Function spaces and contractive extensions in Approach Theory: The role of regularity

TL;DR

This paper extends classical regularity characterizations from convergence spaces to convergence-approach spaces, introducing new notions and improving existing results in the context of approach theory.

Contribution

It generalizes regularity characterizations to convergence-approach spaces and introduces a new concept of strictness, enhancing the theoretical framework.

Findings

Characterizations of regularity and strong regularity in convergence-approach spaces.

Introduction of a new notion of strictness for convergence-approach spaces.

Improved results over existing convergence space theories.

Abstract

Two classical results characterizing regularity of a convergence space in terms of continuous extensions of maps on one hand, and in terms of continuity of limits for the continuous convergence on the other, are extended to convergence-approach spaces. Characterizations are obtained for two alternative extensions of regularity to convergence-approach spaces: regularity and strong regularity. The results improve upon what is known even in the convergence case. On the way, a new notion of strictness for convergence-approach spaces is introduced.