On cofree S-spaces and cofree S-flows

Abstract

[EN] Let S-Tych be the category of Tychonoff S-spaces for a topological monoid S. We study the cofree S-spaces and cofree S-flows over topological spaces and we prove that for any topological space X and a topological monoid S, the function space C(S,X) with the compact-open topology and the action s · f = (t → f(st)) is the cofree S-space over X if and only if the compact-open topology is admissible and Tychonoff. Finally we study injective S-spaces and we characterize injective cofree S-spaces, when the compact-open topology is admissible and Tychonoff. As a consequence of this result, we characterize the cofree S-spaces and cofree S-flows, when S is a locally compact topological monoid.