Topologization of sets endowed with an action of a monoid

Original · EN

Given a set X and a family G of self-maps of X, we study the problem of the existence of a non-discrete Hausdorff topology on X with respect to which all functions f∈ G are continuous. A topology on X with this property is called a G-topology. The answer is given in terms of the Zariski G-topology ζG on X, that is, the topology generated by the subbase consisting of the sets {x∈ X:f(x)≠ g(x)} and {x∈ X:f(x)≠ c}, where f,g∈ G and c∈ X. We prove that, for a countable monoid G⊂ Xˣ, X admits a non-discrete Hausdorff G-topology if and only if the Zariski G-topology ζG is non-discrete; moreover, in this case, X admits 2ᶜ hereditarily normal G-topologies.