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Keywords:
normal space; (weakly) densely normal space; (weakly) $\theta $-normal space; almost normal space; almost $\beta $-normal space; $\kappa $-normal space; (weakly) $\beta $-normal space
Summary:
The notion of $\beta $-normality was introduced and studied by Arhangel'skii, Ludwig in 2001. Recently, almost $\beta $-normal spaces, which is a simultaneous generalization of $\beta $-normal and almost normal spaces, were introduced by Das, Bhat and Tartir. We introduce a new generalization of normality, namely weak $\beta $-normality, in terms of $\theta $-closed sets, which turns out to be a simultaneous generalization of $\beta $-normality and $\theta $-normality. A space $X$ is said to be weakly $\beta $-normal (w$\beta $-normal$)$ if for every pair of disjoint closed sets $A$ and $B$ out of which, one is $\theta $-closed, there exist open sets $U$ and $V$ such that $\overline {A\cap U}=A$, $\overline {B\cap V}=B$ and $\overline {U}\cap \overline {V}=\emptyset $. It is shown that w$\beta $-normality acts as a tool to provide factorizations of normality.
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