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

2022/10/31 by J. de la Nuez González, González, J. de la Nuez
Mathematics · #57N99 #57S05 #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2210.17280

openalex publication_date 2022/10/31 · openalex created_date 2022/11/07 · openalex updated_date 2026/07/28

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.

Related

• The compact-open topology on the diffeomorphism or homeomorphism group of a smooth manifold without boundary is minimal in almost all dimensions
• Minimum Topological Group Topologies
• Non-Roelcke precompactness of groups of surface homeomorphisms
• Order and minimality of some topological groups
• On homeomorphism groups and the set-open topology