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.