Cowellpoweredness of Some Categories of Quasi-Uniform Spaces

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 $$\mathbf{QUnif}$$ .