$CC$-normal topological spaces

Abstract

A topological space X is called CC -normal if there exist a normal space Y and a bijective functionis a homeomorphism for each countably compact subspace A ⊆ X . We will investigate this property and produce some examples to illustrate the relation between CC -normality and other weaker kinds of normality.

S-paracompactness and S2-paracompactness

“…Recall that a space (X,τ) is S-normal if there exist a normal space Y and a function f such that f : X −→ Y is a bijection and f | B : B −→ f (B) is a homeomorphism for every separable subspaces B ⊆ X (see [5]). Since any T 2 paracompact space is normal, then any S 2 -paracompact space is S-normal.…”

Epinormality

“…A topological space X is called C-normal if there exist a normal space Y and a bijective function f : X −→ Y such that the restriction f | C : C −→ f (C) is a homeomorphism for each compact subspace C ⊆ X, [7]. It is clear that any epinormal space is C-normal, but the converse is not true in general.…”

“…Another example is the above Mrówka space Ψ(A), where A ⊂ [ω] ω is mad. It is C-normal because it is locally compact [7].…”

New topologies between the usual and Niemytzki

“…Recall that a topological space X is called C-normal [1] (CC-normal [9], L-normal [11], S-normal [10]) if there exists a normal space Y and a bijective function f : X −→ Y such that the restriction f A : A −→ f (A) is a homeomorphism for each compact (countably compact, Lindelöf, separable) subspace A ⊆ X. A topological space X is called C 2 -paracompact [12] if there exists a T 2 paracompact space Y and a bijective function f : X −→ Y such that the restriction f A : A −→ f (A) is a homeomorphism for each compact subspace A ⊆ X.…”