Cowellpoweredness of Some Categories of Quasi-Uniform Spaces

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Abstract

We study cowellpoweredness in the category \(\mathbf{QUnif}\) of quasi-uniform spaces and uniformly continuous maps. A full subcategory \(\mathcal{A}\) of \(\mathbf{QUnif}\) is cowellpowered when the cardinality of the codomains of any class of epimorphisms in \(\mathcal{A}\), with a fixed common domain, is bounded. We use closure operators in the sense of Dikranjan–Giuli–Tholen which provide a convenient tool for describing the subcategories \(\mathcal{A}\) of \(\mathbf{QUnif}\) and their epimorphisms. Some of the results are obtained by using the knowledge of closure operators, epimorphisms and cowellpoweredness of subcategories of the category \(\mathbf{Top}\) of topological spaces and continuous maps. The transfer is realized by lifting these subcategories along the forgetful functor \(T{:}\mathbf{QUnif}\rightarrow \mathbf{Top}\) and studying when epimorphisms and cowellpoweredness are preserved by the lifting. In other cases closure operators of \(\mathbf{QUnif}\) are used to provide specific results for \(\mathbf{QUnif}\) that have no counterpart in \(\mathbf{Top}\). This leads to a wealth of cowellpowered categories and a wealth of non-cowellpowered categories of quasi-uniform spaces, in contrast with the current situation in the case of the smaller category \(\mathbf{Unif}\) of uniform spaces, where no example of a non-cowellpowered subcategory is known so far. Finally, we present our main example: a non-cowellpowered full subcategory of \(\mathbf{QUnif}\) which is the intersection of two “symmetric” cowellpowered full subcategories of \(\mathbf{QUnif}\).

Keywords

Quasi-uniform space Uniform space (Regular, semiregular) closure operator \(\theta \)-closure Sequential closure \(S(\alpha )\)-space \(\varrho \)-separated space Epimorphism cowellpoweredness Cowellpowered category 

Mathematics Subject Classification

Primary 18A20 54B30 54D10 54E15 Secondary 18B30 54B17 

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Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità di UdineUdineItaly
  2. 2.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

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