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Epimorphisms in categories of separated fuzzy topological spaces. (English) Zbl 0812.54007

Considering Salbany-type closure operator [S. Salbany, Categor. Topol., Proc. Conf. Mannheim 1975; Lect. Notes Math. 540, 548-565 (1976; Zbl 0335.54003)], the authors characterize the epimorphisms in three full subcategories of \(FTS\) (= the category of fuzzy topological spaces and continuous mappings) namely, \(FTS_ 0\) of \(0^*\)-\(T_ 0\)-spaces of P. Wuyts and R. Lowen [Fuzzy Sets Syst. 19, 51-80 (1986; Zbl 0606.54004)], \(FTS_ 1(=FT_ S)\) of \(T_ 1\)-spaces of M. H. Ghanim, E. E. Kerre and A. S. Mashhour [J. Math. Anal. Appl. 102, 189-202 (1984; Zbl 0543.54006)] and \(FTS_{\alpha_ 2}\) of \(\alpha\)-\(T_ 2\)-spaces of S. E. Rodabaugh [Topology Appl. 11, 319-334 (1980; Zbl 0484.54008)].
In dealing with the subcategory \(FTS_ 0\), the authors construct a nice result under Proposition 2.4. Some examples in relation to the work of previous authors are also mentioned in this interesting article.

MSC:

54A40 Fuzzy topology
18B30 Categories of topological spaces and continuous mappings (MSC2010)
54B30 Categorical methods in general topology
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[1] Adámek, J.; Herrlich, H.; Strecker, G. E., Abstract and Concrete Categories (1990), Wiley: Wiley New York · Zbl 0695.18001
[2] Alderton, I. W., Function spaces in fuzzy topology, Fuzzy Sets and Systems, 32, 115-124 (1989) · Zbl 0691.54004
[3] Baron, S., A note on epi in \(T_0\), Canad. Math. Bull., 11, 503-504 (1968) · Zbl 0167.20702
[4] Castellini, G., Closure operators, monomorphisms and epimorphisms in categories of groups, Cahiers Topologie Géom. Différentielle Catégoriques, 27, 2, 151-167 (1986) · Zbl 0592.18001
[5] G. Castellini and D. Hajek, Closure operators and connectedness, Preprint.; G. Castellini and D. Hajek, Closure operators and connectedness, Preprint. · Zbl 0791.54001
[6] G. Castellini, J. Koslowski and G.E. Strecker, Closure operators and polarities, in progress.; G. Castellini, J. Koslowski and G.E. Strecker, Closure operators and polarities, in progress. · Zbl 0815.18001
[7] Chang, C. L., Fuzzy topological spaces, J. Math. Anal. Appl., 24, 182-190 (1968) · Zbl 0167.51001
[8] Dikranjan, D.; Giuli, E., Closure operators induced by topological epireflections, Coll. Math. Soc. J. Bolyai, 41, 233-246 (1983) · Zbl 0601.54016
[9] Dikranjan, D.; Giuli, E., Closure operators I, Topology Appl., 27, 129-143 (1987) · Zbl 0634.54008
[10] Dikranjan, D.; Guili, E.; Tholen, W., Closure operators II, (Proceedings of the Conference in Categorical Topology. Proceedings of the Conference in Categorical Topology, Prague, 1988 (1989), World Scientific: World Scientific Singapore), 297-335
[11] Ghanim, M. H.; Kerre, E. E.; Mashhour, A. S., Separation axioms, subspaces and sums in fuzzy topology, J. Math. Anal. Appl., 102, 189-202 (1984) · Zbl 0543.54006
[12] Giuli, E., Bases of topological epireflections, Topology Appl., 27, 265-273 (1980) · Zbl 0441.18012
[13] Koslowski, J., Closure operators with prescribed properties, (Category Theory and its Applications, 1248 (1988)), 208-220
[14] Lowen, R., Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl., 56, 621-633 (1976) · Zbl 0342.54003
[15] Lowen, R., Initial and final fuzzy topologies and the fuzzy tychonoff theorem, J. Math. Anal. Appl., 58, 11-21 (1977) · Zbl 0347.54002
[16] Lowen, R., On the Existence of Natural Non-topological Fuzzy Topological Spaces (1985), Heldermann: Heldermann Berlin · Zbl 0568.54007
[17] Lowen, R.; Srivastava, A. K., \(FTS_0\): The epireflective hull of the Sierpinski object in FTS, Fuzzy Sets and Systems, 29, 171-176 (1989) · Zbl 0658.54005
[18] Nel, L. D.; Wilson, R. G., Epireflections in the category of \(T_0\) spaces, Fund. Math., 75, 69-74 (1972) · Zbl 0232.54018
[19] Rodabaugh, S. E., The Hausdorff separation axiom for fuzzy topological spaces, Topology Appl., 11, 319-334 (1980) · Zbl 0484.54008
[20] Salbany, S., Reflective subcategories and closure operators, (1975. 1975, Proceedings of the Conference in Categorical Topology, Mannheim, 1975, 540 (1976), Springer L.N.M), 548-565
[21] Srivastava, A. K.; Srivastava, R., Fuzzy Sierpinski space: Another note, Fuzzy Math., 3, 99-103 (1985) · Zbl 0589.54010
[22] Wuyts, P.; Lowen, R., On local and global measures of separation in fuzzy topological spaces, Fuzzy Sets and Systems, 19, 51-80 (1986) · Zbl 0606.54004
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