Topologization of sets endowed with an action of a monoid

Details Topologization of sets endowed with an action of a monoid Topologization of sets endowed with an action of a monoid

2011-12-24

arxiv.org/abs/1112.5729v1

Mathematics

General Topology

7 pages

Scientific paper

Given a set X and a family G of functions from X to X we pose and explore the question of the existence of a non-discrete Hausdorff topology on X such that all functions f in G are continuous. A topology on X with the latter property is called a G-topology. The answer will be given in terms of the Zariski G-topology \zeta_G on X. This 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 $f,g\in G$, $c\in X$. We prove that for a countable submonoid $G\subset X^X$ the $G$-act $X$ admits a non-discrete Hausdorff $G$-topology if and only if the Zariski $G$-topology $\zeta_G$ is not discrete if and only if $X$ admits $2^{\mathfrak c}$ normal $G$-topologies.

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