Int. J. Anal. Appl. (2023), 21:9 Received: Nov. 22, 2022. 2020 Mathematics Subject Classification. 03E72, 54A40. Key words and phrases. topology; fuzzy sets; separation properties; covering properties fuzzy paracompactness. https://doi.org/10.28924/2291-8639-21-2023-9 © 2023 the author(s) ISSN: 2291-8639 1 Regularity and Paracompactness in Fuzzy Topological Spaces Francisco Gallego Lupiáñez* Dpt. Mathematics, Universidad Complutense, Madrid (Spain) *Corresponding author: fg_lupianez@mat.ucm.es ABSTRACT. In this paper we obtain two characterizations of regular fuzzy topological spaces using Luo's and Abd El-Monsef and others' paracompact fuzzy topological spaces. 1. Introduction In this paper we obtain two characterizations of fuzzy regularity as a fuzzy covering property. Indeed, we show that one can characterize fuzzy regularity as a paracompact-type fuzzy property in Luo's and Abd El-Monsef and others' both senses. 2. Definitions and Main Results Definition 1 [1] Let  be a set in a fts ( ),X  and let (   )0,1 , 0,1 ;r s  we define ( ){ }[ ] :r x X x r    = ( ){ }( ) :s x X x s     =  r r r    = Definition 2 [1] Let A be a family of sets and be a set in a fts ( ),X  . We say that A is locally finite (resp. ∗-locally finite) in for each point e in , there exists ( )Q e  such that     https://doi.org/10.28924/2291-8639-21-2023-9 2 Int. J. Anal. Appl. (2023), 21:9 is quasi-coincident (resp. intersects) with at most a finite number of sets of A ; we often omit the word "in  " when X = . Definition 3 [1] A family of sets A is called a Q -cover of a set μ if for each ,( )x supp  there exist a A such that  and are quasi-coincident at x . Let ( 0,1 .r  A is called a r − Q cover of if A is a Q -cover of .r    Definition 4 [1] Let ( 0,1 r  , be a set in a fts . We say that  is r -paracompact (resp. * r -paracompact) if for each -open Q -cover of there exists an open refinement of it which is both locally finite (resp.∗-locally finite) in and a r - Q -cover of . The fuzzy set is called S-paracompact (resp. S*-paracompact) if for every ( 0,1r  , is r -paracompact (resp. *r -paracompact). Definition 5 [2] A family of fuzzy sets U is called an L -cover of a fuzzy set if .      U Definition 6 [2] Let be a fuzzy set in a fts . We say that  is fuzzy paracompact (resp. ∗-fuzzy paracompact) if for each open L -cover B of and for each ( 0,1 ,  there exists an open refinement *B of B which is both locally finite (resp. ∗-locally finite) in and L -cover of  − . We say that a fts is fuzzy paracompact (resp. ∗-fuzzy paracompact) if each constant set in X is fuzzy paracompact (resp. ∗-fuzzy paracompact). Theorem 1 Let a fuzzy Hausdorff fts (in any of Wuyts and Lowen’s definitions that are good extensions of Hausdorffness). Then is fuzzy regular if and only if for each ( 0,1 ,r  for each r -open Q-cover of and for each fuzzy point x  of X there exists an open refinement of it which is both ∗-locally finite in x  and a r -Q-cover of . Proof (⇒) For each ( 0,1 ,r  let U be a r -open Q-cover of ( ),X  , and x be a fuzzy point of X . Then, we have that the family of crisp sets ( )1 { } r U U − U∣ , is an open cover of  ( ),X  , which is Hausdorff and regular ([3], [4]). Then ([5], [6], [7]), it has an open refinement [ ] x V    ( ),X  r        ( ),X    ( ),X  ( ),X  ( ),X  ( ),X  ( ),X  3 Int. J. Anal. Appl. (2023), 21:9 which is a cover of X , and is locally finite in x . For each V x V we have an V U U with ( )1 ( ) . V r V U −  Let .{ }V VU Vx x =  W V∣ Then, x W , is both an open refinement of U and a r Q− -cover of ( ),X  , and also is ∗-locally finite in x  , indeed, because x V is locally finite in x, we have an open neigborhood G of x that G intersects with only finite number of members of x V . Then ( ) G Q x    intersects with only a finite number of members of x W . (⇐) Let U [ ] be an open cover of ( ],[ )X  ; then  U U U∣ is an open Q-cover of 1 , X and, for each ,x X it has an open refinement r x V which is a Q-cover of 1 X and also locally finite in 1 r x − . Let  (1 ) | rr xV V− =W V ; then [ ]W is both a refinement of U and a cover of ( ],[ )X  . Also, W is locally finite in x . Indeed: we take 1 1 ( x ) r O Q −  such that 1 O , is quasi- coincident with only a finite number of members 1 ,..., n V V , of r x V . Let 1 ( ) ( ) r OO = , then [ ].x O   For each V rx V , if (1 )r O V −  , we have a crisp point y X , such that ( ) ( ) ( ) ( )1 1, 1 , 1,O y r V y r O y V y  − +  then 1O qV and  1,..., nV V V . Hence the neighborhood O of x intersects with only a finite number of members ( ) ( ) (1 ) (1 )1 ,..., . rnr V V − − W Theorem 2. Let ( ),X  be a fuzzy Hausdorff fts (in any of Wuyts and Lowen’s definitions that are good extensions of Hausdorffness [3]). Then ( ),X  is fuzzy regular if and only if for each r I , and for each open L -cover B of r , for each ( 0,1 ,  and for each fuzzy point x of X , there exists an open refinement B * of it which is both ∗-locally finite in x  and L -cover of .r − Proof (⇒) For each r I , and for each open L -cover B of r , for each ( 0,1 ,  and for each fuzzy point x  of X , we have that the family of crisp sets ( ){ ,1] } [ ]¹(G Gr  −=  − U B∣ is an open cover of ( ],[ )X  which is Hausdorff and regular ([3, 4]). Then ([5],[6],[7]), it has an open refinement * [ ] x U which is a cover of X and is locally finite in x . For each *xV U , there 4 Int. J. Anal. Appl. (2023), 21:9 exists ,( ]¹ 1( ) V G r  − − U , such that (( ])¹ ,1 V V G r  −  − . So  *|V V xG V   B* = U is refinement of B . Then, there exists * x V U such that x V and ( ) .VG x r  − So, ,( )( ) V V G x r   − and  *| .V V xG V r    − U Since *xU is locally finite in x , there exists  A  with x A , such that intersects with at most a finite number of members of * x U . Then, there exists A   such that A x q   and A  intersects with a finite number of fuzzy sets of B * . (⇐) Let [ ]U be an open cover of X and x X , then { } U U= B U∣ is an open L -cover of ( ),X  and for each, r I is { } U rU   U∣ . For each ( 0,1 ,  there exists an open refinement *B of B which is both locally finite in x  and L -cover of r − . This implies that 1 { } rG G −  B*∣ for all 1 .  Let  1(( ,1])G r G−= −  B *U* , then [ ]U* is an open refinement of U (indeed, for each 1(( ,1])G r − − U * there exists G V U such that ( ( )¹ ,1 )GG r V − −  . Since   1G G r    − B * , then U * is an open refinement of U . And, since x X there exists ( )A Q x which intersects with only 1,.., .nG G B * Since ( ) 1,A x +  we have ( ) 1A x  − , then ( ( )  ¹ 1 ,1 .x A  − −  If ( ( ) ( ( ) ¹ 1 ,1 ¹ ,1A G r − −−  −   , there exists some point z, such that ( ) 1A z  − and ( )G z r  − , so A G   . Then, if the neigborhood ( ( )¹ 1 ,1A − − of x intersects with infinite members of U * , A intersects with infinite members of *B .Thus U * is locally finite in x . This yields that the Hausdorff topological space ( ],[ )X  is regular ([5], [6], [7]) and ( ),X  is fuzzy regular ([4]). 3. Discussion In this paper, fuzzy regularity is characterized as a fuzzy covering property. Future research could obtain characterization of other fuzzy separation properties as fuzzy covering properties. Conflicts of Interest: The author declares that there are no conflicts of interest regarding the publication of this paper. 5 Int. J. Anal. Appl. (2023), 21:9 References [1] M.K. Luo, Paracompactness in Fuzzy Topological Spaces, J. Math. Anal. Appl. 130 (1988), 55–77. https://doi.org/10.1016/0022-247x(88)90386-1. [2] M.E.A. El-Monsef, F.M. Zeyada, S.N. El-Deeb, I.M. Hanafy, Good Extensions of Paracompactness, Math. Japonica. 37 (1992), 195-200. [3] P. Wuyts, R. Lowen, On Separation Axioms in Fuzzy Topological Spaces, Fuzzy Neighborhood Spaces, and Fuzzy Uniform Spaces, J. Math. Anal. Appl. 93 (1983), 27–41. https://doi.org/10.1016/0022- 247x(83)90217-2. [4] Y.M. Liu, M.K. Luo, Fuzzy Topology, World Scientific Publishing, Singapore, 1997. [5] J. Abdelhay, Caracterizacao dos Espacos Topologicos Regulares e Normais por Meio de Coberturas, Gazeta de Matematica (Lisboa) 9 (1948), 8–9. (in Portuguese). [6] J.M. Boyte, Point (Countable) Paracompactness, J. Aust. Math. Soc. 15 (1973), 138–144. https://doi.org/10.1017/s144678870001288x. [7] J. Chew, Regularity as a Relaxation of Paracompactness, Amer. Math. Mon. 79 (1972), 630–632. https://doi.org/10.1080/00029890.1972.11993101. https://doi.org/10.1016/0022-247x(88)90386-1 https://doi.org/10.1016/0022-247x(83)90217-2 https://doi.org/10.1016/0022-247x(83)90217-2 https://doi.org/10.1017/s144678870001288x https://doi.org/10.1080/00029890.1972.11993101