Homeomorphisms of Compact Sets in Certain Hausdorff Spaces

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 [1–3]). We let 𝜔 denote the finite ordinals (i.e., the natural numbers) and ℕ denotes the counting numbers (i.e., ℕ=𝜔⧵{0}). Also, for a given set 𝑋, we denote the collection of all subsets of 𝑋 by 𝒫(𝑋), and we denote the cardinality of 𝑋 by |𝑋|. In other words, |𝑋| is the smallest ordinal number for which a bijection of |𝑋| onto 𝑋 exists.

In this paper, we will only consider compact topologies that are Hausdorff. A topology 𝜏 on a set 𝑋 is compact if and only if 𝒜⊆𝜏 and 𝑋⊆⋃𝒜 imply 𝑋⊆⋃^𝑛_𝑗=1𝑈_𝑗 for some 𝑛∈ℕ and {𝑈₁,…,𝑈_𝑛}⊆𝒜. Therefore, compact topologies need not be Hausdorff.

2. A Class of Hausdorff Spaces

Let 𝑉, 𝑊, and 𝑥₀ be sets such that 𝑊 is infinite and the collection 𝑉,𝑊,𝑥₀ (2.1) is pairwise disjoint. For example, let 𝑉={(𝜈,0)∣𝜈∈𝜔}, 𝑊={(𝜇,1)∣𝜇∈2^𝜔}, and 𝑥₀=2^2𝜔. Unless otherwise stated, we let 𝑋=𝑥₀∪𝑉∪𝑊. (2.2)

Recall that for set 𝑌 and 𝐺⊆𝑌, we have 𝑌⧵𝐺=𝑦∈𝑌∣𝑦∉𝐺. (2.3)

Definition 2.1. Let 𝐴 be an infinite set. Define ℱ𝑟(𝐴)={𝐹∈𝒫(𝐴)∣|𝐴⧵𝐹|∈𝜔}. (2.4) We call ℱ𝑟(𝐴)  the Fréchet filter on 𝐴.

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

Definition 2.2. Consider the collection ℬ₁⊆𝒫(𝑋) defined as follows: ℬ₁=𝒫𝑋⧵𝑥₀∪𝐹∪𝑥₀∣𝐹∈ℱ𝑟(𝑊). (2.5)

Proposition 2.3. The collection ℬ₁ generates a Hausdorff topology 𝜏 on 𝑋.

Proof. Clearly, ℬ₁ is a basis for a topology 𝜏 (see [5, Section  13]).
Let 𝑣,𝑤∈𝑋 such that 𝑣≠𝑤. If 𝑣,𝑤∈𝑋⧵{𝑥₀}, then {𝑣},{𝑤}∈ℬ₁, 𝑣∈{𝑣}, 𝑤∈{𝑤}, and {𝑣}∩{𝑤}=∅. (2.6) If 𝑣=𝑥₀, then either 𝑤∈𝑉 or 𝑤∈𝑊. Assume that 𝑤∈𝑉, and let 𝐹∈ℱ𝑟(𝑊). Since 𝑉∩𝑊=∅, {𝑤}∈𝒫(𝑉)⊆ℬ₁ and 𝐹∪{𝑥₀}∈ℬ₁, we have {𝑤}∩𝐹∪𝑥₀=∅. (2.7) Assume that 𝑤∈𝑊. Note that {𝑤}∈ℬ₁. Also, 𝑊⧵{𝑤}∈ℱ𝑟(𝑊), which implies [𝑊⧵{𝑤}]∪{𝑥₀}∈ℬ₁ and 𝑣=𝑥₀∈[𝑊⧵{𝑤}]∪𝑥₀. (2.8) Observe that, {𝑤}∩[𝑊⧵{𝑤}]∪𝑥₀=∅. (2.9)
We infer that 𝜏 is Hausdorff.

Proposition 2.4. If 𝐴⊆𝑋⧵{𝑥₀}, then 𝐴 is compact in (𝑋,𝜏) if and only if 𝐴 is a finite set.

Proof. Note that finite sets are compact in any topological space. So, assume that 𝐴 is an infinite, and let 𝒜={{𝑎}∣𝑎∈𝐴}⊆𝒫𝑋⧵𝑥₀⊆𝜏, (2.10) which implies 𝐴=𝒜. (2.11) Let 𝒰⊆𝒜 be a nonempty, finite subcollection of 𝒜. Therefore, there exists {𝑎_𝑖}^𝑚_𝑖=1⊆𝐴, for some 𝑚∈ℕ, such that 𝒰=𝑎_𝑖^𝑚_𝑖=1, (2.12) which implies 𝒰=𝑎_𝑖^𝑚_𝑖=1. (2.13) If 𝐴⊆⋃𝒰, then we would have 𝐴⊆{𝑎_𝑖}^𝑚_𝑖=1, contradicting 𝐴 being an infinite set. Consequently, infinite subsets of 𝑋⧵{𝑥₀} are not compact in the topological space (𝑋,𝜏).

Corollary 2.5. The set 𝑊 is not compact in (𝑋,𝜏).

Corollary 2.6. The set 𝑉 is compact in (𝑋,𝜏) if and only if 𝑉 is finite.

Proposition 2.7. Let 𝐴⊆𝑋⧵{𝑥₀}. The set 𝐴∪{𝑥₀} is compact in (𝑋,𝜏) if and only if 𝐴∩𝑉 is a finite set.

Proof. The topology 𝜏 on 𝑋 is generated by ℬ₁ (see Proposition 2.3).
Assume that 𝐴∩𝑉 is an infinite set. Let 𝐹∈ℱ𝑟(𝑊), and let 𝑄=𝐹∪{𝑥₀}. Hence, 𝑥₀∈𝑄,𝑄∈𝜏. (2.14) Let 𝒜_𝐴={{𝑎}∣𝑎∈𝐴}, and let 𝒜={𝑄}∪𝒜_𝐴. Note that 𝒜_𝐴⊆𝜏, 𝒜⊆𝜏, 𝐴=𝒜_𝐴,𝐴∪𝑥₀⊆𝒜. (2.15) Suppose that 𝒰⊆𝒜_𝐴 is a finite subcollection such that 𝐴∪𝑥₀⊆𝒰. (2.16) It can be assumed, without loss of generality, that 𝑄∈𝒰 and 𝒰={𝑈_𝑖}^𝑚_𝑖=0 for 𝑚+1=|𝒰|∈ℕ such that 𝑈₀=𝑄. So, by definition of 𝒜, there exists {𝑎_𝑖}^𝑚_𝑖=1⊆𝐴 such that 𝑈_𝑖={𝑎_𝑖} for 𝑖=1,...,𝑚. Note that 𝑄=𝐹∪{𝑥₀}⊆𝑊, which implies 𝑄∩(𝐴∩𝑉)=𝐴∩(𝑄∩𝑉)=∅. (2.17) So, expressions (2.16) and (2.17) would imply 𝐴∩𝑉⊆^𝑚_𝑖=1𝑈_𝑖=𝑎₁,…,𝑎_𝑚, (2.18) contradicting 𝐴∩𝑉 being an infinite set. We infer that 𝐴∪{𝑥₀} is not compact in (𝑋,𝜏).
Conversely, assume that 𝐴∩𝑉 is a finite set. Let 𝒜⊆𝜏 such that 𝐴∪𝑥₀⊆𝒜. (2.19) Hence, there exists 𝑈∈𝒜 such that 𝑥₀∈𝑈. Since 𝑥₀∉𝐸 for 𝐸∈𝒫(𝑋⧵{𝑥₀}), there exists 𝐹∈ℱ𝑟(𝑊) such that 𝑥₀∈𝐹∪𝑥₀⊆𝑈. (2.20) Assume that 𝐴∩𝑉={𝑣_𝑖}^𝑚_𝑖=1 for some 𝑚∈𝜔 [we define {𝑣_𝑖}⁰_𝑖=1=∅]. Observe that, 𝐴=𝐴∩[𝑉∪𝑊]=([𝐴∩𝐹]∪[𝐴∩𝑊⧵𝐹])∪𝑣_𝑖^𝑚_𝑖=1. (2.21) Since 𝑊⧵𝐹 is finite, we can assume that 𝐴∩𝑊⧵𝐹=𝑎_𝑞^𝑘_𝑞=1, (2.22) for some 𝑘∈𝜔 (we define {𝑎_𝑞}⁰_𝑞=1=∅). Hence, there exists {𝑈_𝑖}^𝑚_𝑖=1⊆𝒜 and {𝑈_𝑞}^𝑘_𝑞=1⊆𝒜 such that 𝑣_𝑖∈𝑈_𝑖,𝑎_𝑞∈𝑈_𝑞, (2.23) for 𝑖=1,…,𝑚 and 𝑞=1,...,𝑘 (again, we define ⋃⁰_𝑖=1𝑈_𝑖=⋃⁰_𝑞=1𝑈_𝑞=∅). Consequently, from expressions (2.19), (2.20), (2.21), (2.22), and (2.23), we have 𝐴∪𝑥₀⊆𝑈∪^𝑚_𝑖=1𝑈_𝑖∪^𝑘_𝑞=1𝑈_𝑞. (2.24) We infer 𝐴∪{𝑥₀} is compact in (𝑋,𝜏).

Corollary 2.8. The set 𝑊∪{𝑥₀} is compact in (𝑋,𝜏).

Proof. Note that 𝑊⊆𝑋⧵{𝑥₀}. Also, 𝑊∩𝑉=∅ implies that 𝑊∩𝑉 is finite. Therefore, 𝑊∪{𝑥₀} is compact in (𝑋,𝜏) by Proposition 2.7.

Corollary 2.9. The set 𝑉∪{𝑥₀} is compact in (𝑋,𝜏) if and only if 𝑉 is finite.

Proof. Note that 𝑉⊆𝑋⧵{𝑥₀} and 𝑉=𝑉∩𝑉. Therefore, the corollary follows from Proposition 2.7.

Corollary 2.10. The topological space (𝑋,𝜏) is compact if and only if 𝑉 is finite.

Proof. Observe that 𝑋=(𝑋⧵{𝑥₀})∪{𝑥₀} and 𝑉=(𝑋⧵{𝑥₀})∩𝑉. Therefore, the corollary follows from Proposition 2.7.

Proposition 2.11. If 𝐾⊆𝑋 is an infinite, compact set (in (𝑋,𝜏)), then 𝐾=𝐴∪{𝑥₀} for some infinite set 𝐴⊆𝑋⧵{𝑥₀} such that 𝐴∩𝑉 is a finite set.

Proof. If 𝑥₀∉𝐾, then we would have 𝐾⊆𝑋⧵{𝑥₀}, contradicting Proposition 2.4, since 𝐾 is an infinite, compact set. Hence, 𝐾=(𝐾⧵{𝑥₀})∪{𝑥₀} and (𝐾⧵{𝑥₀})∩𝑉 is a finite set by Proposition 2.7, since (𝐾⧵{𝑥₀})∪{𝑥₀} is compact. Also, note that 𝐾 being an infinite set implies 𝐾⧵{𝑥₀} is an infinite set. Let 𝐴=𝐾⧵{𝑥₀}.

Notation 2.12. Let 𝑍 be a nonempty set, and let 𝜃 be a Hausdorff topology on 𝑍. For 𝑧∈𝑍, we let 𝒩_𝜃(𝑧) denote the filter of 𝜃-neighborhoods of 𝑧; that is, 𝑈∈𝒩_𝜃(𝑧) if and only if 𝑧∈𝑂⊆𝑈 for some 𝑂∈𝜃.

Recall that for 𝐴⊆𝑋 and 𝑥∈𝑋, 𝑥 is an accumulation point of 𝐴 if and only if for 𝑈∈𝒩_𝜏(𝑥), there exists 𝑎∈𝐴 such that 𝑎≠𝑥 and 𝑎∈𝑈.

Remark 2.13. If 𝐴⊆𝑋 and 𝑥∈𝑋⧵{𝑥₀}, then 𝑥 is not an accumulation point of 𝐴.

Indeed, 𝑥∈𝑋⧵{𝑥₀} implies {𝑥}∈𝒩_𝜏(𝑥). So, if 𝑎∈𝐴 such that 𝑎≠𝑥, then 𝑎∉{𝑥}.

Remark 2.14. The element 𝑥₀ is an accumulation point of 𝑋⧵{𝑥₀} in (𝑋,𝜏).

Indeed, let 𝑈∈𝒩_𝜏(𝑥₀). Hence, {𝑥₀}∪𝐹⊆𝑈 for some 𝐹∈ℱ𝑟(𝑊). Since 𝑊 is infinite and 𝑥₀∉𝑊, we have that 𝐹⧵{𝑥₀}=𝐹≠∅. Let 𝑦∈𝐹. Hence, 𝑦∈𝑋⧵{𝑥₀} and 𝑦∈𝑈.

Consequently, 𝑋⧵{𝑥₀} is not closed [in (𝑋,𝜏)], which implies {𝑥₀} is not open; that is, {𝑥₀}∉𝜏. In fact, 𝑥₀ is the only element of 𝑋 such that {𝑥₀}∉𝜏.

3. Homeomorphisms of Compact Sets in (𝑋,𝜏)

The following proposition is obvious and is stated without proof.

Proposition 3.1. Let 𝜅 be an infinite cardinal. If 𝐷 and 𝐺 are sets such that 𝑥₀∉𝐷∪𝐺 and |𝐷|=𝜅=|𝐺|, (3.1) then there exists a map 𝜁∶𝐷∪{𝑥₀}→𝐺∪{𝑥₀} such that 𝜁(𝑥₀)=𝑥₀ and 𝜁 is a bijection.

Remark 3.2. If 𝐹∈ℱ𝑟(𝑊) and 𝑍⊆𝑋 is a finite set, then 𝐹⧵𝑍∈ℱ𝑟(𝑊).

Lemma 3.3. Let 𝐽 and 𝐾 be nonempty subsets of 𝑋⧵{𝑥₀} such that 𝐽∩𝑉 and 𝐾∩𝑉 are finite sets. Let 𝜑∶𝐽∪{𝑥₀}→𝐾∪{𝑥₀} be a bijection such that 𝜑(𝑥₀)=𝑥₀. If 𝐹∈ℱ𝑟(𝑊), then 𝐾∪𝑥₀∩𝐸∪𝑥₀⊆𝜑𝐽∪𝑥₀∩𝐹∪𝑥₀ (3.2) for some 𝐸∈ℱ𝑟(𝑊).

Proof. Let 𝐹∈ℱ𝑟(𝑊). Note that 𝐽∪𝑥₀=𝐽∪𝑥₀∩𝐹∪𝑥₀(𝐽∩(𝑊⧵𝐹))(𝐽∩𝑉). (3.3) Also, the sets 𝐽∩(𝑊⧵𝐹) and 𝐽∩𝑉 are finite.
Let 𝑍=𝜑(𝐽∩(𝑊⧵𝐹))(𝐽∩𝑉). (3.4) Consequently, 𝑍⊆𝑋⧵{𝑥₀} and 𝑍 is a finite set.
Let 𝐸=𝐹⧵𝑍. (3.5) Therefore, 𝐸∈ℱ𝑟(𝑊) by Remark 3.2. Note that 𝐹∪𝑥₀⧵𝑍=𝐸∪𝑥₀. (3.6) So, 𝐾∪𝑥₀𝐸∪𝑥₀=𝐾∪𝑥₀𝐹∪𝑥₀⧵𝑍. (3.7) Observe that, 𝜑𝐽∪𝑥₀=𝜑𝐽∪𝑥₀∩𝐹∪𝑥₀(((𝑥𝐽∩𝑊⧵𝐹))𝐽∩𝑉)=𝜑𝐽∪₀∩𝐹∪𝑥₀𝜑((𝑥(𝐽∩𝑊⧵𝐹))𝐽∩𝑉)=𝜑𝐽∪₀∩𝐹∪𝑥₀𝑍. (3.8) Therefore, 𝜑𝐽∪𝑥₀⧵𝑍=𝜑𝐽∪𝑥₀∩𝐹∪𝑥₀𝑍=𝜑𝑥⧵𝑍𝐽∪₀∩𝐹∪𝑥₀𝑥⧵𝑍⊆𝜑𝐽∪₀∩𝐹∪𝑥₀. (3.9) Consequently, 𝐾∪𝑥₀∩𝐸∪𝑥₀=𝐾∪𝑥₀∩𝐹∪𝑥₀⊆𝑥⧵𝑍𝐾∪₀=𝜑𝑥⧵𝑍𝐽∪₀𝑥⧵𝑍⊆𝜑𝐽∪₀∩𝐹∪𝑥₀. (3.10)

Proposition 3.4. Let 𝜏 be the Hausdorff topology on 𝑋 generated by ℬ₁. Let 𝐴 and 𝐵 be infinite subsets of 𝑋⧵{𝑥₀} such that 𝐴∩𝑉 and 𝐵∩𝑉 are finite sets. If 𝐴 and 𝐵 have the same cardinality (i.e., |𝐴|=|𝐵|), then any bijection of 𝐴∪{𝑥₀} onto 𝐵∪{𝑥₀} that has 𝑥₀ as a fixed point is a homeomorphism.

Proof. Let 𝜁∶𝐴∪{𝑥₀}→𝐵∪{𝑥₀} be a bijection with 𝜁(𝑥₀)=𝑥₀. Note that the existence of such a bijection is established by Proposition 3.1. Let 𝜁⁻¹∶𝐵∪{𝑥₀}→𝐴∪{𝑥₀} denote the inverse map of 𝜁.
Let 𝜆 be the topology on 𝐴∪{𝑥₀} induced by 𝜏, and let 𝜌 be the topology induced on 𝐵∪{𝑥₀} by 𝜏.
Let 𝑈∈𝜆. We will show that 𝜁[𝑈]∈𝜌.
Let 𝑏∈𝜁[𝑈]. Either 𝑏=𝑥₀ or 𝑏∈𝐵.

Case 1. Assume that 𝑏=𝑥₀. Hence, 𝑥₀=𝜁⁻¹(𝑥₀)=𝜁⁻¹(𝑏)∈𝑈. So, there exists 𝑄∈ℬ₁ such that 𝑥₀∈(𝐴∪{𝑥₀})∩𝑄⊆𝑈. Since 𝑥₀∉𝐷 for any 𝐷∈𝒫(𝑋⧵{𝑥₀}), we have that 𝑄=𝐹∪{𝑥₀} for some 𝐹∈ℱ𝑟(𝑊). Therefore, 𝐴∪𝑥₀∩𝑄=𝐴∪𝑥₀∩𝐹∪𝑥₀, (3.11) which implies 𝑥₀∈𝐴∪𝑥₀∩𝐹∪𝑥₀⊆𝑈, (3.12) which implies 𝑏=𝜁𝑥₀∈𝜁𝐴∪𝑥₀∩𝐹∪𝑥₀⊆𝜁[𝑈]. (3.13) By Lemma 3.3, there exists 𝐸∈ℱ𝑟(𝑊) such that 𝐵∪𝑥₀∩𝐸∪𝑥₀⊆𝜁𝐴∪𝑥₀∩𝐹∪𝑥₀. (3.14) Let 𝐺=(𝐵∪{𝑥₀})∩(𝐸∪{𝑥₀}). Note that 𝐸∪{𝑥₀}∈𝜏. Hence, 𝐺∈𝜌 and 𝑏=𝑥₀∈𝐺⊆𝜁𝐴∪𝑥₀∩𝐹∪𝑥₀⊆𝜁[𝑈]. (3.15)

Case 2. Assume that 𝑏∈𝐵. Consequently, {𝑏}∈𝒫(𝑋⧵{𝑥₀}), which implies {𝑏}∈𝜏, which implies {𝑏}∈𝜌 (since {𝑏}={𝑏}∩[𝐵∪{𝑥₀}]). Therefore, 𝑏∈𝜁[𝑈] implies 𝑏∈{𝑏}⊆𝜁[𝑈]. (3.16)
From expression (3.15) in Case 1 and expression (3.16) in Case 2, we infer 𝜁[𝑈]∈𝜌. (3.17)
Let 𝑆∈𝜌. We will show that 𝜁⁻¹[𝑆]∈𝜆.
Let 𝑎∈𝜁⁻¹[𝑆]. Either 𝑎=𝑥₀ or 𝑎∈𝐴.

Case 3. Assume that 𝑎=𝑥₀. Hence, 𝑥₀=𝜁(𝑥₀)=𝜁(𝑎)∈𝑆. So, there exists 𝑀∈ℬ₁ such that 𝑥₀∈(𝐵∪{𝑥₀})∩𝑀⊆𝑆. Since 𝑥₀∉𝐷 for any 𝐷∈𝒫(𝑋⧵{𝑥₀}), we have that 𝑀=𝐻∪{𝑥₀} for some 𝐻∈ℱ𝑟(𝑊). Therefore, 𝐵∪𝑥₀∩𝑀=𝐵∪𝑥₀∩𝐻∪𝑥₀, (3.18) which implies 𝑥₀∈𝐵∪𝑥₀∩𝐻∪𝑥₀⊆𝑆, (3.19) which implies 𝑎=𝜁⁻¹𝑥₀∈𝜁⁻¹𝐵∪𝑥₀∩𝐻∪𝑥₀⊆𝜁⁻¹[𝑆]. (3.20) By Lemma 3.3, there exists 𝐶∈ℱ𝑟(𝑊) such that 𝐴∪𝑥₀∩𝐶∪𝑥₀⊆𝜁⁻¹𝐵∪𝑥₀∩𝐻∪𝑥₀. (3.21) Let 𝑇=(𝐴∪{𝑥₀})∩(𝐶∪{𝑥₀}). Note that 𝐶∪{𝑥₀}∈𝜏. Hence, 𝑇∈𝜆 and 𝑎=𝑥₀∈𝑇⊆𝜁⁻¹𝐵∪𝑥₀∩𝐻∪𝑥₀⊆𝜁⁻¹[𝑆]. (3.22)

Case 4. Assume that 𝑎∈𝐴. Consequently, {𝑎}∈𝒫(𝑋⧵{𝑥₀}), which implies {𝑎}∈𝜏, which implies {𝑎}∈𝜆 (since {𝑎}={𝑎}∩[𝐴∪{𝑥₀}]). Therefore, 𝑎∈𝜁⁻¹[𝑆] implies 𝑎∈{𝑎}⊆𝜁⁻¹[𝑆]. (3.23)
From expression (3.22) in Case 3 and expression (3.23) in Case 4, we infer that 𝜁⁻¹[𝑆]∈𝜆. (3.24)
Consequently, from expression (3.17) and (3.24), we infer that 𝜁 is a homeomorphism of 𝐴∪{𝑥₀} onto 𝐵∪{𝑥₀}.

Proposition 3.5. Let 𝜏 be the Hausdorff topology on 𝑋 generated by ℬ₁. Let 𝐾⊆𝑋 be a compact set. If 𝑊⊆𝐾, then 𝑊∪{𝑥₀} is homeomorphic to 𝐾.

Proof. Note that 𝑊⊆𝐾 implies |𝑊|≤|𝐾|, which implies 𝐾 in an infinite set (𝑊 is an infinite set), which implies 𝐾=𝐴∪{𝑥₀} for some 𝐴⊆𝑋⧵{𝑥₀} such that 𝐴∩𝑉 is a finite set by Proposition 2.11. Consequently, 𝑊⊆𝐾 and 𝑥₀∉𝑊 imply 𝑊⊆𝐴. So, 𝐴=𝐴∩(𝑊∪𝑉)=𝑊∪(𝐴∩𝑉), (3.25) which implies |𝐴|=|𝑊|+|𝐴∩𝑉|=|𝑊|, (3.26) (see [2, Corollary  2.3, page 162]). Therefore, 𝑊∪{𝑥₀} is homeomorphic to 𝐴∪{𝑥₀}=𝐾 by Proposition 3.4.

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

Theorem 3.7. Let 𝜏 be the Hausdorff topology on 𝑋 generated by ℬ₁, and let 𝐾₁ and 𝐾₂ be nonempty compact subsets of 𝑋. If 𝐾₁ and 𝐾₂ have the same cardinality (i.e., |𝐾₁|=|𝐾₂|), then there exists a homeomorphism of 𝐾₁ onto 𝐾₂.

Proof. Let 𝐾₁ and 𝐾₂ be nonempty compact subsets of 𝑋 such that |𝐾₁|=|𝐾₂|. If 𝐾₁ is a finite set, then 𝐾₂ is a finite set. Hence, 𝜏 induces the discrete topology on 𝐾₁ and 𝐾₂ (see Remark 3.6). Consequently, any bijection of 𝐾₁ onto 𝐾₂ is a homeomorphism.
Assume that 𝐾₁ is an infinite set. Hence, 𝐾₂ is an infinite set. So, 𝐾₁=𝐴∪{𝑥₀} and 𝐾₂=𝐵∪{𝑥₀} such that 𝐴⊆𝑋⧵{𝑥₀}, 𝐵⊆𝑋⧵{𝑥₀}, 𝐴∩𝑉 is a finite set and 𝐵∩𝑉 is a finite set by Proposition 2.11. Observe that 𝐾₁ and 𝐾₂ being infinite sets imply 𝐴 and 𝐵 are infinite subsets of 𝑋⧵{𝑥₀} such that |𝐴|=|𝐵|. Therefore, there exists a homeomorphism of 𝐾₁ onto 𝐾₂ by Proposition 3.4.

4. Examples

Example 4.1. Let 𝑊={1/𝑛}^∞_𝑛=1, 𝑉=∅, and 𝑥₀=0. Let 𝑋=𝑉∪𝑊∪𝑥₀=1𝑛^∞_𝑛=1∪{0}, (4.1) and let ℬ₁1=𝒫𝑛^∞_𝑛=1∪𝐹∪𝑥₀1∣𝐹∈ℱ𝑟𝑛^∞_𝑛=1. (4.2) Consider the Hausdorff space (𝑋,𝜏), where the topology 𝜏 is generated by ℬ₁. Observe that (𝑋,𝜏) is compact (Corollary 2.10, since 𝑉=∅) and 𝑊={1/𝑛}^∞_𝑛=1 is not compact by Corollary 2.5. If 𝐾⊆𝑋 is an infinite compact set, then |𝐾|1=𝜔=𝑛^∞_𝑛=1∪{0}, (4.3) therefore, 𝐾 is homeomorphic to {1/𝑛}^∞_𝑛=1∪{0} (Theorem 3.7). In other words, all infinite, compact subsets of {1/𝑛}^∞_𝑛=1∪{0} are homeomorphic. Note that topology 𝜏 on 𝑋 is induced by the standard euclidean topology on ℝ.

Example 4.2. Let 𝐵 be a set such that |𝐵|=𝜔. Let 𝜑∶𝜔→𝐵 be a bijection. For 𝑛∈𝜔, we will denote 𝜑(𝑛) by 𝑥_𝑛, that is, 𝑥_𝑛=𝜑(𝑛). Therefore, 𝐵={𝑥_𝑛}_𝑛∈𝜔, where 𝑚,𝑛∈𝜔 and 𝑚≠𝑛 imply 𝑥_𝑚≠𝑥_𝑛. Let 𝑊={𝑥_2𝑗}_𝑗∈ℕ and let 𝑉={𝑥_2𝑗+1}_𝑗∈𝜔. Note that the maps 𝑗→𝑥_2𝑗 and 𝑗→𝑥_2𝑗+1 are bijections of ℕ onto 𝑊 and 𝜔 onto 𝑉, respectively. Also, 𝑥₀∉𝑊∪𝑉, 𝑊∩𝑉=∅ and 𝐵={𝑥₀}∪𝑉∪𝑊; consequently, 𝑉=[𝐵⧵{𝑥₀}]⧵𝑊. We can write ℬ₁⊆𝒫(𝐵) (see Definition 2.2) as follows. ℬ₁=𝒫𝑥_𝑛^∞_𝑛=1∪𝐹∪𝑥₀∣𝐹∈ℱ𝑟(𝑊). (4.4) The collection ℬ₁ generates a Hausdorff topology 𝜏 on 𝐵 (see Proposition 2.3). Note that (𝐵,𝜏) is not compact (Corollary 2.10), sets 𝑉 and 𝑊 are not compact (Corollaries 2.6 and 2.5, resp.), 𝑉∪{𝑥₀} is not compact (Corollary 2.9), and 𝑊∪{𝑥₀} is compact (Corollary 2.8). Also, if 𝐾⊆𝐵 is an infinite compact set, then |𝐾|=𝜔=𝑊∪𝑥₀, (4.5) therefore, 𝐾 is homeomorphic to 𝑊∪{𝑥₀}={𝑥_2𝑗}_𝑗∈𝜔 (Theorem 3.7). In other words, all infinite, compact subsets of 𝐵={𝑥_𝑛}_𝑛∈𝜔 are homeomorphic.

Example 4.3. Let 𝜃 be an infinite cardinal and consider the collection {𝜅_𝑛}_𝑛∈𝜔 of infinite cardinals defined as follows. Let 𝜅₀=𝜔 and for 𝑛∈𝜔, let 𝜅_𝑛+1=2^𝜅𝑛. Also, we denote 𝜅_𝜔=⋃_𝑛∈𝜔𝜅_𝑛. Hence, 𝜅_𝜔 is the cardinal number, that is, the supremum of {𝜅_𝑛}_𝑛∈𝜔 and 𝜅_𝑛<𝜅_𝜔 for each 𝑛∈𝜔 (see the Alephs section in [3], page 29). For 𝑛∈𝜔, define 𝑊_𝑛=𝜇,𝑛∣𝜇∈𝜅_𝑛,𝑊_𝜔=𝜇,𝜔∣𝜇∈𝜅_𝜔. (4.6) Observe that |𝑊_𝜔|=𝜅_𝜔, |𝑊_𝑛|=𝜅_𝑛 for each 𝑛∈𝜔, and the collection 𝑊_{𝑛𝑛∈𝜔}∪𝑊_𝜔 (4.7) is pairwise disjoint. Let 𝑊=𝑊_𝜔∪_𝑛∈𝜔𝑊_𝑛,𝑉={(𝜆,𝜔+1)∣𝜆∈𝜃},𝑥₀=2^𝜅𝜔,𝜔+2. (4.8) Note that the collection {{𝑥₀},𝑊,𝑉} is pairwise disjoint. Let 𝑋=𝑥₀∪𝑉∪𝑊 (4.9) and consider the noncompact, Hausdorff space (𝑋,𝜏), where topology 𝜏 is generated by ℬ₁ (see Definition 2.2 and Corollary 2.10). For 𝜂∈𝜔+1, define 𝒦_𝜂=𝐾∈𝒫(𝑋)∣𝐾iscompact,|𝐾|=𝜅_𝜂. (4.10) Observe, 𝑊_𝜂∪{𝑥₀}∈𝒦_𝜂 for 𝜂∈𝜔+1 by Proposition 2.7 (since 𝑊_𝜂∩𝑉=∅) and the fact that |𝑊_𝜂∪{𝑥₀}|=𝜅_𝜂. Also, the collection {𝒦_𝜂}_{𝜂∈𝜔+1} is pairwise disjoint and Theorem 3.7 implies that all of the sets in 𝒦_𝜂 are homeomorphic to each other. Since 𝜅_𝑛<𝜅_𝜔 for each 𝑛∈𝜔, we have |𝑊|=𝑊_𝜔∪_𝑛∈𝜔𝑊_𝑛=𝜅_𝜔+𝜅_𝜔=𝜅_𝜔, (4.11) (see [2, Corollary  2.3, page 162]), which implies |𝑊∪{𝑥₀}|=𝜅_𝜔. Consequently, 𝑊∪{𝑥₀}∈𝒦_𝜔 (see Corollary 2.8), which implies 𝑊∪{𝑥₀} is homeomorphic to 𝑊_𝜔∪{𝑥₀}. If we let [𝑉]^<𝜔 denote the set of all finite subsets of 𝑉, then 𝑊_𝜂∪𝐸∪𝑥₀∈𝒦_𝜂for𝐸∈[𝑉]^<𝜔,𝜂∈𝜔+1, (4.12) (see Proposition 2.7). Consequently, the size of 𝜃 can affect the cardinality of 𝒦_𝜂 for 𝜂∈𝜔+1 (e.g., let 𝜃 be a Mahlo cardinal ([3, Chapter 8, page 95])). If the generalized continuum hypothesis is assumed, then the collection {𝒦_𝜂}_{𝜂∈𝜔+1} is a partition of the collection of all infinite, compact subsets of 𝑋.

Acknowledgment

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