Group action on \(\mathbb R\times \mathbb Q\) and fine group topologies (Q1009745)

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 \(\mathbb R\times\mathbb Q\).