Epimorphisms and cowellpoweredness of epireflective subcategories of TOP. (English) Zbl 0588.54017

Dikranjan, D.; Giuli, E.

Suppl. Rend. Circ. Mat. Palermo, II. Ser. 6, 121-136 (1984).

For an epireflective subcategory C of TOP and \(X\in TOP\) let \(F_ C(X)\) be the topological space obtained by supplying the underlying set of X with the topology for which the collection of all C-closed subsets of X forms a subbase for the closed sets, where \(A\subset X\) is C-closed iff for each \(x\in X\setminus A\) there are continuous mappings g,h: \(X\to Z\), \(Z\in C\), such that \(g| A=h| A\) and g(x)\(\neq h(x)\). Since the continuity of \(f: X\to Y\) implies the continuity of \(F_ C(f)=f: F_ C(X)\to F_ C(Y)\), for each epireflective subcategory \(\subseteq\) of TOP a functor \(F_ C: TOP\to TOP\) is obtained. It is shown how these functors can be used to describe the epimorphisms in some familiar epireflective subcategories of TOP. Moreover, they turn out to be useful to prove that certain subcategories of TOP are not cowellpowered.

Reviewer: H.Brandenburg

MSC:

54B30	Categorical methods in general topology
18B30	Categories of topological spaces and continuous mappings (MSC2010)
54D10	Lower separation axioms (\(T_0\)–\(T_3\), etc.)

Keywords:

cowellpoweredness; epireflective subcategory; C-closed subsets