Unification approach to the separation axioms between $T_0$ and completely Hausdorff

Abstract

The aim of this paper is to introduce a new weak separation axiom that generalizes the separation properties between $T_1$ and completely Hausdorff. We call a topological space $(X,\tau)$ a $T_{\kappa,\xi}$-space if every compact subset of $X$ with cardinality $\leq \kappa$ is $\xi$-closed, where $\xi$ is a general closure operator. We concentrate our attention mostly on two new concepts: kd-spaces and $T_{1/3}$-spaces.