On Homeomorphism Groups and the Set-Open Topology

Abstract

In this paper we focus on the set-open topologies on the group H(X) of all selfhomeomorphisms 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 B consists of regular open sets and the closure of every set from B is contained in the finite union of connected sets from B. Also we proved that the zero-cozero topology of H(X) is the relativisation to H(X) of the compact-open topology of H(βX) 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.