Homeomorphisms of Compact Sets in Certain Hausdorff Spaces

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 ⊆ ⋃nj 1 Uj for some n ∈ N and {U1, . . . , Un} ⊆ A. Therefore, compact topologies need not be Hausdorff.


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 1 U j for some n ∈ N and {U 1 , . . ., U n } ⊆ A. Therefore, compact topologies need not be Hausdorff.

A Class of Hausdorff Spaces
Let V , W, and x 0 be sets such that W is infinite and the collection {V, W, {x 0 }} 2.1 2 International Journal of Mathematics and Mathematical Sciences is pairwise disjoint.For example, let V { ν, 0 | ν ∈ ω}, W { μ, 1 | μ ∈ 2 ω }, and x 0 2 2 ω .Unless otherwise stated, we let Recall that for set Y and G ⊆ Y , we have

2.3
Definition 2.1.Let A be an infinite set.Define 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 B 1 ⊆ P X defined as follows:

2.9
We infer that τ is Hausdorff.
International Journal of Mathematics and Mathematical Sciences 3 Proof.Note that finite sets are compact in any topological space.So, assume that A is an infinite, and let

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

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 \ {x 0 } 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.
Proof.The topology τ on X is generated by B 1 see Proposition 2.3 .
Assume that A ∩ V is an infinite set.Let F ∈ Fr W , and let Q F ∪ {x 0 }.Hence,

2.15
Suppose that U ⊆ A A is a finite subcollection such that

2.16
It can be assumed, without loss of generality, that Q ∈ U and 2.17 So, expressions 2.16 and 2.17 would imply

2.19
Hence, there exists

2.24
We infer Therefore, the corollary follows from Proposition 2.7.

Corollary 2.10. The topological space X, τ is compact if and only if
Therefore, the corollary follows from Proposition 2.7.
Notation 2.12.Let Z be a nonempty set, and let θ be a Hausdorff topology on Z.For z ∈ Z, we let 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 \ {x 0 }, then x is not an accumulation point of A.
Remark 2.14.The element x 0 is an accumulation point of

Homeomorphisms of Compact Sets in X, τ
The following proposition is obvious and is stated without proof.Lemma 3.3.Let J and K be nonempty subsets of X \ {x 0 } such that J ∩ V and K ∩ V are finite sets.Let ϕ : J ∪ {x 0 } → K ∪ {x 0 } be a bijection such that ϕ x 0 x 0 .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 Therefore, E ∈ Fr W by Remark 3.2.Note that Observe that,

3.9
International Journal of Mathematics and Mathematical Sciences 7 Consequently, Let λ be the topology on A ∪ {x 0 } induced by τ, and let ρ be the topology induced on B ∪ {x 0 } by τ.

3.13
By Lemma 3.3, there exists E ∈ Fr W such that

3.16
From expression 3.15 in Case 1 and expression 3.16 in Case 2, we infer ζ U ∈ ρ.
Let a ∈ ζ −1 S .Either a x 0 or a ∈ A.
Case 3. Assume that a x 0 .Hence,

3.20
By Lemma 3.3, there exists C ∈ Fr W such that

3.23
From expression 3.22 in Case 3 and expression 3.23 in Case 4, we infer that

3.24
Consequently, from expression 3.17 and 3.24 , we infer that ζ is a homeomorphism of A ∪ {x 0 } onto B ∪ {x 0 }.Proposition 3.5.Let τ be the Hausdorff topology on X generated by Proof.Let K 1 and K 2 be nonempty compact subsets of X such that If K 1 is a finite set, then K 2 is a finite set.Hence, τ induces the discrete topology on K 1 and K 2 see Remark 3.6 .Consequently, any bijection of K 1 onto K 2 is a homeomorphism.
Assume that K 1 is an infinite set.Hence, K 2 is an infinite set.So, K 1 A ∪ {x 0 } and K 2 B ∪ {x 0 } such that A ⊆ X \ {x 0 }, B ⊆ X \ {x 0 }, A ∩ V is a finite set and B ∩ V is a finite set by Proposition 2.11.Observe that K 1 and K 2 being infinite sets imply A and B are infinite subsets of X \ {x 0 } such that |A| |B|.Therefore, there exists a homeomorphism of K 1 onto K 2 by Proposition 3.4.

Examples
Example 4.1.Let W {1/n} ∞ n 1 , V ∅ , and x 0 0. Let and let Consider the Hausdorff space X, τ , where the topology τ is generated by B 1 .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 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, We can write B 1 ⊆ P B see Definition 2.2 as follows. The 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.

Proposition 3 . 1 .
Let κ be an infinite cardinal.If D and G are sets such that x 0 / ∈ D ∪ G and |D| κ |G|, 3.1 then there exists a map ζ : D ∪ {x 0 } → G ∪ {x 0 } such that ζ x 0 x 0 and ζ is a bijection.International Journal of Mathematics and Mathematical Sciences Remark 3.2.If F ∈ Fr W and Z ⊆ X is a finite set, then F \ Z ∈ Fr W .

3 . 10 Proposition 3 . 4 .
Let τ be the Hausdorff topology on X generated by B 1 .Let A and B be infinite subsets of X \{x 0 } 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∪{x 0 } onto B∪{x 0 } that has x 0 as a fixed point is a homeomorphism.Proof.Let ζ : A ∪ {x 0 } → B ∪ {x 0 } be a bijection with ζ x 0 x 0 .Note that the existence of such a bijection is established by Proposition 3.1.Let ζ −1 : B ∪ {x 0 } → A ∪ {x 0 } denote the inverse map of ζ.
.26 see 2, Corollary 2.3, page 162 .Therefore, W ∪ {x 0 } is homeomorphic to A ∪ {x 0 } 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. Let τ be the Hausdorff topology on X generated by B 1 , and let K 1 and K 2 be nonempty compact subsets of X.If K 1 and K 2 have the same cardinality (i.e., |K 1 | |K 2 |), then there exists a homeomorphism of K 1 onto K 2 .