University of Hertfordshire

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Topological transformation monoids

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Original languageEnglish
Number of pages21
JournalFundamenta Mathematicae
Publication statusSubmitted - 1 Aug 2019

Abstract

We investigate semigroup topologies on the full transformation monoid ΩΩ of an
infinite set Ω. We show that the standard pointwise topology is the weakest Hausdorff
semigroup topology on ΩΩ, show that this topology is the unique Hausdorff semigroup
topology on ΩΩ that induces the pointwise topology on the group Sym(Ω) of all permutations of Ω, and construct |Ω| distinct Hausdorff semigroup topologies on ΩΩ. In
the case where Ω is countable, we prove that the pointwise topology is the only Polish
semigroup topology on ΩΩ. We also show that every separable semigroup topology on

Ω is perfect, describe the compact sets in an arbitrary Hausdorff semigroup topology on ΩΩ, and show that there are no locally compact perfect Hausdorff semigroup
topologies on ΩΩ when |Ω| has uncountable cofinality.

ID: 19471568