A Hofmann-Mislove theorem for $c$-well-filtered spaces

TL;DR

This paper introduces $c$-well-filtered spaces and extends the Hofmann-Mislove theorem to these spaces, establishing new relationships between filters, compact sets, and sobriety in topology.

Contribution

It defines $c$-well-filtered spaces and proves a Hofmann-Mislove theorem analogue for them, expanding the understanding of filter-compact set correspondences.

Findings

A retract of a $c$-well-filtered space remains $c$-well-filtered.

Locally Lindelöf and $c$-well-filtered $P$-spaces are countably sober.

A Hofmann-Mislove theorem for $c$-well-filtered spaces is established.

Abstract

The Hofmann-Mislove theorem states that in a sober space, the nonempty Scott open filters of its open set lattice correspond bijectively to its compacts saturated sets. In this paper, the concept of $c$-well-filtered spaces is introduced. We show that a retract of a $c$-well-filtered space is $c$-well-filtered and a locally Lindel\"{o}f and $c$-well-filtered $P$-space is countably sober. In particular, we obtain a Hofmann-Mislove theorem for $c$-well-filtered spaces.