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 ζ G 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 ⊂ X X , X admits a non-discrete Hausdorff G-topology if and only if the Zariski G-topology ζ G is non-discrete; moreover, in this case, X admits 2 c hereditarily normal G-topologies.