The compact-open topology for semigroups of continuous selfmaps (Q1086365)

All spaces in this discussion are assumed to be Hausdorff. Let \(S(X)\), the semigroup of all continuous selfmaps of the topological space \(X\) have the compact-open topology. It is well known that \(S(X)\) is a topological semigroup if \(X\) is locally compact. It does not seem to be known that the converse is ``almost true'' in the sense that there is an extensive class of spaces with the property that for any such space \(X\), if \(S(X)\), with the compact-open topology, is a topological semigroup, then \(X\) must necessarily be locally compact. In the main result, we establish this fact and we discuss an example which shows that there do exist spaces \(X\) such that \(S(X)\) is a topological semigroup in spite of the fact that \(X\) is not locally compact.