Inserting \(L\)-fuzzy real-valued functions. (English) Zbl 0815.54006
In General Topology the Katětov-Tong insertion theorem has been obtained by some different techniques and the Tietze-Urysohn theorem as a simple consequence of it.
By considering Hutton’s \(L\)-fuzzy normality, the second author obtained, in a previous work, an \(L\)-fuzzy version of this insertion theorem by using the same method utilized by Katětov in the ordinary case. And the fuzzification of the Tietze-Urysohn theorem follows as a consequence.
In the present paper, the same results are obtained but with a different technique, used already in General Topology. \(L\) is assumed to be a frame with an order-reversing involution, but the infinite distribution law is not necessary in its full. It is sufficient to demand \(L\) to be meet- continuous with an order reversing involution. The advantage of the method used in this work is the fact that we can consider \(\sigma\)-rings of subsets instead of \(L\)-fuzzy topologies.
By considering Hutton’s \(L\)-fuzzy normality, the second author obtained, in a previous work, an \(L\)-fuzzy version of this insertion theorem by using the same method utilized by Katětov in the ordinary case. And the fuzzification of the Tietze-Urysohn theorem follows as a consequence.
In the present paper, the same results are obtained but with a different technique, used already in General Topology. \(L\) is assumed to be a frame with an order-reversing involution, but the infinite distribution law is not necessary in its full. It is sufficient to demand \(L\) to be meet- continuous with an order reversing involution. The advantage of the method used in this work is the fact that we can consider \(\sigma\)-rings of subsets instead of \(L\)-fuzzy topologies.
Reviewer: M.W.Warner (London)
MSC:
| 54A40 | Fuzzy topology |
References:
| [1] | Alexandrov, Mat. Sbornik (N.S.) 8 pp 307– (1940) |
| [2] | and , Introduction to Dimension Theory, Nauka, Moskow, 1973 (Russian) |
| [3] | Blair, Czechoslovak Math. J. 31 pp 63– (1981) |
| [4] | Bonan, C.R. Acad. Sc. Paris 270 pp 1226– (1970) |
| [5] | Chang, J. Math. Anal. Appl. 24 pp 182– (1968) |
| [6] | General Topology, Polish Sci. Publ., Warszawa, 1977 |
| [7] | Gantner, J. Math. Anal. Appl. 62 pp 547– (1978) |
| [8] | and , Rings of Continuous Functions, Springer-Verlag, New York, 1976 |
| [9] | Goguen, J. Math. Anal. Appl. 43 pp 734– (1973) |
| [10] | Hutton, J. Math. Anal. Appl. 50 pp 74– (1975) |
| [11] | Katetov, Fund. Math. 38 pp 85– (1951) |
| [12] | Fund. Math. 38 pp 40– (1953) |
| [13] | Katsaras, J. Math. Anal. Appl. 84 pp 44– (1981) |
| [14] | Lattice morphisms, sobriety, and Urysohn lemmas, in: Applications of Category Theory to Fuzzy Subsets (, and , Eds.), Kluwer Academic Publ., Dordrecht, 1992, pp. 258–274 · Zbl 0765.54005 |
| [15] | Kotzé, Rendiconti di Matematica (1) 4 pp 138– (1984) |
| [16] | and , Insertion of a measurable function, J. Austral. Math. Soc. (submitted) · Zbl 0851.54018 |
| [17] | Kubiak, Comment. Math. Univ. Carolinae 34 (1993) |
| [18] | Kubiak, J. Math. Anal. Appl. 125 pp 141– (1987) |
| [19] | Around I(L), in: First Joint IFSA-EC and EURO-WG Workshop on Progress in Fuzzy Sets in Europe, Nov. 25–27, 1986, Warsaw (J. Kacprzyk and A. Straszak, Eds.), Warszawa, 1988, pp. 175–179 |
| [20] | Kubiak, Math. Japon. 31 pp 875– (1986) |
| [21] | The topological modification of the L-fuzzy unit interval, in: Applications of Category Theory to Fuzzy Subsets (, and , Eds.), Kluwer Academic Publ., Dordrecht, 1992, pp. 276–305 · Zbl 0766.54006 |
| [22] | Insertion of a continuous function, Topology Proc. 4 (1979) 463–478 · Zbl 0443.54012 |
| [23] | Mrówka, Nieuw Archief voor Wiskunde (3) 16 pp 94– (1968) |
| [24] | Rodabaugh, J. Math. Anal. Appl. 131 pp 25– (1988) |
| [25] | Rodabaugh, Fuzzy Sets Syst. 11 pp 163– (1983) · Zbl 0525.54002 |
| [26] | Rodabaugh, Fuzzy Sets Syst. 40 pp 297– (1991) |
| [27] | Scott, Amer. Math. Monthly 85 pp 192– (1978) |
| [28] | Real Functions, Vol. I, P.W.N., Warszawa, 1958 (polish) |
| [29] | Speed, J. Austral. Math. Soc. 16 pp 185– (1973) |
| [30] | Sultan, J. Austral. Math. Soc 20 pp 359– (1975) |
| [31] | Tong, Duke Math J. 19 pp 289– (1952) |
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