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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 \(\mathbf{QUnif}\).

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206, 33100, Udine, Italy

    Dikran Dikranjan

  2. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, South Africa

    Hans-Peter A. Künzi

Authors
  1. Dikran Dikranjan
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  2. Hans-Peter A. Künzi
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Correspondence to Hans-Peter A. Künzi.

Additional information

Communicated by M. M. Clementino.

Dedicated to the memory of Sérgio de Ornelas Salbany.

The second author would like to thank the South African National Research Foundation for partial financial support (CPR20110610000019344 and IFR1202200082) as well as Programma SIR 2014 by MIUR (Project GADYGR, Number RBSI14V2LI) partially supporting his visits to Udine University.

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Dikranjan, D., Künzi, HP.A. Cowellpoweredness of Some Categories of Quasi-Uniform Spaces. Appl Categor Struct 26, 1159–1184 (2018). https://doi.org/10.1007/s10485-018-9523-5

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