The compact-open topology on the homeomorphism group of a surface without boundary is minimal

TL;DR

This paper proves that the compact-open topology is the minimal Hausdorff topology on the homeomorphism group of a boundaryless surface, establishing its uniqueness under certain conditions.

Contribution

It demonstrates the minimality and uniqueness of the compact-open topology on the homeomorphism group of surfaces without boundary, extending known automatic continuity results.

Findings

The compact-open topology is minimal among Hausdorff group topologies.

It is the unique Hausdorff separable topology under specified surface conditions.

The result applies to surfaces with certain finite or Cantor set complements.

Abstract

We show that the homeomorphism group of a surface without boundary does not admit a Hausdorff group topology strictly coarser than the compact-open topology. In combination with known automatic continuity results, this implies that the compact-open topology is the unique Hausdorff separable group topology on the group if the surface is closed or the complement in a closed surface of either a finite set or the union of a finite set and a Cantor set.