Measure of compactness for filters in product spaces: Kuratowski-Mr\`owka in CAP revisited

TL;DR

This paper revisits a measure of compactness in convergence approach spaces, providing new characterizations of maps, product theorems, and extending classical results like Mr owka-Kuratowski to this broader setting.

Contribution

It introduces a measure of compactness for families of sets in convergence approach spaces and characterizes various maps as those respecting this measure, extending classical theorems.

Findings

New product theorems for spaces and maps

Characterizations of quotient, closed, and perfect maps

Extension of Mr owka-Kuratowski results to approach spaces

Abstract

The first author introduced a measure of compactness for families of sets, relative to a class of filters, in the context of convergence approach spaces. We characterize a variety of maps (types of quotient maps, closed maps, and variants of perfect maps) as those respecting this measure of compactness under one form or another. We establish a product theorem for measure of compactness that yields as instances new product theorems for spaces and maps, and new product characterizations of spaces and maps, thus extending existing results from the category of convergence spaces to that of convergence approach spaces. In particular, results of the Mr\`owka-Kuratowski type are obtained, shedding new light on existing results for approach spaces.