Sober approach spaces are firmly reflective for the class of epimorphic embeddings. (English) Zbl 1120.54009
From the abstract: The main result of this paper states that the subconstruct of sober approach spaces, introduced by B. Banaschewski, R. Lowen and C. Van Olmen [Topology Appl. 153, 3059–3070 (2006; Zbl 1114.54007)], is firmly \(\mathcal U\)-reflective in the construct of \(T_0\) approach spaces for the class \(\mathcal U\) of epimorphic embeddings. This result generalizes the fact that the subconstruct of sober topological spaces is firmly reflective for the class of b-dense embeddings in the construct of \(T_0\) spaces.
Reviewer: Dexue Zhang (Chengdu)
MSC:
| 54B30 | Categorical methods in general topology |
| 18A40 | Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) |
| 18B35 | Preorders, orders, domains and lattices (viewed as categories) |
Citations:
Zbl 1114.54007References:
| [1] | Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. Wiley, New York (1990) · Zbl 0695.18001 |
| [2] | Banaschewski, B., Lowen, R., Van Olmen, C.: Sober approach spaces. Topology Appl. to appear |
| [3] | Brümmer, G.C.L., Giuli, E.: A categorical concept of completion of objects. Comment. Math. Univ. Carolin. 33(1), 131–147 (1992) · Zbl 0760.18005 |
| [4] | Brümmer, G.C.L., Giuli, E., Herrlich, H.: Epireflections which are completions. Cahiers Topologie Géom. Différentielle Catég. XXXIII, 71–93 (1992) · Zbl 0760.18004 |
| [5] | Claes, V., Colebunders, E., Gerlo, A.: Epimorphisms and Cowellpoweredness for separated metrically generated theories. Acta Math. Hungar. to appear · Zbl 1121.54023 |
| [6] | Giuli, E.: Zariski closure, completeness and compactness. Topology Appl. to appear · Zbl 1105.18003 |
| [7] | Hoffmann, R.-E.: Charakterisierung nüchterner Räume. Manuscr. Math. 15(2), 185–191 (1975) · Zbl 0306.54027 · doi:10.1007/BF01213787 |
| [8] | Johnstone, P.T.: Stone Spaces, Cambridge Studies in Advanced Math. 3. Cambridge University Press, Cambridge (1982) |
| [9] | Lowen, R.: Approach Spaces: The Missing Link in the Topology–Uniformity–Metric Triad, Oxford Mathematical Monographs. Oxford University Press (1997) · Zbl 0891.54001 |
| [10] | Pultr, A.: Frames. In: Hazewinkel, M. (ed.) Handbook of Algebra, vol. 3, pp. 791–857. Elsevier (2003) |
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