Group action on R×Q and fine group topologies

Let X be a Tychonoff space, H(X) the group of all self-homeomorphisms of X and the evaluation map. Let LH(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 LH(X) has been proven for T2 locally compact spaces, for T2 rim-compact and locally connected spaces and for products of T2 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 LH(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 LH(R×Q) admits a least element.