Some Remarks Concerning Semi-$T_{\frac{1}{2}}$ Spaces

Abstract

In this paper we prove that each subspace of an Alexandroff $T_0$-space is semi-$T_\frac{1}{2}$. In particular, any subspace of the folder $X^n$, where $n$ is a positive integer and $X$ is either the Khalimsky line $(\mathbb Z, \tau_K)$, the Marcus-Wyse plane $(\mathbb Z^2, \tau_{MW})$ or any  partially ordered set with the upper topology is  semi-$T_\frac{1}{2}$. Then we study the basic properties of spaces possessing the axiom semi-$T_\frac{1}{2}$ such as  finite productiveness and  monotonicity.