Group action on and fine group topologies

Let X be a Tychonoff space, H ( X ) the group of all self-homeomorphisms of X and e : ( f , x ) ∈ H ( X ) × X → f ( x ) ∈ X the evaluation map. Let L H ( X ) be the upper-semilattice of all group topologies on H ( X ) with the additional property that the evaluation map is continuous (ordered by inclusion). The existence of a least element in L H ( X ) has been proven for T 2 locally compact spaces, for T 2 rim-compact and locally connected spaces and for products of T 2 zero-dimensional spaces satisfying the property: any two non-empty clopen subspaces are homeomorphic. We show that X being rim-compact is not a necessary condition in order for L H ( X ) to have a least element. Let R and Q be the sets of the real and rational numbers respectively, both carrying the Euclidean topology. It is known that R × Q is not rim-compact. We prove that L H ( R × Q ) admits a least element.