${\mathcal{S}}_X$-convergence and locally hypercompact spaces

Abstract

In this paper, we give a topological version of Scott convergence theorem for locally hypercompact spaces. We introduce the notion of ${\mathcal{S}}_X^{\ast }$-convergence on a $T_0$ topological space $X$, and define the notion of finitely approximated spaces. Monotone determined spaces are natural topological extensions of dcpos. The main results are: (1) A monotone determined space $X$ is a locally hypercompact space iff ${\mathcal{S}}_X^{\ast }$-convergence is topological. (2) For a $T_0$ space $X$, ${\mathcal{S}}_X^{\ast }$-convergence is topological iff $X$ is a finitely approximating space. (3) If the Lawson topology on a monotone determined space $X$ is compact, then $X$ is a dcpo endowed with the Scott topology.