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On homeomorphism groups and the set-open topology

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 $\mathcal{H}(βX)$ 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.