal / Chaos, Solitons and Fractals 42 (2009) 3242–3246
3243
[15]], where a space X is called paracompact iff every open cover of X has an open locally finite refinement, which is also a cover of X. A topology r on X is called a subtopology of a space ðX; sÞ if r s. 2. Characterizations of various disconnectedness H
Theorem 2.1. A topological space ðX; sÞ is disconnected iff it has an open cover U consisting of proper subsets of X such that U < U. Proof. Let ðX; sÞ be disconnected. Then there exist U; V 2 s such that U – ;; V – /; U [ V ¼ X and U \ V ¼ ;. So, U ¼ fU; Vg is H UÞ ¼ V V. an open cover of X consisting of proper subsets of X. Here U < U, because StðU; UÞ ¼ U U and StðV; H Conversely, let U ¼ fU b : b 2 Ig be an open cover of X consisting of proper subsets of X, i.e. U b – X; 8b 2 I and let U < U. Because of the result that a cover V of a space Y star refines itself iff for some partition Q of Y; V < Q < V [exercise 36 A (3), p. 249 [15]], we have a partition P of X such that U < P < U. As P < U and U is consisting of proper subsets, P is not a trivial partition, i.e. P – fXg. Let P ¼ fAa : a 2 Kg, where Aa – /; Aa – X; 8a 2 K. We shall show that each Aa is open. Let Aa 2 P and x 2 Aa . Then as U is a cover of X there exists a U bx 2 U such that x 2 U bx . Now as U < P; U bx Aa0 , for some Aa0 2 P. We shall show that a ¼ a0 . If not, then x 2 U bx Aa0 and x 2 Aa . So, Aa ; Aa0 are distinct members of the partition P but Aa0 \ Aa – / – a contradiction. Hence a ¼ a0 . Thus Aa ¼ Aa0 and hence x 2 U bx Aa , and it is true for all x 2 Aa . Hence Aa ¼ [x2Aa U bx is open. Now as P is a non trivial partition, so P ¼ fAa : a 2 Kg contains at least two non empty members say Aa1 ; Aa2 . Now let K1 K be such that a1 2 K1 and a2 2 K K1 ; then U ¼ [a2K1 Aa ; V ¼ [a2KK1 Aa are open sets with U – /; V – /, U [ V ¼ X and U \ V ¼ /. So, ðX; sÞ is disconnected. h Theorem 2.2. If ðY; sÞ is a metrizable Lindeloff space then the following are equivalent: (i) ðY; sÞ is zero-dimensional. (ii) Every open cover U of Y is a normally open cover with a normal sequence of open covers U ¼ U1 ; U2 , . . . such that Ui consists of clopen subsets of ðY; sÞ, for at least one
ði P 2Þ
ðÞ
(iii) Every open cover U of ðY; sÞ has a refinement V consisting of clopen sets only. Proof. We shall first prove that ðiÞ () ðiiÞ. Let ðiÞ holds, i.e. ðY; sÞ is zero-dimensional. Let U ¼ fU a : a 2 Kg be an open cover of Y. As ðY; sÞ is Lindeloff, U has a countable subcover U0 ¼ fU i : i ¼ 1; 2; . . .g, finite or infinite. So,
U0 < U
ðIÞ 1
Now by the reduction theorem [Theorem 1 , p. 279, [12]], there exists a sequence of disjoint clopen sets H1 ; H2 ; . . . . . ., which is finite (when U0 is finite) or infinite (when U0 is infinite) such that [Hi ¼ Y and Hi U i ; 8i. Let V ¼ fHi : i ¼ 1; 2; . . . . . .g. Clearly V is a partition of Y and so H
V
ðIIÞ
Now for each Hi 2 V; StðHi ; VÞ ¼ Hi U i 2 U0 and so H
V < U0
ðIIIÞ
Then (I) and (III) imply H
V
ðIVÞ
Now we construct the sequence U1 ; U2 ,. . . of open covers of Y by taking U1 ¼ U; Ui ¼ V; 8i P 2. (II) and (IV) imply that this is a normal sequence, where Ui ¼ V consists of disjoint clopen sets for all i P 2. So ðiÞ ) ðiiÞ. Conversely, let ðiiÞ holds. Let l0 be the collection of all normally open covers in ðY; sÞ. Now as ðY; sÞ is completely regular, so is uniformizable. Let l be the fine uniformity (having a base l0 ) on Y compatible with s. So, bx ¼ fStðx; UÞ : U 2 l0 g is a nbd. base at each x 2 ðY; sÞ. Now, let G 2 s. Then for x 2 G; Stðx; UÞ G for some U 2 l0 . So, by ðiiÞ there exists a normal sequence of open covers U1 ; U2 ,. . .. . .. such that U ¼ U1 and for at least one iðP 2Þ; Ui consists of clopen sets of ðY; sÞ. If the value of i be n (say), then H x 2 U for some U 2 Un . Now as Un < U1 ¼ U, so x 2 StðU; Un Þ V for some V 2 U and also V Stðx; UÞ. So, x 2 U StðU; Un Þ V Stðx; UÞ G. Therefore x 2 U G and U is a clopen subset of ðY; sÞ. Hence ðY; sÞ is zero-dimensional, i.e. ðiiÞ ) ðiÞ is established. Next, we shall prove that ðiiiÞ () ðiÞ.
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First let ðiÞ holds and U ¼ fU a : a 2 Kg be an open cover of Y. Then by the result that in a zero-dimensional metric space every open set is the union of a sequence of disjoint clopen sets (Corollary 1a, p. 277, [12]), every open set U a 2 U, can be expressed as the disjoint union of clopen sets. So ðiiiÞ follows trivially by taking an cover U0 consisting of such disjoint clopen sets for each U a whose union is U a but not taking U a (which clearly shows U0 < U). So, ðiÞ ) ðiiiÞ holds. Conversely, let ðiiiÞ holds. Let l be the fine uniformity having base l0 of all normally open covers of ðY; sÞ compatible with the topology s of Y. Consequently, bx ¼ fStðx; UÞ : U 2 l0 g forms a local base at x in ðY; sÞ, for each x 2 Y. Let G 2 s and x 2 G. Then x 2 Stðx; UÞ G for some U 2 l0 . Now by (iii), U has a refinement U0 , consisting of clopen sets of ðY; sÞ and hence we have x 2 Stðx; U0 Þ Stðx; UÞ G. As U0 is a cover of X, so x 2 U for some U 2 U0 and hence x 2 U Stðx; U0 Þ G, where U is clopen. Therefore ðY; sÞ is zero-dimensional, i.e. ðiiiÞ ) ðiÞ. h Definition 2.3. A space X is called strongly zero-dimensional iff for every pair of disjoint zero sets of the space, there is a clopen set containing one zero set and missing the other. Theorem 2.4. For a T 2 space X, if bX has no proper non trivial pseudometrizable subtopology then the following are equivalent: (i) X is totally disconnected, (ii) X is strongly zero-dimensional, (iii) bX satisfies property (*). Proof. ðbX; sÞ is obviously Lindeloff.Also ðbX; sÞ is metrizable.Indeed, if possible, let ðbX; sÞ be non-metrizable.Since in a paracompact T 2 space, every open cover has an open star refinement, so every open cover is a normally open cover.As ðbX; sÞ is T 2 , for all x – y 2 bX, there exist disjoint open sets Ox and Oy containing x and y, respectively.Clearly Ox and Oy are proper open subsets of X, for all x – y 2 bX.Fix y0 ; x0 2 bX and take U ¼ fOx ; xð – x0 Þ 2 bX such that Ox0 \ Ox ¼ ;; x0 2 Ox0 ; x 2 Ox ; Ox0 ; Ox 2 sg [ fOx0 2 s : Ox0 \ Oy0 ¼ ;; x0 2 Ox0 ; y0 2 Oy0 ; Oy0 2 sg; then U is an s-open cover of bX.So, U is a normally open cover.Let U ¼ U1 ; U2 ,. . ..be the corresponding normal sequence of open covers.Then l0 ¼ fU1 ; U2 ; . . .g, is a base for some uniformity l on bX.Let sl be the corresponding uniform topology induced by the uniformity l on bX.Then sl is pseudometrizable as l has a countable base l0 .The family bx ¼ fStðx; Ui Þ : i ¼ 1; 2; 3; . . . :g is a nbd.base H at x 2 bX in ðbX; sl Þ.Here it is to be noted that all members of all U;i s are proper open subsets of bX [as Uk < Uk1 and U1 conH sists of proper open subsets of bX, so Uk < U1 , 8k > 1 and hence U 2 Uk ) 9V 2 U1 such thatStðU; Uk Þ V – bX]. Now for Stðx; Ui Þ 2 bx ; x 2 X; Stðx; Ui Þ # StðU; Ui Þ for some U 2 Ui such that x 2 U. Also StðU; Ui Þ V for some V 2 Ui1 and as V – bX, then Stðx; Ui Þ V – bX. So, bx contains many proper subsets of bX. Therefore sl contains proper nonempty open sets, and hence sl is a non-trivial topology on bX. Now if l01 be the collection of all open covers of bX then l01 is the base for the fine uniformity lF on bX [as ðbX; sÞ is paracompact T 2 , so every open cover of ðbX; sÞ is normally open cover, and as the family of all normally open covers of the uniformizable space ðbX; sÞ, forms a base for the fine uniformity lF on bX which induces the topology s ] which gives the s as the uniform topology. Now l0 l01 ) sl # s ¼ slF . As s is not pseudometrizable and sl is pseudometrizable, topology so sl – s – a contradiction. Since X is strongly zero-dimensional iff bX is zero-dimensional, then theorem follows by the direct application of Theorem 2.2. h Theorem 2.5. If every open cover of a T 2 space ðX; sÞ has a refinement which star refines itself then ðX; sÞ is totally disconnected. Proof. Let Y be a subset of X consisting at least two points. Let x 2 Y and keep it fixed. Then for each y 2 Y with x – y, there exist open sets U x (y) and U y ðxÞ containing x and y, respectively, such that U x ðyÞ \ U y ðxÞ ¼ /. So, U x ðyÞ; U y ðxÞ are proper open sets in X and hence U x ðyÞ \ Y and U y ðxÞ \ Y are proper open subsets of Y for all yð – xÞ 2 Y. Now we shall consider the points of X Y. Case I: Suppose cardðX YÞ > 1. For the fixed point x 2 Y and for each y0 2 X Y, there exist open sets U x ðy0 Þ and U y0 ðxÞ containing x and y0 , respectively, such that U x ðy0 Þ \ U y0 ðxÞ ¼ ;. So, fU y ðxÞ : yð – xÞ 2 Yg [ fU x ðyÞ: for some yð – xÞ 2 Yg [ fU y0 ðxÞ : y0 2 X Yg constitutes an open cover of X consisting of proper open subsets of X and intersection of members of this cover with Y are also proper open subsets of Y. Case II: Suppose cardðX YÞ ¼ 1, i.e. X Y ¼ fy0 g. Since ðX; sÞ is T 2 , then as before the cover fU y ðxÞ : yð – xÞ 2 Yg [ fU x ðyÞ: for some yð – xÞ 2 Yg [ fU y0 ðxÞg is consisting of proper open subsets of X; and the intersection of members of this cover with Y are also proper open subsets of Y. So, for each case we obtain an open cover U ¼ fU a : a 2 Kg of X such that / – U a – X and U a \ Y are proper subsets of Y; 8a 2 K. Now, by the given condition U has a open refinement V ¼ fHb : b 2 K0 g such that V < V. Then 0 0 VY ¼ fHb \ Y : b 2 K g is an open cover of Y, where Hb \ Y are proper open subsets in Y for all b 2 K [as Hb U a , for some U a 2 U and U a \ Y – Y hence so is Hb \ Y]. H Now we shall show that VY < VY . Indeed, we always have StðHb \ Y; VY Þ StðHb ; VÞ. But StðHb ; VÞ Hb0 for some Hb0 2 V as V < V. So, StðHb \ Y; VY Þ Hb0 . Again, StðHb \ Y; VY Þ Y. Therefore, StðHb \ Y; VY Þ Hb0 \ Y, where Hb0 \ Y 2 VY . Hence VY < VY . So, Y has an open cover consisting of proper open subsets of Y which star refines itself. Therefore, by Theorem 2.1, Y is disconnected. Hence X is totally disconnected. h
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Problem 2.6. Whether the sufficient condition of Theorem 2.5 is actually a characterization of totally disconnectedness is still unknown to the authors. Theorem 2.7. A space X is extremally disconnected if and only if for every open set U, the cover fU; X Ug of X has an open refinement. Proof. Let for every open set U, the cover U ¼ fU; X Ug of X has an open refinement, say V. Now for each x 2 U, there exists a V x 2 V such that x 2 V x (as V is a cover X). Since V < U, there exists a W x 2 U such that x 2 V x W x . Hence W x is either U or X U. But W x cannot be X U as x R X U. So, W x ¼ U. Thus, x 2 V x U and this holds for each x 2 U. Therefore U is open and hence X is extremally disconnected. Conversely, let X be extremally disconnected. Then for every open set U, the open cover fU; X Ug itself satisfies the condition that required. h Theorem 2.8. Let ðX; sÞ be a separable zero-dimensional space in which every F r set is closed, then ðX; sÞ is extremally disconnected. Proof. Let U be any open set. Now as open subspace of a separable space is separable, X U is separable. Let A ¼ fx1 ; x2 ; . . . ; xi ; . . .g be the countable dense subset of X U. So, A ¼ X U in ðX U; sXU Þ. Now as, X is zero-dimensional so for xi 2 X U, there exists clopen set in ðX; sÞ say, U xi such that xi 2 U xi X U. Now 1 1 [xi 2A fxi g [1 i¼1 U xi X U, i.e. A [i¼1 U xi X U. Now, sXU – closure ðAÞ sXU – closure ð[i¼1 U xi Þ. Therefore 1 1 1 U Þ s – closure ð[ U Þ [ s – closure ðU Þ ¼ [ U X U. Hence X U X U sXU – closure ([1 x x x x i i i i i¼1 i¼1 i¼1 i¼1 1 [1 i¼1 U xi X U. So X U ¼ [i¼1 U xi . As every F r set is closed, so X U is a closed set. Therefore, U is open. Hence X is extremally disconnected. h Corollary 2.9. If in a separable zero-dimensional space every F r set is closed then, every open set is closed. Corollary 2.10. Let ðX; sÞ be a regular separable space in which every F r set is closed, then X is zero dimensional if and only if X is extremally disconnected. Proof. As an extremally disconnected regular space is zero dimensional, hence the result follows.
h
3. Conclusion As far as application of topology in various other branches of mathematics and physics is concerned, the study of disconnectedness in topology plays a predominant role. For example, M.H. Stone [14] asserts that every Boolean algebra has the topological representation as the Boolean algebra COðXÞ of clopen sets for some totally disconnected space X (while the Boolean algebra COðXÞ of clopen sets for a connected space X is not so interesting as it consists only of ; and X). It has been observed that there is a close resemblance of fractals and dimension theory with various types of disconnectedness like total disconnectedness, zero-dimensionality and extremal disconnectedness. Further, analysis of several well-known physical phenomena related to electromagnetism have also been simplified in recent times (2008) by the help of the concept of disconnectedness [8]. We believe, more and more investigations in the arena of disconnectedness, specially, offering new characterizations of such concept, may have possible applications in physics like fractals and quantum mechanics as well as in dimension theory. Acknowledgement The authors are grateful to the learned referee for his valuable comments which improved the paper a great extent. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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