Cowellpoweredness of Some Categories of Quasi-Uniform Spaces

摘要

We study cowellpoweredness in the category QUnif of quasi- uniform spaces and uniformly continuous maps. A full subcategory A of QUnif is cowellpowered when the cardinality of the codomains of any class of epimorphisms inA, 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 A of 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 Top of topological spaces and continuous maps. The transfer is realized by lifting these subcategories along the forgetful functor T : QUnif. Top and studying when epimorphisms and cowellpoweredness are preserved by the lifting. In other cases closure operators of QUnif are used to provide specific results for QUnif that have no counterpart in 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 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 QUnif which is the intersection of two " symmetric" cowellpowered full subcategories of QUnif.