Topologization of sets endowed with an action of a monoid

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.

10 pages

References in corpus (5)

• The Markov-Zariski topology of an abelian group
• On topologizable and non-topologizable groups
• Reflection principle characterizing groups in which unconditionally closed sets are algebraic
• The number of non-solutions to an equation in a group and non-topologizable torsion-free groups
• Algebraically determined topologies on permutation groups

Cited by in corpus (3)

• Algebraically determined topologies on permutation groups
• Topological Transformation Monoids
• Hindman's finite sums theorem and its application to topologizations of algebras