Strongly semi-continuous functions and $\delta$-stratifiable spaces

Abstract

In [6], a sufficient condition for a space to be $\delta$-stratifiable was presented and it was asked whether the condition is necessary. Jin et al [4] gave a negative answer to the question by showing that a space with the condition is zerodimensional. In this paper, we show that a space with the condition is precisely an almost discrete space. Moreover, we introduce the notions of strongly lower (upper)
semi-continuous functions, with which the characterizations of $\delta$-stratifiable spaces are presented.