Weakly and Strongly Reversible Spaces

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.