Set-valued maps and some generalized metric spaces

TL;DR

This paper characterizes various generalized metric spaces, including stratifiable and semi-metrizable spaces, using set-valued maps with strictly increasing closed covers, extending the understanding of their structure and related real-valued functions.

Contribution

It introduces new characterizations of generalized metric spaces via set-valued maps with strictly increasing closed covers, broadening the theoretical framework.

Findings

Characterization of stratifiable and semi-metrizable spaces using set-valued maps.

Extension of characterizations to spaces with generalized real-valued functions.

Application of these characterizations to understand boundedness and structure of such spaces.

Abstract

To give characterizations of monotonically countably paracompact spaces with set-valued maps, Yamazaki [22] introduced the notion of strictly increasing closed cover of a topological space with which the boundedness of a set-valued map was defined. In this paper, we show that most of generalized metric spaces such as stratifiable spaces, semi-metrizable spaces can be characterized with set-valued maps with values into the family of all closed nonempty subsets of a space which has a strictly increasing closed cover. Moreover, as an application, we use the results obtained to give characterizations of the corresponding spaces with generalized real-valued functions.