Article

 Title: A new form of fuzzy $\alpha$-compactness (English) Author: Shi, Fu-Gui Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 131 Issue: 1 Year: 2006 Pages: 15-28 Summary lang: English . Category: math . Summary: A new form of $\alpha$-compactness is introduced in $L$-topological spaces by $\alpha$-open $L$-sets and their inequality where $L$ is a complete de Morgan algebra. It doesn’t rely on the structure of the basis lattice $L$. It can also be characterized by means of $\alpha$-closed $L$-sets and their inequality. When $L$ is a completely distributive de Morgan algebra, its many characterizations are presented and the relations between it and the other types of compactness are discussed. Countable $\alpha$-compactness and the $\alpha$-Lindelöf property are also researched. (English) Keyword: $L$-topology Keyword: compactness Keyword: $\alpha$-compactness Keyword: countable $\alpha$-compactness Keyword: $\alpha$-Lindelöf property Keyword: $\alpha$-irresolute map Keyword: $\alpha$-continuous map MSC: 54A40 MSC: 54D35 idZBL: Zbl 1108.54009 idMR: MR2211000 . Date available: 2009-09-24T22:23:43Z Last updated: 2012-06-18 Stable URL: http://hdl.handle.net/10338.dmlcz/134081 . Reference: [1] H. Aygün: $\alpha$-compactness in $L$-fuzzy topological spaces.Fuzzy Sets Syst. 116 (2000), 317–324. MR 1792325 Reference: [2] K. K. Azad: On fuzzy semicontinuity, fuzzy almost continuity and fuzzy weak continuity.J. Math. Anal. Appl. 82 (1981), 14–32. MR 0626738 Reference: [3] S. Z. Bai: Fuzzy strongly semiopen sets and fuzzy strong semicontinuity.Fuzzy Sets Syst. 52 (1992), 345–351. Zbl 0795.54009, MR 1198195 Reference: [4] S. Z. Bai: The SR-compactness in $L$-fuzzy topological spaces.Fuzzy Sets Syst. 87 (1997), 219–225. Zbl 0912.54009, MR 1442416 Reference: [5] P. Dwinger: Characterization of the complete homomorphic images of a completely distributive complete lattice. I.Indagationes Mathematicae (Proceedings) 85 (1982), 403–414. Zbl 0503.06012, MR 0683528 Reference: [6] G. Gierz, et al.: A Compendium of Continuous Lattices.Springer, Berlin, 1980. Zbl 0452.06001, MR 0614752 Reference: [7] S. R. T. Kudri: Compactness in $L$-fuzzy topological spaces.Fuzzy Sets Syst. 67 (1994), 329–336. Zbl 0842.54011, MR 1304566 Reference: [8] S. G. Li, S. Z. Bai, N. Li: The near SR-compactness axiom in $L$-topological spaces.Fuzzy Sets Syst. 147 (2004), 307–316. MR 2089294 Reference: [9] Y. M. Liu, M. K. Luo: Fuzzy Topology.World Scientific, Singapore, 1997. MR 1643076 Reference: [10] R. Lowen: A comparison of different compactness notions in fuzzy topological spaces.J. Math. Anal. Appl. 64 (1978), 446–454. Zbl 0381.54004, MR 0497443 Reference: [11] S. N. Maheshwari, S. S. Thakur: On $\alpha$-irresolute mappings.Tamkang J. Math. 11 (1981), 209–214. MR 0696921 Reference: [12] S. N. Maheshwari, S. S. Thakur: On $\alpha$-compact spaces.Bull. Inst. Math. Acad. Sinica 13 (1985), 341–347. MR 0866569 Reference: [13] O. Njåstad: On some classes of nearly open sets.Pacific J. Math. 15 (1965), 961–970. MR 0195040 Reference: [14] A. S. B. Shahana: On fuzzy strong semicontinuity and fuzzy precontinuity.Fuzzy Sets Syst. 44 (1991), 303–308. MR 1140864 Reference: [15] F.-G. Shi: Fuzzy compactness in $L$-topological spaces.Fuzzy Sets Syst., submitted. Zbl 1080.54004 Reference: [16] F.-G. Shi: Countable compactness and the Lindelöf property of $L$-fuzzy sets.Iran. J. Fuzzy Syst. 1 (2004), 79–88. Zbl 1202.54007, MR 2138608 Reference: [17] F.-G. Shi: Theory of $L_\beta$-nested sets and $L_\alpha$-nested and their applications.Fuzzy Systems and Mathematics 4 (1995), 65–72. (Chinese) MR 1384670 Reference: [18] S. S. Thakur, R. K. Saraf: $\alpha$-compact fuzzy topological spaces.Math. Bohem. 120 (1995), 299–303. MR 1369688 Reference: [19] G. J. Wang: Theory of $L$-fuzzy Topological Spaces.Shanxi Normal University Press, Xian, 1988. (Chinese) .

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