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Topologizing homeomorphism groups of rim-compact spaces. (English) Zbl 1116.54010
Let \(X\) be a Tychonoff space, \(H(X)\) the group of all self-homeomorphisms of \(X\) with the usual composition and \(e:(f,x)\in H(X)\times X\rightarrow f(x)\in X\) the evaluation function. Topologies on \(H(X)\) providing continuity of the evaluation function are called \(admissible\). Topologies on \(H(X)\) compatible with the group operations are called \(group\) \(topologies\). Whenever \(X\) is locally compact \(T_2\), the minimum among all admissible group topologies on \(H(X)\) exists. The aim of this paper is to find conditions on \(X\) not involving local compactness for the existence of a minimal admissible group topology on \(H(X)\). First the authors show that whenever \(\gamma (X)\) is a \(T_2\)-compactification of \(X\) to which any self-homeomorphism of \(X\) continuously extends then the relativization to \(H(X)\) of the compact-open topology in \(H(\gamma(X))\) is an admissible group topology whose two-sided uniformity is complete. Then the authors ask if one can weaken local compactness into rim-compactness and they show that rim-compactness alone is not enough to assure the admissible group topology to be minimal. As a counterexample they consider the space \(\mathbb N\) of natural numbers whose Freudenthal compactification agrees with Stone-Čech compactification. Then they show that whenever \(X\) agrees with the rational number space \(\mathbb Q\) equipped with the Euclidean topology, then the admissible group topology induced on \(H(\mathbb Q)\) by \(F(\mathbb Q)=\beta \mathbb Q\), which is just the closed-open topology, realizes the minimum.

MSC:
54C35 Function spaces in general topology
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
54H99 Connections of general topology with other structures, applications
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