C-normal topological property

Abstract

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

C -Paracompactness and C 2 -paracompactness

“…ω 1 is C -paracompact because it is C 2 -paracompact, being T 2 locally compact (see Theorem 2.12 below), but not paracompact because it is countably compact noncompact. The following theorem can be proved in a similar way as in [3]. Theorem 2.7 If X is a T 1 space such that the only compact subsets are the finite subsets, then X is C 2paracompact.…”

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.…”

“…A topological space ( X , τ ) is called submetrizable if there exists a metric d on X such that the topology τ d on X generated by d is coarser than τ , i.e., τ d ⊆ τ , see [5]. Since submetrizabilty implies both C-normality [1] and C 2 -paracompactness [12], we conclude that any H-space ( X , U A N ) is both C-normal and C 2 -paracompact being submetrizable by the usual metric. By the theorem "If X is T 3 separable L-normal and of countable tightness, then X is normal."…”

Results about the Alexandroff duplicate space

“…Gen. Topol. 17,no. 2 that the restriction f |C : C −→ f (C) is a homeomorphism for each compact subspace C ⊆ X.…”

“…It was proved in [2] that if X is C-normal, then so is its Alexandroff Duplicate. A topological space ( X , τ ) is called epinormal [2] if there is a coarser topology τ ′ on X such that ( X , τ ′ ) is T 4 .…”