Epi-Partial Normality

Abstract

The main aim of this work is to study a new version of normality called epi-partial normality, which lies between epi-almost normality and epi-mild normality. A space (X, T) is called an epi-partially normal space if there exists a topology T’, which is coarser than T, such that (X, T’) is Hausdorff partially normal. In this work, we investigate this property and present some examples that illustrate the relationships between epi-partial normality and other weaker kinds of both normality and regularity. We show that this property is a topological, a semi regularization and an additive property. Some properties and relationships of epi-partial normality are presented and proved.

Epi-completely Regular Topological Spaces

“…At the beginning of 2020, Alshammari studied the notion of epi-almost normality [3]. Thabit studied the notion of epi-partial normality in 2021 [32]. At the end of 2021, Thabit and others studied the notion of epiquasi normality [31].…”

“…A space (X, T ) is said to be epi-normal [15] (resp. epi-mildly normal [18], epi-almost normal [3], epi-regular [5], epi-quasi normal [31], epi-partially normal [32]), if there exists a topology T ′ on X coarser than T such that (X, T ′ ) is a T 4 (resp. Hausdorff mildly-normal, Hausdorff almost-normal, T 3 , Hausdorff-quasi-normal, Hausdorff partially-normal) space.…”

“…Tychonoff) space Y and a bijective function f : X → Y such that the restriction function f | A : A → f (A) is a homeomorphism for each Lindelöf subspace A ⊆ X. The basic definitions and any undefined terms in this article can be found in [31] and [32].…”

C-Almost Normality and L-Almost Normality

“…Then, Alzahrani studied the notions of C-regularity, L-regularity, C-Tychonoff and L-Tychonoff in 2018, see [8,9]. Thabit studied the notion of epi-partial normality in 2021, see [34]. At the end of 2021, Thabit and others studied the notion of epi-quasi normality [33].…”

“…A space (X, T ) is said to be an epi-regular space [7], if there exists a topology T ′ on X coarser than T such that (X, T ′ ) is T 3 (regular and T 1 ). A space (X, T ) is said to be an epi-partially normal space [34], if there exists a topology T ′ on X coarser than T such that (X, T ′ ) is Hausdorff partially normal. A space (X, T ) is said to be an epi-quasi normal space [33], if there exists a topology T ′ on X coarser than T such that (X, T ′ ) is Hausdorff quasi normal.…”