A note on Hausdorff groups

Abstract

It is well known that in the category of Hausdorff groups continuous homomorphisms having dense image are epic. That these homomorphisms are precisely the epics is a conjecture. If one could prove that proper closed subgroups (endowed with the subspace topology) cannot have epic inclusions, then the conjecture would be shown to be true. Closed subgroups fall into two classifications:(a) open (and hence closed) subgroups and(b) nowhere dense closed subgroups.In this note it is shown that proper open subgroups cannot have epic inclusion. To do so an algebraic construction of Peter J. Hilton is topologized. The nowhere dense case remains open.