Weakly and Strongly Reversible Spaces

TL;DR

This paper investigates different notions of reversibility in topological spaces, characterizing classes of spaces where bijections or condensations are homeomorphisms, and identifying which spaces fall into these classes.

Contribution

The paper introduces and compares weakly and strongly reversible spaces, providing characterizations and showing their relationships with known classes like sequential and metrizable spaces.

Findings

Weakly reversible non-reversible spaces are disjoint from certain sequential spaces.

Strongly reversible spaces are limited to discrete, antidiscrete, and cofinite-like topologies.

Abstract

A topological space ${\mathcal X}$ is reversible iff each continuous bijection (condensation) $f: {\mathcal X} \rightarrow {\mathcal X}$ is a homeomorphism; weakly reversible iff whenever ${\mathcal Y}$ is a space and there are condensations $f:{\mathcal X} \rightarrow {\mathcal Y}$ and $g:{\mathcal Y} \rightarrow {\mathcal X}$, there is a homeomorphism $h:{\mathcal X} \rightarrow {\mathcal Y}$; strongly reversible iff each bijection $f: {\mathcal X} \rightarrow {\mathcal X}$ is a homeomorphism. We show that the class of weakly reversible non-reversible spaces is disjoint from the class of sequential spaces in which each sequence has at most one limit (containing e.g. metrizable spaces). On the other hand, the class of strongly reversible topologies contains only discrete topologies, antidiscrete topologies and natural generalizations of the cofinite topology.