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        Topological transformation monoids

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        Author
        Mesyan, Zak
        Mitchell, James D.
        Peresse, Yann
        Attention
        2299/23926
        Abstract
        We investigate semigroup topologies on the full transformation monoid T(X) of an infinite set X. We show that the standard pointwise topology is the weakest Hausdorff semigroup topology on T(X), show that the pointwise topology is the unique Hausdorff semigroup topology on T(X) that induces the pointwise topology on the group of all permutations of X, and construct |X| distinct Hausdorff semigroup topologies on T(X). In the case where X is countable, we prove that the pointwise topology is the only Polish semigroup topology on T(X). We also show that every separable semigroup topology on T(X) is perfect, describe the compact sets in an arbitrary Hausdorff semigroup topology on T(X), and show that there are no locally compact perfect Hausdorff semigroup topologies on T(X) when |X| has uncountable cofinality.
        Publication date
        2018-09-12
        License
        Unspecified
        Other links
        http://hdl.handle.net/2299/23926
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