Topologizations of a set endowed with an action of a monoid

摘要

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.