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Topologizations of a set endowed with an action of a monoid. (English) Zbl 1294.54020

In the article under review the authors explore the following problem: given a set \(X\) endowed with the action \(G \times X \to X\), \((g,x) \mapsto gx\) of a monoid \(G\), find conditions under which \(X\) admits a non-discrete Hausdorff \(G\)-topology, i.e., a topology with respect to which all shifts \(g: X\to X\), \(g: x \mapsto gx\) are continuous. This problem goes back to Markov’s criterion for topologizability of countable groups [A. A. Markov, Mat. Sb. 18, No. 60, 3–28 (1946; Zbl 0061.04208)]. For actions over a group this problem has been considered by the first two authors of the paper under review in [T. Banakh and I. Protasov, J. Math. Sci., New York 188, No. 2, 77–84 (2013; Zbl 1347.54052)].
The Zariski \(G\)-topology \(\zeta_G\) on a \(G\)-act \(X\) is determined by the subbase \(\tilde{\zeta}_G\) of open sets \(\{x\in X: fx\neq gx\}\), \(\{x\in X: fx \neq c\}\), where \(f,g\in G\), \(c\in X\). The main result (Theorem 5.1) gives conditions under which a \(G\)-act \(X\) over a monoid \(G\) admits hereditary normal \(G\)-topologies. Namely, for every \(G\)-act \(X\) over a monoid \(G\) such that \(|G| \leq \psi(x_0, \zeta_G)\), where \(\psi(x_0, \zeta_G)\) is a pseudocharacter of a family \(\tilde{\zeta}_G\) at a point \(x_0\in X\), and for every infinite cardinal \(\kappa\), \(|G| \leq \kappa \leq \psi (x_0, \zeta_G)\) and every infinite cardinal \(\lambda \leq \text{cf} (\kappa)\), the \(G\)-act admits \(2^{2^{\kappa}}\) hereditary normal \(G\)-topologies with pseudocharacter \(\lambda\) at the point \(x_0\). The criterion (Theorem 5.5) for the \(G\)-topologizability of a given \(G\)-act \(X\) in the case of countable monoids follows: a countable monoid \(G\) admits a non-discrete Hausdorff \(G\)-topology if and only if the Zariski \(G\)-topology \(\zeta_G\) is non-discrete.

MSC:

54H15 Transformation groups and semigroups (topological aspects)
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