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

TL;DR

This paper extends Scott convergence to locally hypercompact spaces using a new topological convergence notion, characterizing these spaces and their relation to dcpos and Scott topology.

Contribution

It introduces $ abla$-convergence and finitely approximated spaces, providing new characterizations of locally hypercompact and monotone determined spaces.

Findings

Monotone determined spaces are locally hypercompact iff $ abla$-convergence is topological.

A $T_0$ space has topological $ abla$-convergence iff it is finitely approximating.

If the Lawson topology on a monotone determined space is compact, then it is a Scott domain.

Abstract

In this paper, we give a topological version of Scott convergence theorem for locally hypercompact spaces. We introduce the notion of $\mathcal{S}^*_X$-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 $\mathcal{S}^*_X$-convergence is topological. (2) For a $T_0$ space $X$, $\mathcal{S}^*_X$-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.