Epinormality

Abstract

A topological space ( X , τ ) is called epinormal if there is a coarser topology τ on X such that ( X , τ ) is T 4 . We investigate this property and present some examples to illustrate the relationships between epinormality and other weaker kinds of normality.

C-Tychonoff and L-Tychonoff Topological Spaces

“…Recall that a topological space ( X , τ ) is called epinormal if there is a coarser topology τ on X such that ( X , τ ) is T 4 [3]. By a similar proof as that of Theorem 1 above, we can prove the following corollary: Corollary 6.…”

C -Paracompactness and C 2 -paracompactness

“…We have to mention that Corollary 2.9 of [2], of the second author, is incorrect; the condition of cardinality less than continuum must be added to its hypothesis. Observe that Example 2.18 shows that C 2 -paracompactness does not imply the Lindelöf property.…”

“…ω 1 + 1 is an example of C 2 -paracompact that is not submetrizable. Recall that a topological space (X, τ ) is called epinormal if there is a coarser topology τ ′ on X such that (X, τ ′ ) is T 4 [2]. Epinormality implies C -normality [3].…”

“…Suppose that K 2 is infinite but a ̸ ∈ E . The open cover (2), that A i ∩ E is finite and for each i ∈ K 3 \K 1 we have, by (2)…”

“…(called also an Urysohn space[7]), if for each distinct element a, b ∈ X there exist two open sets U and V such that a ∈ U , b ∈ V , and U ∩ V = ∅. Since epinormality implies T 21 2[2], we have the following corollary.Corollary 2.17Any C 2 -paracompact Fréchet space is completely Hausdorff, (T 2 1 2 ).Any finite complement topology on an infinite set is C -paracompact and Fréchet but not Hausdorff; thus, Theorem 2.16 is not always true for C -paracompactness. The next example is an application for Theorem 2.Recall that two countably infinite sets are said to be almost disjoint[18] if their intersection isfinite.…”