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Keywords:
factorization system; closure operator; separation axioms; prereflection; reflection
factorization system; closure operator; separation axioms; prereflection; reflection
Summary:
In an abstract category with suitable notions of subobject, closure and point, we discuss the separation axioms $T_0$ and $T_1$. Each of the arising subcategories is reflective. We give an iterative construction of the reflectors and present characteristic examples.
References:
[1] Arhangel'skiĭ, Wiegandt R.: Connectednesses and disconnectednesses in topology. Topology Appl. 5 (1975), 9-33. MR 0367920
[2] Börger R.: Kategorielle Beschreibungen von Zusammenhangsbegriffen. Ph.D. Thesis, Fernuniversität Hagen, 1981.
[3] Brümmer G.C.L.: A categorical study of initiality in uniform topology. Ph.D. Thesis, University of Cape Town, 1971.
[4] Čech E.: Topological spaces. revised by Z. Frolík and M. Katětov (Academia, Praha, 1966). MR 0211373
[5] Clementino M.M.: Ph.D. Thesis, Universidade de Coimbra (in preparation).
[6] Dikranjan D., Giuli E.: Closure operators I. Topology Appl. 27 (1987), 129-143. MR 0911687 | Zbl 0634.54008
[7] Dikranjan D., Giuli E., Tholen W.: Closure operators II, Categorical topology and its relations to analysis, algebra and combinatorics. Proc. Int. Conf. Prague, 1988 (World Scientific, Singapore-New Jersey-London-Hong Kong), 297-335. MR 1047909
[8] Giuli E.: Bases of topological epireflections. Topology Appl. 11 (1980), 265-273. MR 0585271
[9] Harvey J.M.: $T_0$-separation in topological categories. Quaest. Mathematicae 2 (1977), 177-190. MR 0486050 | Zbl 0384.18002
[10] Hoffmann R.-E.: $(E,M)$-universally topological functors. Habilitationsschrift, Universität Düsseldorf, 1974.
[11] Hušek M., Pumplün D.: Disconnectedness. Quaest. Mathematicae 13 (1990), 449-459. MR 1084754
[12] Hušek M., Pumplün D.: Note on separation in categories. preprint, 1991. MR 1276007
[13] MacLane S.: Categories for the working mathematician. Springer-Verlag, Berlin-Heidelberg- New York, 1971. MR 0354798 | Zbl 0705.18001
[14] Mantovani S.: Connessione e totale reflessività. Ph.D. Thesis, Università di Milano, 1988.
[15] Marny T.: On epireflective subcategories of topological categories. Gen. Top. Appl. 10 (1979), 175-181. MR 0527843 | Zbl 0415.54007
[16] Preuß G.: Connection properties in topological categories and related topics. Lecture Notes in Math. 719 (Springer-Verlag, Berlin-Heidelberg-New York, 1979), 293-305. MR 0544654
[17] Salicrup G., Vasquez R.: Connection and disconnection. Lecture Notes in Math. 719 (SpringerVerlag, Berlin-Heidelberg-New York, 1979), 326-344. MR 0544657
[18] Tholen W.: Factorizations, localizations and the orthogonal subcategory problem. Math. Nachr. 114 (1983), 63-85. MR 0745048 | Zbl 0553.18003
[19] Tholen W.: Factorizations, fibres and connectedness. Categorical Topology, Proc. Conference Toledo, Ohio, 1983 (Heldermann Verlag, Berlin, 1984), 549-566. MR 0785035 | Zbl 0547.18001
[20] Tholen W.: Prereflections and reflections. Comm. Algebra 14 (1986), 717-740. MR 0830816 | Zbl 0587.18002
[21] Tiller J.A.: Component subcategories. Quaest. Mathematicae 4 (1980), 19-40. MR 0573837 | Zbl 0431.18001
[22] Weck-Schwarz S.: Cartesish abgeschlossene monotopologische Hüllen und die Rolle der $T_0$-Reflexionen. Ph.D. Thesis, Universität Hannover, 1988.
[23] Weck-Schwarz S.: $T_0$-objects and separated objects in topological categories. Quaest. Mathematicae 14 (1991), 315-325. MR 1123910
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