Topologization of sets endowed with an action of a monoid

TL;DR

This paper investigates conditions under which a set with a monoid action admits a non-discrete Hausdorff topology making all monoid functions continuous, introducing the Zariski G-topology as a key tool.

Contribution

It characterizes the existence of non-discrete Hausdorff G-topologies using the Zariski G-topology and shows the abundance of such topologies for countable monoids.

Findings

A non-discrete Hausdorff G-topology exists iff the Zariski G-topology is non-discrete.

Countable monoids admit $2^{ ext{cardinality}}$ hereditarily normal G-topologies.

The Zariski G-topology is generated by sets distinguishing points via G-functions.

Abstract

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 $f\in G$ 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 $\zeta_G$ on $X$, that is, the topology generated by the subbase consisting of the sets $\{x\in X:f(x)\ne g(x)\}$ and $\{x\in X:f(x)\ne c\}$, where $f,g\in G$ and $c\in X$. We prove that, for a countable monoid $G\subset X^X$, $X$ admits a non-discrete Hausdorff $G$-topology if and only if the Zariski $G$-topology $\zeta_G$ is non-discrete; moreover, in this case, $X$ admits $2^{\mathfrak c}$ hereditarily normal $G$-topologies.