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Alexandroff topologies and monoid actions. (English) Zbl 1440.54001

In this paper, the statement “Any Alexandroff topology may be obtained through a monoid action” is provided. Several topological properties for Alexandroff spaces bearing in mind specific examples of monoid actions are introduced. A specific notion of dependence based on the union of subsets is introduced for an Alexandroff space \(X\) with associated topological closure operator \(\sigma \). The authors study the family \(\mathcal{A}_{\sigma ,X}\) of closed subsets \(Y\) of \(X\) such that, for any \(y_{1}\), \(y_{2}\in Y\), there exists a third element \(y\in Y\) whose closure contains both \(y_{1}\) and \(y_{2}\). A decomposition theorem regarding an Alexandroff space as the union (not necessarily disjoint) of a pair of closed subsets characterized by such a dependence is provided.

MSC:

54A05 Topological spaces and generalizations (closure spaces, etc.)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
20M30 Representation of semigroups; actions of semigroups on sets
20M15 Mappings of semigroups
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