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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2011
- DOI:
- 10.48550/arXiv.1112.5729
- arXiv:
- arXiv:1112.5729
- Bibcode:
- 2011arXiv1112.5729B
- Keywords:
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- Mathematics - General Topology
- E-Print:
- 10 pages