On (pre-) approach spaces within convergence approach spaces

TL;DR

This paper explores the parallels between (pre)topologies and (pre)approach spaces within convergence approach spaces, providing insights into their structure and the reflector from convergence approach spaces to approach spaces.

Contribution

It introduces a more complete parallel between topologies and approach spaces in the context of convergence approach spaces, and characterizes approach spaces via pointfree convergence frames.

Findings

A new perspective on the relationship between convergence approach spaces and approach spaces.

Characterization of approach spaces among convergence approach spaces.

Insights into the structure of approach spaces and their reflectors.

Abstract

The purpose of this note is to illustrate a parallel between (pre)topologies when seen among convergence spaces and (pre)approach spaces when seen among convergence approach spaces, that appears to be a more complete parallel than in the traditional presentation of these approach structures. This sheds some light on the reflector from the category of convergence approach spaces to that of approach spaces and even on the structure of approach spaces as such. This point of view allows for a characterization of approach spaces among convergence approach spaces represented as pointfree convergence frames as in [18].