QFS-space and its properties

TL;DR

This paper introduces QFS-spaces based on quasi-finitely separating maps, explores their properties, and establishes their relationships with quasicontinuous spaces, including closure properties and upper powerspace results.

Contribution

It defines QFS-spaces and quasicontinuous maps, providing new characterizations and properties, including closure under subspaces, retracts, and powerspaces.

Findings

QFS-spaces are quasicontinuous spaces

Closed subspaces of QFS-spaces are QFS-spaces

Upper powerspaces of continuous QFS-spaces are FS-spaces

Abstract

In this paper, the concept of quasi-finitely separating map and quasiapproximate identity are introduced. Based on these concepts, QFS-spaces and quasicontinuous maps are defined. Properties and characterizations of QFS-spaces are explored. Main results are: (1) Each QFS-space is quasicontinuous space; (2) Closed subspaces, quasicontinuous projection spaces of QFS-spaces are QFS-spaces; (3) Continuous retracts of QFS-spaces are QFS-spaces and a kind of retracts of QFS-spaces are constructed; (4) Upper powerspaces of continuous QFS-spaces are FS-spaces.