In this paper we prove that each subspace of an Alexandroff $T_0$-space is semi-$T_{\fac12}$. In particular, any subspace of the folder $X^n$, where $n$ is a positive integer and $X$ is either the Khalimsky line $(\Bbb Z,\tau_K)$, the Marcus--Wyse plane $(\Bbb Z^2,\tau_{MW})$ or any partially ordered set with the upper topology is semi-$T_\frac12$. Then we study the basic properties of spaces possessing the axiom semi-$T_\frac12$ such as finite productiveness and monotonicity.