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# Regular Homeomorphisms of Finite Order on Countable Spaces

Published:2009-06-01
Printed: Jun 2009
• Yevhen Zelenyuk
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## Abstract

We present a structure theorem for a broad class of homeomorphisms of finite order on countable zero dimensional spaces. As applications we show the following. \begin{compactenum}[\rm(a)] \item Every countable nondiscrete topological group not containing an open Boolean subgroup can be partitioned into infinitely many dense subsets. \item If $G$ is a countably infinite Abelian group with finitely many elements of order $2$ and $\beta G$ is the Stone--\v Cech compactification of $G$ as a discrete semigroup, then for every idempotent $p\in\beta G\setminus\{0\}$, the subset $\{p,-p\}\subset\beta G$ generates algebraically the free product of one-element semigroups $\{p\}$ and~$\{-p\}$. \end{compactenum}
 Keywords: Homeomorphism, homogeneous space, topological group, resolvability, Stone-\v Cech compactification
 MSC Classifications: 22A30 - Other topological algebraic systems and their representations 54H11 - Topological groups [See also 22A05] 20M15 - Mappings of semigroups 54A05 - Topological spaces and generalizations (closure spaces, etc.)

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