Homeomorphisms of Compact Sets in Certain Hausdorff Spaces

Definition 2.1. Let A be an infinite set. Define

We call ℱr(A)  the Fréchet filter on A.

Definition 2.2. Consider the collection ℬ₁⊆𝒫(X) defined as follows:

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

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 Assume that w ∈ W. Note that {w} ∈ ℬ₁. Also, W∖{w} ∈ ℱr(W), which implies [W∖{w}] ∪ {x₀} ∈ ℬ₁ and Observe that,

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

which implies Let 𝒰⊆𝒜 be a nonempty, finite subcollection of 𝒜. Therefore, there exists $$, for some m ∈ ℕ, such that which implies If A⊆⋃𝒰, then we would have $$, 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,

Let 𝒜_A = {{a}∣a ∈ A}, and let 𝒜 = {Q} ∪ 𝒜_A. Note that 𝒜_A⊆τ, 𝒜⊆τ, Suppose that 𝒰⊆𝒜_A is a finite subcollection such that It can be assumed, without loss of generality, that Q ∈ 𝒰 and $$ for m + 1 = |𝒰| ∈ ℕ such that U₀ = Q. So, by definition of 𝒜, there exists $$ such that U_i = {a_i} for i = 1, …, m. Note that Q = F ∪ {x₀}⊆W, which implies So, expressions (2.16) and (2.17) would imply 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

Hence, there exists U ∈ 𝒜 such that x₀ ∈ U. Since x₀ ∉ E for E∈𝒫(X∖{x₀}), there exists F ∈ ℱr(W) such that Assume that $$ for some m ∈ ω [we define $$]. Observe that, Since W∖F is finite, we can assume that for some k ∈ ω (we define $$). Hence, there exists $$ and $$ such that for i = 1, …, m and q = 1, …, k (again, we define $$). Consequently, from expressions (2.19), (2.20), (2.21), (2.22), and (2.23), we have 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.

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

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

Proposition 3.1. Let κ be an infinite cardinal. If D and G are sets such that x₀ ∉ D ∪ G and

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

for some E ∈ ℱr(W).

Proof. Let F ∈ ℱr(W). Note that

Also, the sets J∩(W∖F) and J∩V are finite.

Let

Consequently, Z⊆X∖{x₀} and Z is a finite set.

Let

Therefore, E ∈ ℱr(W) by Remark 3.2. Note that So, Observe that, Therefore, Consequently,

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,

which implies which implies By Lemma 3.3, there exists E ∈ ℱr(W) such that Let G = (B ∪ {x₀})∩(E ∪ {x₀}). Note that E ∪ {x₀} ∈ τ. Hence, G ∈ ρ and

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

From expression (3.15) in Case 1 and expression (3.16) in Case 2, we infer

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,

which implies which implies By Lemma 3.3, there exists C ∈ ℱr(W) such that Let T = (A ∪ {x₀})∩(C ∪ {x₀}). Note that C ∪ {x₀} ∈ τ. Hence, T ∈ λ and

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

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 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,

which implies (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.

Example 4.1. Let $$, V = ∅, and x₀ = 0. Let

and let 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 $$ is not compact by Corollary 2.5. If K⊆X is an infinite compact set, then therefore, K is homeomorphic to $$ (Theorem 3.7). In other words, all infinite, compact subsets of $$ 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, $$, where m, n ∈ ω and m ≠ n imply x_m ≠ x_n. Let $$ and let $$. 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.

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 therefore, K is homeomorphic to $$ (Theorem 3.7). In other words, all infinite, compact subsets of $$ are homeomorphic.

Example 4.3. Let θ be an infinite cardinal and consider the collection $$ of infinite cardinals defined as follows. Let κ₀ = ω and for n ∈ ω, let $$. Also, we denote κ_ω = ⋃_n∈ωκ_n. Hence, κ_ω is the cardinal number, that is, the supremum of $$ and κ_n < κ_ω for each n ∈ ω (see the Alephs section in [3], page 29). For n ∈ ω, define

Observe that |W_ω| = κ_ω, |W_n| = κ_n for each n ∈ ω, and the collection is pairwise disjoint. Let Note that the collection {{x₀}, W, V} is pairwise disjoint. Let 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 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 $$ is pairwise disjoint and Theorem 3.7 implies that all of the sets in 𝒦_η are homeomorphic to each other. Since κ_n < κ_ω for each n ∈ ω, we have (see [2, Corollary  2.3, page 162]), which implies |W ∪ {x₀}| = κ_ω. Consequently, W ∪ {x₀} ∈ 𝒦_ω (see Corollary 2.8), which implies W ∪ {x₀} is homeomorphic to W_ω ∪ {x₀}. If we let [V]^<ω denote the set of all finite subsets of V, then (see Proposition 2.7). Consequently, the size of θ can affect the cardinality of 𝒦_η for η ∈ ω + 1 (e.g., let θ be a Mahlo cardinal ([3, Chapter 8, page 95])). If the generalized continuum hypothesis is assumed, then the collection $$ is a partition of the collection of all infinite, compact subsets of X.