International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 493290 | 11 pages | https://doi.org/10.1155/2011/493290

Homeomorphisms of Compact Sets in Certain Hausdorff Spaces

Academic Editor: Vladimir Mityushev
Received05 Jan 2011
Accepted18 Apr 2011
Published11 Aug 2011

Abstract

We construct a class of Hausdorff spaces (compact and noncompact) with the property that nonempty compact subsets of these spaces that have the same cardinality are homeomorphic. Also, it is shown that these spaces contain compact subsets that are infinite.

1. Introduction

In this paper, we construct a class of Hausdorff spaces with the property that nonempty compact subsets of these spaces that have the same cardinality are homeomorphic (Theorem 3.7). Conditions are given for these spaces to be compact (Corollary 2.10). Also, it is shown that these spaces contain compact subsets that are infinite (Corollary 2.10).

This paper uses the Zermelo-Fraenkel axioms of set theory with the axiom of choice (see [13]). We let denote the finite ordinals (i.e., the natural numbers) and denotes the counting numbers (i.e., ). Also, for a given set , we denote the collection of all subsets of by , and we denote the cardinality of by . In other words, is the smallest ordinal number for which a bijection of onto exists.

In this paper, we will only consider compact topologies that are Hausdorff. A topology on a set is compact if and only if and imply for some and . Therefore, compact topologies need not be Hausdorff.

2. A Class of Hausdorff Spaces

Let , , and be sets such that is infinite and the collection is pairwise disjoint. For example, let , , and . Unless otherwise stated, we let

Recall that for set and , we have

Definition 2.1. Let be an infinite set. Define We call   the Fréchet filter on .

Note that being infinite implies that is a filter (see [4, Definition  3.1, page 48]).

Definition 2.2. Consider the collection defined as follows:

Proposition 2.3. The collection generates a Hausdorff topology on .

Proof. Clearly, is a basis for a topology (see [5, Section  13]).
Let such that . If , then , , , and If , then either or . Assume that , and let . Since , and , we have Assume that . Note that . Also, , which implies and Observe that,
We infer that is Hausdorff.

Proposition 2.4. If , then is compact in if and only if is a finite set.

Proof. Note that finite sets are compact in any topological space. So, assume that is an infinite, and let which implies Let be a nonempty, finite subcollection of . Therefore, there exists , for some , such that which implies If , then we would have , contradicting being an infinite set. Consequently, infinite subsets of are not compact in the topological space .

Corollary 2.5. The set is not compact in .

Corollary 2.6. The set is compact in if and only if is finite.

Proposition 2.7. Let . The set is compact in if and only if is a finite set.

Proof. The topology on is generated by (see Proposition 2.3).
Assume that is an infinite set. Let , and let . Hence, Let , and let . Note that , , Suppose that is a finite subcollection such that It can be assumed, without loss of generality, that and for such that . So, by definition of , there exists such that for . Note that , which implies So, expressions (2.16) and (2.17) would imply contradicting being an infinite set. We infer that is not compact in .
Conversely, assume that is a finite set. Let such that Hence, there exists such that . Since for , there exists such that Assume that for some [we define ]. Observe that, Since is finite, we can assume that for some (we define ). Hence, there exists and such that for and (again, we define ). Consequently, from expressions (2.19), (2.20), (2.21), (2.22), and (2.23), we have We infer is compact in .

Corollary 2.8. The set is compact in .

Proof. Note that . Also, implies that is finite. Therefore, is compact in by Proposition 2.7.

Corollary 2.9. The set is compact in if and only if is finite.

Proof. Note that and . Therefore, the corollary follows from Proposition 2.7.

Corollary 2.10. The topological space is compact if and only if is finite.

Proof. Observe that and . Therefore, the corollary follows from Proposition 2.7.

Proposition 2.11. If is an infinite, compact set (in ), then for some infinite set such that is a finite set.

Proof. If , then we would have , contradicting Proposition 2.4, since is an infinite, compact set. Hence, and is a finite set by Proposition 2.7, since is compact. Also, note that being an infinite set implies is an infinite set. Let .

Notation 2.12. Let be a nonempty set, and let be a Hausdorff topology on . For , we let denote the filter of -neighborhoods of ; that is, if and only if for some .

Recall that for and , is an accumulation point of if and only if for , there exists such that and .

Remark 2.13. If and , then is not an accumulation point of .

Indeed, implies . So, if such that , then .

Remark 2.14. The element is an accumulation point of in .

Indeed, let . Hence, for some . Since is infinite and , we have that . Let . Hence, and .

Consequently, is not closed [in ], which implies is not open; that is, . In fact, is the only element of such that .

3. Homeomorphisms of Compact Sets in

The following proposition is obvious and is stated without proof.

Proposition 3.1. Let be an infinite cardinal. If and are sets such that and then there exists a map such that and is a bijection.

Remark 3.2. If and is a finite set, then .

Lemma 3.3. Let and be nonempty subsets of such that and are finite sets. Let be a bijection such that . If , then for some .

Proof. Let . Note that Also, the sets and are finite.
Let Consequently, and is a finite set.
Let Therefore, by Remark 3.2. Note that So, Observe that, Therefore, Consequently,

Proposition 3.4. Let be the Hausdorff topology on generated by . Let and be infinite subsets of such that and are finite sets. If and have the same cardinality (i.e., ), then any bijection of onto that has as a fixed point is a homeomorphism.

Proof. Let be a bijection with . Note that the existence of such a bijection is established by Proposition 3.1. Let denote the inverse map of .
Let be the topology on induced by , and let be the topology induced on by .
Let . We will show that .
Let . Either or .

Case 1. Assume that . Hence, . So, there exists such that . Since for any , we have that for some . Therefore, which implies which implies By Lemma 3.3, there exists such that Let . Note that . Hence, and

Case 2. Assume that . Consequently, , which implies , which implies (since ). Therefore, implies
From expression (3.15) in Case 1 and expression (3.16) in Case 2, we infer
Let . We will show that .
Let . Either or .

Case 3. Assume that . Hence, . So, there exists such that . Since for any , we have that for some . Therefore, which implies which implies By Lemma 3.3, there exists such that Let . Note that . Hence, and

Case 4. Assume that . Consequently, , which implies , which implies (since ). Therefore, implies
From expression (3.22) in Case 3 and expression (3.23) in Case 4, we infer that
Consequently, from expression (3.17) and (3.24), we infer that is a homeomorphism of onto .

Proposition 3.5. Let be the Hausdorff topology on generated by . Let be a compact set. If , then is homeomorphic to .

Proof. Note that implies , which implies in an infinite set ( is an infinite set), which implies for some such that is a finite set by Proposition 2.11. Consequently, and imply . So, which implies (see [2, Corollary  2.3, page 162]). Therefore, is homeomorphic to by Proposition 3.4.

Remark 3.6. Let be a nonempty set and let be a Hausdorff topology on . If is a nonempty finite set, then induces the discrete topology on .

Theorem 3.7. Let be the Hausdorff topology on generated by , and let and be nonempty compact subsets of . If and have the same cardinality (i.e., ), then there exists a homeomorphism of onto .

Proof. Let and be nonempty compact subsets of such that . If is a finite set, then is a finite set. Hence, induces the discrete topology on and (see Remark 3.6). Consequently, any bijection of onto is a homeomorphism.
Assume that is an infinite set. Hence, is an infinite set. So, and such that , , is a finite set and is a finite set by Proposition 2.11. Observe that and being infinite sets imply and are infinite subsets of such that . Therefore, there exists a homeomorphism of onto by Proposition 3.4.

4. Examples

Example 4.1. Let , , and . Let and let Consider the Hausdorff space , where the topology is generated by . Observe that is compact (Corollary 2.10, since ) and is not compact by Corollary 2.5. If is an infinite compact set, then therefore, is homeomorphic to (Theorem 3.7). In other words, all infinite, compact subsets of are homeomorphic. Note that topology on is induced by the standard euclidean topology on .

Example 4.2. Let be a set such that . Let be a bijection. For , we will denote by , that is, . Therefore, , where and imply . Let and let . Note that the maps and are bijections of onto and onto , respectively. Also, , and ; consequently, . We can write (see Definition 2.2) as follows. The collection generates a Hausdorff topology on (see Proposition 2.3). Note that is not compact (Corollary 2.10), sets and are not compact (Corollaries 2.6 and 2.5, resp.), is not compact (Corollary 2.9), and is compact (Corollary 2.8). Also, if is an infinite compact set, then therefore, is homeomorphic to (Theorem 3.7). In other words, all infinite, compact subsets of are homeomorphic.

Example 4.3. Let be an infinite cardinal and consider the collection of infinite cardinals defined as follows. Let and for , let . Also, we denote . Hence, is the cardinal number, that is, the supremum of and for each (see the Alephs section in [3], page 29). For , define Observe that , for each , and the collection is pairwise disjoint. Let Note that the collection is pairwise disjoint. Let and consider the noncompact, Hausdorff space , where topology is generated by (see Definition 2.2 and Corollary 2.10). For , define Observe, for by Proposition 2.7 (since ) and the fact that . Also, the collection is pairwise disjoint and Theorem 3.7 implies that all of the sets in are homeomorphic to each other. Since for each , we have (see [2, Corollary  2.3, page 162]), which implies . Consequently, (see Corollary 2.8), which implies is homeomorphic to . If we let denote the set of all finite subsets of , then (see Proposition 2.7). Consequently, the size of can affect the cardinality of for (e.g., let be a Mahlo cardinal ([3, Chapter 8, page 95])). If the generalized continuum hypothesis is assumed, then the collection is a partition of the collection of all infinite, compact subsets of .

Acknowledgment

This work was partially supported by NSA Grant no. H98230-09-1-0125.

References

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Copyright © 2011 Arthur D. Grainger. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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