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.

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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.

In this paper, we present some new results about the Alexandroff Duplicate Space. We prove that if a space X has the property P , then its Alexandroff Duplicate space A(X) may not have P , where P is one of the following properties: extremally disconnected, weakly extremally disconnected, quasi-normal, pseudocompact. We prove that if X is α-normal, epinormal, or has property wD, then so is A(X). We prove almost normality is preserved by A(X) under special conditions. 2010 MSC: 54F65; 54D15; 54G20.