International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020
Fuzzy *Irreducible Spaces
M. Rowthri, B. Amudhambigai
in (X,T) such that 1 . That is., cl() 1 in Abstract: In this paper, the concept of * operator on a (X,T) . family of fuzzy open sets in a fuzzy topological space is introduced. Also, the concepts of fuzzy * irreducible Definition 2.6 [7] A fuzzy set in a fuzzy topological space spaces, fuzzy generic sets and fuzzy (X,T) is called fuzzy nowhere dense if there exists no quasi-Sober spaces are initiated and some properties non-zero fuzzy open set in (X,T) such that cl() . are discussed. That is., int(cl()) 0 in (X,T) .
Keywords : operator on FO(X,) , fuzzy generic * sets , fuzzy irreducible spaces, fuzzy III. FUZZY IRREDUCIBLE SPACES quasi-Sober spaces Throughout this paper, fuzzy topological space is shortly 2010 AMS Subject Classification: 54A40, 03E72.. denoted by fts. Then FO(X,) , FC(X,) and I. INTRODUCTION X) denote set of all fuzzy open sets, fuzzy closed sets in (X,) and fuzzy points over L.A. Zadeh[8] initiated fuzzy set in 1965. In 1968, Chang [2] characterized fuzzy topological space. Njastad [5] X respectively. introduced open sets. In the same sprit Bin Shahna [1] Definition 3.1 Let (X,) be a fts. A fuzzy operator defined fuzzy open sets and fuzzy closed sets. The *: FO(X,) I X idea of an irreducible or hyperconnected topological space is defined as, if for each FO(X,) with 0 , has been studied by T. Thompson. In this paper, the concept X of operator on a family of fuzzy open sets in a fts is F int() *() and *(0 X ) 0 X . introduced. Also, the concepts of fuzzy irreducible Remark 3.1 It is easy to check that some examples of fuzzy spaces, fuzzy generic sets and fuzzy operators on FO(X,) are the well known fuzzy operators quasi-Sober spaces are initiated and some viz. F int , F int(Fcl) , Fcl(F int) , Fint(Fcl(Fint)) and properties are discussed. Fcl(F int(Fcl)) . Definition 3.2 Let (X,) be a fts and * be a fuzzy II. PRELIMINARIES operator on . Then any FO(X,) is called This section contains basic definitions and preliminary fuzzy * open if *() . Then 1 is called results needed for this paper. X fuzzy closed set. Definition 2.1 [4] Let X be a topological space. * (i) X is irreducible, if X , and whenever Notation 3.1 The family of all fuzzy open (resp.
X Z1 Z2 with Zi closed, X Z1 or X Z2 . fuzzy closed) sets in is notated by (ii) Z X is an irreducible component of X if Z is a F *O(X,) (resp. F *C(X,) ). maximal irreducible subset of X. Definition 3.3 For any X in a fts and be a Definition 2.2 [3] A subset F of a topological space is I irreducible if, F A B where A and B then F A or fuzzy operator on , the fuzzy interior of F B . (briefly, F *int() ) is defined by Definition 2.3 [4] A topological space X is said to be F *int() { : and F *O(X,) }. quasi-sober if for every irreducible closed subset has a generic Definition 3.4 For any in a fts and be a point. fuzzy operator on , the fuzzy closure of Definition 2.4 [6] A fuzzy set A is quasi-coincident with (briefly, F *cl() ) is defined by the fuzzy set B iff x X such that A (x) B (x) 1 . Definition 2.5 [7] A fuzzy set in a fuzzy topological space F *cl() { : and F *C(X,) }. (X,T) is called fuzzy dense if there exists no fuzzy closed set Definition 3.5 Any fts is said to be a fuzzy irreducible space, where is a fuzzy operator on
if, for any 1, 2 F *O(X,) where
and . Revised Manuscript Received on April 15, 2020. 1 0 X , 2 0 X 1q2 * Correspondence Author Definition 3.6 Any I X in M. Rowthri*, Department of Mathematics, Sri Sarada College for Women(Autonomous), Salem, India. Email: [email protected] a fts is said to be fuzzy Dr. B. Amudhambigai, Department of Mathematics, Sri Sarada College irreducible, for Women(Autonomous), Salem, India.. Email: [email protected]
Published By: Retrieval Number: F9132038620/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.F9132.059120 208 & Sciences Publication Fuzzy * Irreducible Spaces
where * is a fuzzy operator on FO(X,) if, 0 X and , the only possibility is
(1 2 ) where 1, 2 F *C(X,) , then either F *cl(). Hence F *C(X,) .
1 or 2 . Then the set of all fuzzy Proposition 3.4 Let in fts , * irreducible sets is noted by F *I(X,) where be a fuzzy operator on . If Definition 3.7 Any I X is called a fuzzy F *I(X,) , then . maximal irreducible set of fts (X,) , where is Proof. Let F *I(X,) . Let be the set of all X a fuzzy operator on , if there is no fuzzy irreducible sets j I such that F *I(X,) such that . Then collection of j , j J where J is an indexed set. Let be the all fuzzy maximal irreducible sets is denoted by set of fuzzy irreducible sets k , k 1, 2,...,l such F *MI(X, ) that 1 2 ... l . Definition 3.8 Let be a fuzzy operator on Let ' k k , k 1, 2,...,l . It is enough to show FO(X1,1) and FO(X 2 , 2 ) in a ftss (X1,1) and that 'F *I(X,) . Let ' 1 2 , where (X 2 , 2 ) respectively. Any function 1, 2 F *C(X,) . Since each f : (X1,1) (X 2 , 2 ) is said to be a fuzzy k F *I(X,) ,i 1, 2,...,l , we get i 1 or continuous function if for every F *O(X , ) , 2 2 Suppose that for some i J , and 1 i 2 . 0 f ()F *O(X1,1) . , then for all Thus Therefore Proposition 3.1 Let be a fuzzy operator on and ' F *MI(X,) . and in a ftss and Proposition 3.5 If is fuzzy irreducible, respectively. Let f : (X , ) (X , ) be a 1 1 2 2 then for any F *O(X,) , F *cl() 1X . bijective and fuzzy continuous function. If Proof. Since is fuzzy irreducible, for any F *I(X , ) then f ()F *I(X , ) 1 1 2 2 two , F *O(X,) where 0 X and 0 X , Proof. Let , F *C(X , ) such that 1 2 2 2 q . Let 1 F *O(X,) such that 1 0 X . If f () ( ) . Then 1 2 1 1X , then F *cl(1) 1X . Assume that 1 1X f 1( f ()) f 1( ) , which implies that and be such that x . Let 1 2 t1 1 f 1( ) . Then f 1( f ()) , since f is one-one F *O(X,) and 0 such that x . By 1 2 1 1 X t1 1 Thus f 1( ) f 1( ) . Since F *I(X , ) , hypothesis, q . That is, there exist some x where 1 2 1 1 1 1 ti 1 f 1( ) or f 1( ) Thus f () f ( f 1( )) i 1and i J, J is an indexed set such that x with 1 2 1 ti 1 ~ xt qxt . Since 1 is an arbitrary fuzzy open set in 1 1 i or f () f ( f (2 )) and so f () 2 . Therefore , it follows that every fuzzy open 1 in . is such that x i 1 and i J such that ti 1 Remark 3.2 For any in a fts and be a xt 1 with . Therefore is a fuzzy limit point of fuzzy operator on , F *cl() . i . So every fuzzy point over X is a fuzzy limit point of . Proof. Proof is obvious from the definition of fuzzy 1 Thus, every is such that x F *cl() , closure of a fuzzy set . ti Proposition 3.2 If F *I(X,) in fts and F *O(X,) . Hence, fuzzy closure of be a fuzzy operator on , then every fuzzy open set is 1X . F *cl() F *I(X,) . Proof. Assume that and IV. FUZZY QUASI SOBER SPACE
F *cl() 1 2 where 1, 2 F *C(X,) . Definition 4.1 Let be a fuzzy operator on X Since F *cl() and F *I(X,) , 1 or in a fts . Let be a fuzzy I . Then F *cl() or F *cl() . X 2 1 2 irreducible closed set. Any fuzzy set I with Hence , is said to be a fuzzy generic set of if Proposition 3.3. If , then F *MI(X,) F *cl() . F *C(X,) . Proof. Let . Then, there is no F *I(X,) such that . By Proposition 3.2,
F *cl() F *I(X,) . Since
Published By: Retrieval Number: F9132038620/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.F9132.059120 209 & Sciences Publication International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-9 Issue-1, May 2020
Definition 4.2 Let * be a fuzzy operator on FO(X,) Proof. Let . Let in a fts . Then is said to be a fuzzy (X,) 1, 2 F *C(X 2 , 2 ) such that * quasi-Sober space, if for every fuzzy . Since irreducible closed set there exists a fuzzy , . As is injective, . Then generic set. . Since is a fuzzy Remark 4.1 Let be a fuzzy operator on in a continuous function, fts . Let Y X and (Y, ) be a fuzzy subspace of Y f 1( ), f 1( ) F *C(X , ) . This implies that . Then be a fuzzy operator on FO(Y, ) . 1 2 1 1 Y , F *C(X , ) . Since Definition 4.3 Let be a fuzzy operator on 1 1 , or in a fts . If for X with there exists a , I q . Thus or . F *C(X,) such that either , q or Therefore, or . This implies that , q , then is called fuzzy or . Hence Kolmogorov. F *IC(X 2 , 2 ) . Proposition 4.1 Let be a fuzzy operator on Proposition 4.4 Let be a fuzzy operator on in a fts . Let and be a fuzzy subspace in a fts . Let and be a fuzzy of . If is fuzzy Kolmogorov, then open subspace of . If is fuzzy is fuzzy Kolmogorov. quasi-Sober, then is fuzzy
Proof. Let with . Then there exists a quasi-Sober. such that , or , Proof. Let F *IC(Y, Y ) . Then . Let with q . Since F *IC(X,) . Since is fuzzy quasi-Sober, there exists a fuzzy generic , F *C(Y, Y ) . Also, since , and , , with q set such that F *cl() . or with q . Hence is fuzzy Then F *cl X () Y Y . Since
Kolmogorov. F *IC(Y, Y ) , and . Notation 4.1 Let be a fuzzy operator on Hence is fuzzy s quasi-Sober. and where and (Y,) are any two ftss. V. CONCLUSION For any I X , I Y , F *cl() with respect to and are denoted by F *cl() and In this paper, we explored operator on a family F *cl() respectively. Then the collection of all fuzzy of fuzzy open sets in a fuzzy topological spaces . We can irreducible closed in id denoted by extend operator on fuzzy homotopy and fuzzy spectral spaces. F *IC(X,) Proposition 4.2 Let be a fuzzy operator on REFERENCES in a fts . Let and be a fuzzy 1. A. S. Bin Shahna, On fuzzy strong semicontinuity and fuzzy closed subspace of . If is fuzzy precontinuity, Fuzzy Sets and Systems, vol.44, no.2, 1991, pp. 303– 308. quasi-Sober, then is fuzzy 2. C. L. Chang, “Fuzzy topological spaces”, J. Math. Anal., Appl. 24, quasi-Sober. 1968, 182-190. 3. Z. Dondsheng, H. W. Kin, “On Topologies defined by Irreducible Sets”, Proof. Let F *IC(Y, Y ) . Then Journal of Logical and Algebraic Methods in Programming, 84, 2015, F *IC(X,) . Since is fuzzy 185-195. 4. https://stacks.math.columbia.edu/download/topology.pdf quasi-Sober, there exists a fuzzy -generic set 5. O. Njastad, “On some classes of nearly open sets”, Pacific Journal of Mathematics, vol.15, 1965 , 961–970. such that F *cl() . Thus 6. P. M. Pu and Y. M Liu, “Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence”, J. Math. Anal Appl. 76 , F *cl() 1Y 1Y . This implies that 1980, 571-599. F *cl() 1Y . Therefor1e F *cl() . 7. G. Thangaraj and G. Balasubramanian, On somewhat fuzzy continuous Hence is fuzzy quasi-Sober. functions, The Journal of Fuzzy Mathematics, vol. 11, no. 2, 2003, pp. 725-736. Proposition 4.3 Let be a fuzzy operator on 8. L. A. Zadeh, Fuzzy sets, Information and control, 8, 1965, 338-353.
FO(X1,1) and FO(X 2 , 2 ) where and
are any two ftss. Let be a injective and fuzzy continuous function. If
F *IC(X1,1) , then
F *IC(X 2 , 2 ) .
Published By: Retrieval Number: F9132038620/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.F9132.059120 210 & Sciences Publication Fuzzy * Irreducible Spaces
AUTHORS PROFILE M. Rowthri M.Sc.,M.Phil., is a research scholar in Mathematics in Sri Sarada College for Women(Autonomous), Salem, Tamilnadu, India. She has published three articles in national and international journals. She is doing research in fuzzy * operator on fuzzy topological
spaces.
Dr. B. Amudhambigai, M.Sc., MPhil, Ph.D is working as an Assistant Professor in Mathematics at Sri Sarada College for Women(Autonomous), Salem, Tamilnadu, India. She has published several articles in many reputed journals and guiding M.Phil and Ph.D students. Her research area includes fuzzy Cryptography and Mathematical Modelling.
Published By: Retrieval Number: F9132038620/2020©BEIESP Blue Eyes Intelligence Engineering DOI:10.35940/ijrte.F9132.059120 211 & Sciences Publication