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.

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A topological space X is C-κ-normal (C-mildly normal ) if there exist a κ-normal (mildly 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 subspace A ⊆ X. We present new results about those two topological properties and use a discrete extension space to solve open problems regarding C2-paracompactness and α-normality

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