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Group action on $$\mathbb R\times \mathbb Q$$ and fine group topologies. (English) Zbl 1163.54011
Under a number of conditions on the base space $$X$$, for example T$$_2$$ rim-compact and locally connected, the lattice of admissible group topologies on the space $$\mathcal H(X)$$ of homeomorphisms, i.e. those topologies for which the evaluation $$\mathcal H(X)\times X\to X$$ is continuous, has a least element. A converse, whether non-rim-compact spaces may have this property, is answered by identifying a least element of the lattice of admissible topology on the non-rim-compact space $$\mathbb R\times\mathbb Q$$.

##### 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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