On homeomorphism groups and the set-open topology

TL;DR

This paper investigates the set-open topologies on homeomorphism groups, establishing their properties, generalizing Dijkstra's theorem, and relating zero-cozero and compact-open topologies for various spaces.

Contribution

It extends the understanding of set-open topologies on homeomorphism groups and connects different topologies, providing new representations for homogeneous zero-dimensional spaces.

Findings

Zero-cozero topology equals the relativization of the compact-open topology for Tychonoff spaces.

Homogeneous zero-dimensional spaces can be represented as quotient spaces of topological groups.

Generalization of Dijkstra's theorem to broader classes of topologies.

Abstract

In this paper we focus on the set-open topologies on the group $\mathcal{H}(X)$ of all self-homeomorphisms of a topological space $X$ which yield continuity of both the group operations, product and inverse function. As a consequence, we make the more general case of Dijkstra's theorem. In this case a homogeneously encircling family $\mathcal{B}$ consists of regular open sets and the closure of every set from $\mathcal{B}$ is contained in the finite union of connected sets from $\mathcal{B}$. Also we proved that the zero-cozero topology of $\mathcal{H}(X)$ is the relativisation to $\mathcal{H}(X)$ of the compact-open topology of $\mathcal{H}(\beta X)$ for any Tychonoff space $X$ and every homogeneous zero-dimensional space $X$ can be represented as the quotient space of a topological group with respect to a closed subgroup.