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Group action on zero-dimensional spaces. (English) Zbl 1118.54011
For a Tychonoff space \(X\), a group topology on the group \(\mathcal H(X)\) of self-homeomorphisms is admissible provided that the evaluation map \(\mathcal H(X)\times X\to X\) is continuous. It is shown that if \(X\) is the product of a family of zero-dimensional spaces in which any two non-empty clopen subspaces of each factor are homeomorphic then the collection of admissible group topologies on \(X\) is a complete lattice. The minimum element is identified and is a familiar compactification in some cases.

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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