Inclusions preserving epimorphisms. (English) Zbl 0820.54003

Dikranjan, Dikran; Giuli, Eraldo

Nuriev, B. R. (ed.), Baku international topological conference held at Baku (USSR), October 3-9, 1987. Proceedings. Baku: Ehlm, 220-230 (1989).

R.-E. Hoffmann [Lect. Notes Math. 915, 148-170 (1982; Zbl 0509.18004)], in a more general context, proved that for every reflective subcategory \({\mathbf R}\) of \({\mathbf {Top}}\) there exists a largest epireflective subcategory \({\mathcal D} ({\mathbf R})\) of \({\mathbf {Top}}\), such that \({\mathbf R}\) is epireflective in \({\mathcal D} ({\mathbf R})\) and the inclusion \({\mathbf R}\to {\mathcal D} ({\mathbf R})\) preserves the epimorphisms. In the case when \({\mathbf R}= {\mathbf {HComp}}\) (the subcategory of all compact Hausdorff spaces) he showed that \({\mathcal D} ({\mathbf R})\) coincides with the subcategory consisting of the spaces \(X\) such that, for each \(K\in {\mathbf {HComp}}\) and each map \(f: K\to X\times X\), \(f^{-1} (\Delta_ X)\) is closed in \(K\). He expressed the hope that this may hold true for all closed-hereditary reflective \({\mathbf R} \subset {\mathbf {Top}}\). In particular he asked if there \({\mathcal D} ({\mathbf {Haus}}) = {\mathcal D} ({\mathbf {HComp}})\) (\({\mathbf {Haus}}\) – the category of Hausdorff spaces) and the precise relation between \({\mathcal D} ({\mathbf {HComp}})\) and other subcategories of \({\mathbf {Top}}\) for which a better topological description is known [see also R.-E. Hoffmann, Arch. Math. 32, 487-504 (1979; Zbl 0463.54016), 4.2].
In this paper we give a description of \({\mathcal D} ({\mathbf R})\) in terms of \({\mathbf R}\)-closure, introduced in [S. Salbany, Lect. Notes Math. 540, 548-565 (1976; Zbl 0335.54003)] and studied in [the authors, Colloq. Math. Soc. János Bolyai 41, 233-246 (1985; Zbl 0601.54016), Rend. Circ. Mat. Palermo, II. Ser. Suppl. 6, 121-136 (1984; Zbl 0588.54017), Commentat. Math. Univ. Carol. 27, 395-417 (1986; Zbl 0627.54005), Topology Appl. 28, 59-74 (1988; Zbl 0658.54015); the second author with M. Hušek, Ann. Mat. Pura Appl., IV. Ser. 145, 337-346 (1986; Zbl 0617.54006) and A. Tozzi, Rend. Circ. Mat. Palermo, II. Ser., Suppl. 12, 291-300 (1986; Zbl 0599.54016)]. This enables us to carry out the program of R.-E. Hoffmann [Lect. Notes Math. 915, loc. cit.] by replacing closedness in the above characterization of \({\mathcal D} ({\mathbf {HComp}})\) by closedness with respect to an appropriate closure operator determined by \({\mathbf R}\). Making use of this description we compute \({\mathcal D} ({\mathbf R})\) in many cases, and in particular we show that \({\mathcal D} ({\mathbf {Haus}})= {\mathbf {Haus}}\).
In the last part of the paper examples are provided to answer negatively the questions posed in [R.-E. Hoffmann, Arch. Math., loc. cit.].
For the entire collection see [Zbl 0742.00083].

MSC:

54B30	Categorical methods in general topology
18B30	Categories of topological spaces and continuous mappings (MSC2010)

Keywords:

epireflective subcategory

Citations:

Zbl 0509.18004; Zbl 0463.54016; Zbl 0335.54003; Zbl 0601.54016; Zbl 0588.54017; Zbl 0627.54005; Zbl 0658.54015; Zbl 0617.54006; Zbl 0599.54016