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Ultracompanions of subsets of a group

I. Protasov; S. Slobodianiuk — 2014

Commentationes Mathematicae Universitatis Carolinae

Let $G$ be a group, $β G$ be the Stone-Čech compactification of $G$ endowed with the structure of a right topological semigroup and $G^{*} = β G ∖ G$ . Given any subset $A$ of $G$ and $p \in G^{*}$ , we define the $p$ -companion $Δ_{p} (A) = A^{*} \cap G p$ of $A$ , and characterize the subsets with finite and discrete ultracompanions.

Almost all submaximal groups are paracompact and σ-discrete

O. Alas; I. Protasov; M. Tkačenko; V. Tkachuk; R. Wilson; I. Yaschenko — 1998

Fundamenta Mathematicae

We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.

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