Fuzzifying Tritopological Spaces

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1 International Mathematical Forum, Vol., 08, no. 9, 7-6 HIKARI Ltd, On α-continuity and α-openness in Fuzzifying Tritopological Spaces Barah M. Sulaiman and Tahir H. Ismail Mathematics Department College of Computer Science and Mathematics University of Mosul, Iraq Copyright 08 Barah M. Sulaiman and Tahir H. Ismail. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We study the concepts of α continuity and α openness mappings and their properties in fuzzifying tritopological spaces. Keywords: Fuzzifying topology, fuzzifying tritopological space, α continuity, α openness Introduction In [0,, ], Ying introduced the fuzzifying topologyˮ. In 965 [8], Njåstad defined α-open set in in general topologyˮ. In 98 [7], Mashhour et al. studied the α-continuity and α-openness of mappingˮ. In 99 [9], Singal and Rajvanshi used this concept in fuzzy topology. In 99, [5, 6], Kumar studied the notions of fuzzy α-continuity and pre-continuity in fuzzy bitopological spacesˮ. In [, ], Khedr et al. studied semi-continuity, csemi-continuity, α- continuity and cα-continuity in fuzzifying topologyˮ. Also, Allam et al. studied semi-continuity, semi-openness, α-continuity and α-openness in fuzzifying bitopological spacesˮ in []. Preliminaries We will use throughout this paper the main formulas of fuzzy logic and corresponding set theoretical notationsˮ in accordance with Ying, and some definitions, results and theorems in fuzzifying topology from [0, ]. We now give some definitions introduced in [] which are helpful in this paper.

2 8 Barah M. Sulaiman and Tahir H. Ismail Definition. [] If (X, τ, τ, τ ) is a fuzzifying tritopological space, then (i) The family of fuzzifying (,,)-α open sets is symbolized as ατ (,,) I(P(X)), where I(P(X)) is the family of all fuzzy sets in P(X), is given by E ατ (,,) x (x E x int (cl (int (E)))), i.e., ατ (,,) (E) = (int (cl (int (E))))(x). x E (ii) The family of fuzzifying (,,)-α closed sets, symbolized as αf (,,) I(P(X)) is given by: F αf (,,) X~F ατ (,,), where X~F is a complement of F. (iii) The (,,)-α nbh. system of x, symbolized as αn (,,) x I(P(X)) is given (,,) by: E αn x F (F ατ(,,) x F E); i.e. αn (,,) x (E) = ατ (,,) (F). x F E (iv) The (,,)-α derived set of E symbolized as αd (,,) (E) is given by: x αd (,,) (E) F (F αn x (,,) F (E {x}) ), i.e., αd (,,) (E)(x) = F (E {x}) ( αn x (,,) (F)). (v) The (,,)-α closure set of E symbolized as αcl (,,) (E) is given by: x αcl (,,) (E) F (F E) (F αf (i,j,k) ) x F), i.e., αcl (,,) (E)(x) = ( αf (,,) (F)). x F E (vi) The (,,)-α interior set of E X symbolized as αint (,,) (E) is given by: αint (,,) (E)(x) = αn x (,,). (vii) The (,,)-α exterior set of E X symbolized as αext (,,) (E) is given by: x αext (,,) (E) x αint (,,) (X~E)(x), i.e. αext (,,) (E)(x) = αint (,,) (X~E)(x). (viii) The (,,)-α boundary set of E X symbolized as αb (,,) (E) is given by: x αb (,,) (E) (x αint (,,) (E)) (x αint (,,) (X~E)), i.e. αb (,,) (E)(x) min( αint (,,) (E)(x)) ( αint (,,) (X~E)(x)). Continuous and open mappings in fuzzifying tritopological spaces Definition.[] If (X, τ ) and (Y, σ) are fuzzifying topological spaces. (i) A unary fuzzy predicate C I(Y X ), called fuzzy continuious, is given by: f C H (H σ f (H) τ), i.e.,c(f) = H P(Y) min (, σ(h) + τ(f (H))). (ii) A unary fuzzy predicate O I(Y X ), called fuzzy open, is given as: f O: = G (G τ f(g) σ). i.e., O(f) = min (, τ(g) + σ(f(g))).

3 On α-continuity and α-openness in fuzzifying tritopological spaces 9 Theorem. [] If (X, τ ) and (Y, σ) are fuzzifying topological spaces and f: (X, τ ) (Y, σ), then f C H (f (int(h)) int(f (H)). Theorem. [] If (X, τ ) and (Y, σ) are fuzzifying topological spaces and f: (X, τ ) (Y, σ) is a fuzzy open mapping, we define (i) f O = H x (f (H) N x H N f(x) ); (ii) f O = H (f (cl(h)) cl(f (H))); (iii) f O = H (int(f (H)) f (int(h))); (iv) f O = G (f(int(g)) int(f(g))). Then O(f) O n (f), n =,,,. Definition. If (X, τ, τ, τ ) and (Y, σ, σ, σ ) are fuzzifying tritopological spaces. The mapping f: (X, τ, τ, τ ) (Y, σ, σ, σ ) is called pairwise fuzzy continuous (respectively pairwise fuzzy open) if f: (X, τ ) (Y, σ ), f: (X, τ ) (Y, σ ) and f: (X, τ ) (Y, σ ) are fuzzy continuous mappings (respectively fuzzy open mappings). Theorem.5 If (X, τ, τ, τ ) and (Y, σ, σ, σ ) are fuzzifying tritopological spaces and f: (X, τ, τ, τ ) (Y, σ, σ, σ ) is pairwise fuzzy continuous mapping and pairwise fuzzy open mapping, then G P(X) and H P(Y ), we have (i) G ατ (,,) f(g) ασ (,,), (ii) H ασ (,,) f (H) ατ (,,). Proof. (i) By using Theorem (.) and Lemma (.) in [] and (iv) of Theorem (.), we get [G ατ (,,) )] = [G int (cl (int (A)))] [f(g) f(int (cl (int (G))))] [f(g) int (f(cl (int (G))))] [f(g) int (cl (f(int (G))))] [f(g) int (cl (int (f(g))))] = [f(g) ασ (i,j,k) ]. (ii) By using Lemma (.) in [], Theorem (.) and (ii) of the Theorem (.), we get [H ασ (,,) )] = [H int (cl (int (H)))] [f (H) f (int (cl (int (H))))] [f (H) int (f (cl (int (H))))] [f (H) int (cl (f (int (H))))] [f (H) int (cl (int (f (H))))] = [f (H) ατ (,,) ]. α-continuity in fuzzifying tritopological spaces Definition. If (X, τ, τ, τ ) and (Y, σ, σ, σ ) are fuzzifying tritopological spaces.

4 0 Barah M. Sulaiman and Tahir H. Ismail A unary fuzzy predicate αc (,,) I(Y X ), called fuzzy (,,)-α continuous, is given by: f αc (,,) (f): = H (H ασ (,,) f (H) ατ (,,) ), i.e., αc (,,) (f) = min(, ασ (,,) (H) + ατ (,,) (f (H))). Theorem. If (X, τ, τ, τ ) and (Y, σ, σ, σ ) are fuzzifying tritopological spaces and f Y X, we put Y (i) f αc (,,) : = H (H αf (,,) f X X (H) αf (,,) ), where αf (,,) is τ the family of all fuzzifying (,,)-α closed sets in X and αf (,,) is the family of all fuzzifying (,,)-α closed sets in Y. (,,) (ii) f αc (,,) : = x H (H αn f(x) G (x G f (H) G ατ (,,) )); (iii) f αc (,,) : = x H (H αn (,,) f(x) G (f(g) H x G G ατ (,,) )); (iv) f αc (,,) : = H (cl (int (cl (f (H)))) f (αcl (,,) (H))); 5 (v) f αc (,,) : = G (f(cl (int (cl (G)))) αcl (,,) (f(g))); 6 (vi) f αc (,,) : = H (f (αint (,,) (H)) int (cl (int (f (H))))). n Then f αc (,,) f αc (,,), n =,,,, 5, 6. Proof. (a) We prove that [f αc (,,) ] = [f αc (,,) ]. Y X [f αc (,,) ] = min(, αf (,,) (H) + αf (,,) (f (H))) = = = G P(Y) = [f αc (,,) ]. min(, ασ (,,) (Y~H) + ατ (,,) (X~f (H))) min(, ασ (,,) (Y~H) + ατ (,,) (f (Y~H))) min(, ασ (,,) (G) + ατ (,,) (f (G))) (b) Now, we prove that [f αc (,,) ] = [f αc (,,) ]. [f αc (,,) ] = min (, αn (,,) f(x) (H) + αn (,,) x (f (H))) If αn (,,) f(x) (H) αn (,,) x (f (H)), then [f αc (,,) ] [f αc (,,) ]. Assume that αn (,,) f(x) (H) > αn (,,) x (f (H)), we have if f(x) E H, then x f (E) f (H). So [f αc (,,) ] = min (, αn (,,) f(x) (H) + αn (,,) x (f (H))) min (, ασ (,,) (E) + = min (, f(x) E H f(x) E H ασ (,,) (E) + ατ (,,) (F) ) x F f (H) ατ (,,) (f (E)) ) f(x) E H

5 On α-continuity and α-openness in fuzzifying tritopological spaces min (, f(x) E H (ασ (,,) (E) + ατ (,,) (f (E)))) f(x) E H min (, ασ (,,) (E) + ατ (,,) (f (E))) min (, ασ (,,) (H) + ατ (,,) (f (H))) = [f αc (,,) ] () Now, we prove that [f αc (,,) ] [f αc (,,) [αc (,,) (f)] = = min(, min(, ], as follows min(, ασ (,,) (H) + ατ (,,) (f (H))) f(x) H αn (,,) f(x) (H) + αn f(x) x f (H) x f (H) (,,) (H) + x f (H) min(, αn (,,) (,,) f(x) (H) + αn x (f (H))) αn x (,,) (f (H))) αn x (,,) (f (H))) = [αc (,,) (f)] () From () and () we get [αc (,,) (f)] = [αc (,,) (f)]. (c) We prove that [f αc (,,) ] = [f αc (,,) ], with f(f (H)) H, we have [f αc (,,) ] = min(, αn (,,) f(x) (H) + αn (,,) x (G)),f(G) H min(, αn (,,) f(x) (H) + αn (,,) x (f (H))) V P(Y) = [f αc (,,) ] () Since if f(g) H then G f (H), from Theorem (.5) in [], we have αn (,,) x (G) αn (,,) x (f (H)). So [f αc (i,j,k) ] = min(, αn (,,) f(x) (H) + αn (,,) x (U)),f(G) H G P(Y) min(, αn (,,) f(x) (G) +,f(g) H αn x (,,) (f (H))) = min(, αn (,,) f(x) (H) + αn (,,) x (f (H))) = [f αc (,,) ] () From () and () we have [f αc (,,) ] = [f αc (,,) ]. 5 (d) To prove that [f αc (,,) ] = [f αc (,,) ] G X, H Y, such that f(g) = H, then G f (H) and G f (f(g)).

6 Barah M. Sulaiman and Tahir H. Ismail We have H P(Y ), [f(f (H)) H] =, then [αcl (,,) (f(f (H))) αcl (,,) (H)] =. So from Lemma (.) in [], we have [f (αcl (,,) (f(f (H)))) f (αcl (,,) (H))] =, moreover [cl (int (cl (f (H)))) f (f(cl (int (cl (f (H))))))] =. Thus αc (,,) (f) = [cl (int (cl (f (H)))) f (αcl (,,) (H))] [f (f(cl (int (cl (f (H)))))) f (αcl (,,) (f(f (H))] 5 [f(cl (int (cl (f (H))))) αcl (,,) (f(f (H))))] [f(cl (int (cl (G)))) αcl (,,) (f(g))] = αc (,,) (f)) (5) Now, we have G X, H Y, such that f(g) = H, then G f (H). Hence 5 (f) = [f(cl (int (cl (G)))) αcl (,,) (f(g))] αc (,,) [f(cl (int (cl (f (H))))) αcl (,,) (f(f (H)))] [f(cl (int (cl (f (H))))) f(αcl (,,) (f (H)))] [cl (int (cl (f (H)))) αcl (,,) (f (H))] = αc (i,j,k) (f) (6) From (5) and (6) we have [f αc (,,) (e) We can easily prove that [f αc (,,) 5 ] = [f αc (,,) ]. 6 ] = [f αc (,,) ]. (f) Now, we prove [f C (,,) ] = [f αc (,,) ] [f αc (,,) ] (,,) = [ x H (H αn f(x) G (x G f (H) G ατ (,,) ))] = = min(, αn f(x) min(, = min(, αn f(x) 6 (,,) (H) + (,,) (H) + αn (,,) f(x) (H) = min(, αn f(x) min(, = x G f (H) x G f (H) + ~f (H) (,,) (H) + αn (,,) f(x) (H) ~f (H) ατ (,,) (G)) x f (H) x f (H) ατ (,,) (G)) ατ (,,) (G)) x G f (H) ατ (,,) (f (H))) + int (cl (int (f (H))))(x)) ~f (H) x f (H)

7 On α-continuity and α-openness in fuzzifying tritopological spaces = min(, αint (,,) (H)(f(x)) + int (cl (int (f (H))))(x)) = min(, f (αint (,,) (H)(f(x))) 6 = [f αc (i,j,k) ]. + int (cl (int (f (H))))(x)) Theorem. If (X, τ, τ, τ ), (Y, σ, σ, σ ) and (Z, γ, γ, γ ) are fuzzifying tritopological spaces then for any f Y X, g Z Y (i) αc (,,) (f) (αc (,,) (g) αc (,,) (g f)); (ii) αc (,,) (g) (αc (,,) (f) αc (,,) (g f)). Proof. (i) If [αc (,,) (g)] [αc (,,) (g f)], then its holds. else, if [αc (,,) (g)] > [αc (,,) (g f)], then [αc (,,) (g)] [αc (,,) (g f)] = W P(Z) αγ (,,) (W) + ατ (,,) (((g f) (W))) min(, αγ (,,) (W) + ασ (,,) (f (W))) (ασ (,,) (g (W)) + ατ (,,) (((g f) (W)) W P(Z) (ασ (,,) (H) + ατ (,,) ((f (H))). min(, W P(Z) Therefore [αc (,,) (g) αc (,,) (g f)] = min(, αc (,,) (g) + αc (,,) (g f)) (, ασ (,,) (H) + ατ (,,) ((f (H))) = αc (,,) (f). (ii)[αc (,,) (g) (αc (,,) (f) αc (,,) (g f))] = [ (αc (,,) (g). (αc (,,) (f). αc (,,) (g f)))] = [αc (,,) (f) (αc (,,) (g) αc (,,) (g f))]. 5 α-open mapping in fuzzifying tritopological spaces Definition 5. If (X, τ, τ, τ ) and (Y, σ, σ, σ ) are fuzzifying tritopological spaces. A unary predicate αo (,,) I(Y X ), called fuzzy (,,)-α open, is given by: f αo (,,) : = U (U ατ (,,) f(u) ασ (,,) ), i.e., αo (,,) (f) = U P(X) min(, ατ (,,) (U) + ασ (,,) (f(u))). Theorem 5. If (X, τ, τ, τ ) and (Y, σ, σ, σ ) are fuzzifying tritopological spaces and f Y X, we define

8 Barah M. Sulaiman and Tahir H. Ismail (i) f αo (,,) (ii) f αo (,,) (iii) f αo (,,) n Then f αo (,,) G (f(αint (,,) (G)) int (cl (int (f(g)))). H (αint (,,) (f (H)) f (int (cl (int (H))))); H (f (int (cl (int (H)))) αcl (,,) (f (H))); f αo (,,), n =,,. Proof. () [f αo (,,) ] = [ G (G ατ (,,) f(g) ασ (,,) ] = [G int (cl (int (G)))) f(g) int (cl (int (f(g))))] [f(g) f(int (cl (int (G))))) f(g) int (cl (int (f(g))))] = min(, f(int (cl (int (G)))))(y) y f(g) + int (cl (int (f(g))))(y))) y f(g) min(, f(αn (,,) y (G)))(y) + int (cl (int (f(g))))(y))) y Y = min(, f(αint (,,) (G)))(y) + int (cl (int (f(g))))(y))) y Y = [f αo (,,) ]. () To prove that [f αo (,,) ] = [f αo (,,) ] We have H P(Y ), [f(f (H)) H] =, then [αint (,,) (f(f (H))) αint (,,) (H)] =. So from Lemma (.) in [], we have [f (αint (,,) (f(f (H)))) f (αint (,,) (H))] =, moreover [cl (int (cl (f (H)))) f (f(cl (int (cl (f (H))))))] =, and we have G X, H Y, such that f(g) = H, then G f (H) and G f (f(g)). Hence (f) = [f(αint (,,) (G) int (cl (int (f(g))))] αo (,,) [f(αint (,,) (f (H))) int (cl (int (H))))] [f (f(αint (,,) (f (H)))) f (int (cl (int (H)))))] [αint (,,) (f (H)) f (int (cl (int (H)))))] = αo (,,) (f) (7) and αo (,,) (f) = [αint (,,) (f (H)) f (int (cl (int (H))))] [f(αint (,,) (f (H))) f(f (int (cl (int (H)))))] [f(αint (,,) (f (H))) int (cl (int (H)))]

9 On α-continuity and α-openness in fuzzifying tritopological spaces 5 [f(αint (,,) (G)) int (cl (int (f(g))))] = αo (,,) (f) () From () and () we get [f αo (,,) ] = [f αo (,,) ]. () We can easily prove that [f αo (,,) ] = [f αo (,,) ]. Theorem 5. If (X, τ, τ, τ ), (Y, σ, σ, σ ) and (Z, γ, γ, γ ) are fuzzifying tritopological spaces then for any f Y X, g Z Y (i) αo (,,) (g) (αo (,,) (f) αo (,,) (g f)); (ii) αo (,,) (f) (αo (,,) (g) αo (,,) (g f)). Proof. (i) If [αo (,,) (f)] [αo (,,) (g f)], then it holds. else, if [αo (,,) (f)] > [αo (,,) (g f)], then [αo (,,) (f)] [αo (,,) (g f)] = ατ (,,) (G) + αγ (,,) (((g f)(g))) min(, ατ (,,) (G) + ασ (,,) (f(g))) (ασ (,,) (f (G)) + αγ (,,) (((g f)(g)) (ασ (,,) (H) + αγ (,,) ((f(h))). min(, Therefore [αo (,,) (f) αo (,,) (g f)] = min(, αo (,,) (f) + αo (,,) (g f)) (, ασ (,,) (H) + αγ (,,) ((f(h))) = αo (,,) (g). (ii) [αo (,,) (f) (αo (,,) (g) αo (,,) (g f))] = [ (αo (,,) (f). (αo (,,) (g). αo (,,) (g f)))] = [αo (,,) (g) (αo (,,) (f) αo (,,) (g f))]. References [] A.A. Allam, A.M. Zahran, A.K. Mousa and H.M. Binshahnah, New types of continuity and openness in fuzzifying bitopological spaces, Journal of the Egyptian Mathematical Society, (06), [] B.M. Sulaiman and T.H. Ismail, On tri α-open sets in fuzzifying tritopological spaces, (submitted for publication). [] F.H. Khedr, F.M. Zeyada and O.R. Sayed, Fuzzy semi-continuity and fuzzy csemi-continuity in fuzzifying topology, J. Math. Inst., 7 (999), no., 05-.

10 6 Barah M. Sulaiman and Tahir H. Ismail [] F.H. Khedr, F.M. Zeyada and O.R. Sayed, α-continuity and cα-continuity in fuzzifying topology, Fuzzy Sets and Systems, 6 (000), [5] S.S. Kumar, On fuzzy pairwise α-continuity and fuzzy pairwise precontinuity, Fuzzy Sets Systems, 6 (99), 8. [6] S.S. Kumar, Semi-open sets, semi-continuity and semi-open mappings in fuzzy bitopological spaces, Fuzzy Sets Systems, 6 (99), 6. [7] A.S. Mashhour, I.A. Hasain and S.N. El-Deeb, α-continuous and α-open mappings, Acta Math. Hungar., (98), [8] O. Njåstad, On some classes of nearly open sets, Pacific J. Math., 5 (965), [9] M.K. Singal and N. Rajvanshi, Fuzzy alpha-sets and alpha-continuous maps, Fuzzy Sets and Systems, 8 (99), [0] M.S. Ying, A new approach for fuzzy topology (I), Fuzzy Sets and Systems, 9 (99), [] M.S. Ying, A new approach for fuzzy topology (II), Fuzzy Sets and Systems, 7 (99), -. [] M.S. Ying, A new approach for fuzzy topology (III), Fuzzy Sets and Systems, 55 (99), Received: September, 08; Published: October 5, 08