0ec6da70-6c24-4184-840a-e261391dd28520210803081053034naun:naunmdt@crossref.orgMDT DepositInternational Journal of Pure Mathematics2313-057110.46300/91019http://www.naun.org/cms.action?id=6985292021292021810.46300/91019.2021.8https://www.naun.org/cms.action?id=23293Francisco GallegoLupiáñezDept.of Mathematics Universidad Complutense Madrid, Spainhttps://orcid.org/0000-0002-9821-8424The aim of this paper is to study fuzzy extensions of some covering properties defined by L. Kalantan as a modification of some kinds of paracompactness-type properties due to A.V.Arhangels'skii and studied later by other authors. In fact, we obtain that: if (X,T) is a topological space and A is a subset of X, then A is Lindelöf in (X,T) if and only if its characteristic map χ_{A} is a Lindelöf subset in (X,ω(T)). If (X,τ) is a fuzzy topological space, then, (X,τ) is fuzzy Lparacompact if and only if (X,ι(τ)) is L-paracompact, i.e. fuzzy L-paracompactness is a good extension of L-paracompactness. Fuzzy L₂-paracompactness is a good extension of L₂- paracompactness. Every fuzzy Hausdorff topological space (in the Srivastava, Lal and Srivastava' or in the Wagner and McLean' sense) which is fuzzy locally compact (in the Kudri and Wagner' sense) is fuzzy L₂-paracompact83202183202138405https://www.naun.org/main/NAUN/puremath/2021/a102019-005(2021).pdf10.46300/91019.2021.8.5https://www.naun.org/main/NAUN/puremath/2021/a102019-005(2021).pdfS.E. Abbas: "Some good types of paracompactness in Šostak's fuzzy topology", J. Fuzzy Math. 13 (2005), no. 2, 395--405. M.E.Abd-Elmonsef, F.M.Zeyada, S.N.El Deeb and I.M.Hanafy: "Good extensions of paracompactness", Math. Japonica 37 (1992), 195-200.S.P. Arya and P. Jha: "Countable fuzzy paracompactness, fuzzy metacompactness and the extension theorem for point-finite fuzzy covers", J. Indian Math. Soc. (N.S.) 63 (1997), no. 1-4, 53--61. A.Bülbül: "Some results on paracompactness of fuzzy topological spaces", J. Karadeniz Tech. Univ., Fac. Arts Sci., Ser. Math.-Phys. 11, (1988), 113-117 . 10.1016/0165-0114(93)90131-zA.Bülbül and M.W.Warner: "On the goodness of some type of fuzzy paracompactness", Fuzzy Sets Syst. 55 (1993), 187-191. 10.1016/0165-0114(93)90405-7Y.X. Chen: "Paracompactness on L-fuzzy topological spaces", Fuzzy Sets Syst. 53 (1993), 335-342 M.E. El-Shafei: "Paracompactness in L-fuzzy topological spaces", J. Fuzzy Math. 13 (2005), no. 3, 637--645 M.A. Fath Alla: "A note on fuzzy paracompact spaces", J. Inst. Math. Comput. Sci. Math. Ser. 3 (1990), no. 2, 159--165. Min Qiang Gu and Zhi Bin Liu: "Relative fuzzy paracompactness in L-fuzzy topological spaces" (Chinese) J. Yunnan Univ. Nat. Sci. 29 (2007), no. 3, 229--233 Qutaiba Ead Hassan: "A survey on fuzzy paracompactness", Far East J. Math. Sci. 13 (2004), no. 1, 119--136 Qutaiba Ead Hassan: "Characterizations of fuzzy paracompactness", Int. J. Pure Appl. Math. 25 (2005), no. 1, 123--133. Cheng-ming Hu: "C-structure of FTS: fuzzy paracompact spaces". I. J. Fuzzy Math. 11 (2003), no. 4, 1005--1020. Kai Hu and Guang Wu Meng: "A new paracompactness in L-fuzzy topological spaces" (Chinese) Mohu Xitong yu Shuxue 21 (2007), no. 5, 59--63 Ji Guang Jiang: "Fuzzy paracompact spaces" (Chinese) Sichuan Daxue Xuebao (1981), no. 3, 47--51. 10.15672/hjms.2018.548L.Kalantan: "L-paracompactness and L₂-paracompactness", Hacettepe J.Math. Stat. 48 (3) (2019), 1-6. 10.1016/0165-0114(94)00204-kS. R.T. Kudri: "Paracompactness in L-fuzzy topological spaces", Fuzzy Sets Syst. 70 (1995), 119-123 10.1016/0165-0114(94)90261-5S.R.T.Kudri and M.W.Warner; "L-fuzzy local compactness", Fuzzy Sets Syst. 67 (1994), 337-345. Yao Yao Lan and Qing Guo Li: "The m-paracompactness in fuzzy topological spaces" (Chinese) Mohu Xitong yu Shuxue 21 (2007), no. 4, 47--52 Fu Min Lei: "A generalization of Michael's theorem to fuzzy paracompact spaces" (Chinese) Sichuan Daxue Xuebao 32 (1995), no. 5, 483--485. Liu Y.-M.and Luo M.-K.:"Fuzzy Topology", World Scientific Pub.(Singapore, 1997) M.K. Luo: "A note on fuzzy paracompactness and fuzzy metric spaces" (Chinese) Sichuan Daxue Xuebao (1985), no. 4, 141--150 M.K. Luo: "Weak paracompactness and compactness in fuzzy topological spaces" (Chinese, English summary) Chinese Ann. Math. Ser. A 7 (1986), no. 5, 590--596. M.K. Luo: "Paracompactness and compactness in fuzzy topological spaces" (Chinese) Acta Math. Sinica 30 (1987), no. 4, 548--552. M.K.Luo: "Paracompactness in fuzzy topological spaces", J.Math. Anal.Appl. 130 (1988), 55-77. 10.1016/s0165-0114(96)00353-3F.G.Lupiáñez: "Fuzzy perfect maps and fuzzy paracompactness", Fuzzy Sets Syst. 98 (1998), 137-140. 10.1016/s0165-0114(97)00096-1F.G. Lupiáñez: "On fuzzy relative paracompactness", Fuzzy Sets Syst. 101 (1999), 485-488. F.G. Lupiáñez: "On some paracompactness-type properties of fuzzy topological spaces", AIP Conference Proceedings, 2116 (2019), 450001 10.1007/s00500-019-03905-8F.G. Lupiáñez: "On some paracompactness-type properties of fuzzy topological spaces", Soft Computing (2019), doi: 10.1007/s00500-019- 03905-8 Guangwu Meng, Zhongping Yao, and Shuili Chen: "Strato-closed operator and L-fuzzy paracompactness", J. Fuzzy Math. 9 (2001), no. 3, 637--642 Guang Wu Meng: "Sheaf paracompact sets in L-fuzzy topological spaces" (Chinese) Mohu Xitong yu Shuxue 9 (1995), no. 2, 45--50, 58. Dao Wen Qiu and Jian Lan Ma: "Fuzzy paracompactness of covering type" (Chinese) Acta Sci. Natur. Univ. Sunyatseni 37 (1998), no. 5, 107--109. 10.3906/mat-1804-54M.M.Saed, L.Kalantan and H.Alzumi: "C-paracompactness and C₂- paracompactness", Turkish J.Math.43 (2019), 9-20 Jizhong Shen: "Paracompactness in fuzzifying topology", J. Fuzzy Math. 4, No. 2, (1996), 277-296. Fu Gui Shi: "Remarks on paracompactness of L-fuzzy subsets" (Chinese) Mohu Xitong yu Shuxue 12 (1998), no. 3, 6--9. 10.1016/s0165-0114(01)00230-5Fu Gui Shi and Chong You Zheng: "A new kind of strong fuzzy paracompactness in L-fuzzy topological spaces" (Chinese) Mohu Xitong yu Shuxue 9 (1995), no. 3, 40--48. 10.1016/s0165-0114(01)00230-5Fu-Gui Shi and Chong-You Zheng: "Paracompactness in Ltopological spaces", Fuzzy Sets Syst. 129, (2002), 29-37. Jun Na Sun: "Almost countably paracompact sets in L-fuzzy topological spaces" (Chinese) Mohu Xitong yu Shuxue 25 (2011), no. 2, 49--53 D. Susha: "Fuzzy α-paracompactness in fuzzy box products", J. Fuzzy Math. 14 (2006), no. 1, 223--229. 10.1016/0022-247x(81)90078-0R.Srivastava, S.N.Lal and A.K.Srivastava: "Fuzzy Hausdorff topological spaces", J.Math. Anal.Appl. 81 (1981), 497-506 Bao Ming Wang, Zhong Ping Yao, Guang Wu Meng: "Stratified Lfuzzy topologies and L-fuzzy paracompactness" (Chinese) Qufu Shifan Daxue Xuebao Ziran Kexue Ban 27 (2001), no. 3, 30--32. 10.1016/0165-0114(93)90190-sM.W.Warner and R.G.McLean: "On compact Hausdorff L-fuzzy spaces", Fuzzy Sets Syst. 56 (1993), 103-110 Jie Zhang, Fu-Gui Shi, and Chong-You Zheng: "A certain strong fuzzy paracompactness", J. Fuzzy Math. 8 (2000), no. 3, 757--767 10.1016/s0165-0114(99)00043-3Xingfang Zhang, Guangwu Meng, Enrui Guan, and Shuili Chen: "Stratiform L-fuzzy topologies and L-fuzzy paracompactness", J. Fuzzy Math. 9 (2001), no. 2, 421--427. Jia Yin Zhao: "A note on layer paracompactness of L-fuzzy subsets" (Chinese) Natur. Sci. J. Harbin Normal Univ. 17 (2001), no. 2, 34--36. Mei Xiang Zhao, Guang Wu Meng, and Zhi Gang: "Paracompactness in L-fuzzy interior spaces" (Chinese) Mohu Xitong yu Shuxue 20 (2006), no. 2, 88--92