Topologizations of a set endowed with an action of a monoid

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.