1.
Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2251-2255 ON FUZZY TYCHONOFF THEOREM AND INITIAL AND FINAL TOPOLOGIES Dr. H.S.Nayal, Manoj Kabadwal Department of Mathematics,Govt.Postgradute College,Ranikhet (Uttrakhand)Abstract: Much of topology can be done in a setting Hold for all ai bi L; and all index sets I.where open sets have fuzzy boundaries. To render 1.2 Definition- An Let A on a set X is a functionthis precise; the paper first describes cl∞ - monoids; A:X→ L. X is called the carrier of A : L iswhich are used to measure the degree of membership called the truth set of A; and for xX; A(x)of points in sets. Then L- or fuzzy sets are defined; is called the degree of membership of x inand suitable collections of these are called L- A.Topological spaces. 1.3 Definition- An L-topological (or just L-) A number of examples and result for such space space is a pair <X; œ > such that X is a set ; œ are given. Perhaps most interesting is a version of the LX andTychonoff theorem which given necessary andsufficient conditions on L for all collection with (1) ع œ عœgiven cardinality of compact L- spaces to have (2) A: B œ A B œcompact product. (3) OX ; 1X œ .We know Tychonoff Theorem as Arbitrary productsof compact spaces are compact. Introduction-There 1.4 Theorem of Fuzzy Tychonoff Theorem- Forseems to be use in certain infinite valued logics [2] in solving fuzzy Tychonoff theorem we firstlyconstructive analysis [1]; and in mathematical define the following lemma.psychology [6] for notions of proximity less 1.5 Lemma- For a:b elements of a cl∞ - monoidrestrictive than those found in ordinary topology L;This paper develops some basic theory for spaces in a b ≤ a b.which open sets are fuzzy. First we need sufficiently Proof- a ≤ 1 , so 1 b = (a 1 ) bgeneral sets of truth values with which to measuredegree of membership. The main thing is no get = a b 1 benough algebraic structure. That is b = (ab) b;1.1 Definition- A cl∞ - monoid is a complete lattice L and therefore a b ≤ b. with an additional associative binary operation similarly a b ≤ a. 1.6 Theorem- Every product of α -compact L- such that the lattice zero 0 is a zero for . The spaces is compact iff 1 is α-isolated in L.” lattice infinity 1 is a identity for and the complete distributive laws Proof – with the help of theorem, if φ is a sub-base for <x: œ > and every cover of X by sets in φ has a finite sub-cover; then < X. œ > is compact. Now we have to proof fuzzy-Tychonoff theorem :- We have < Xi œi > compact for i I; |I| ≤ α and And < x: œ > = πiI. <Xi: œi >. Where œ has the sub-base. φ = {p1i (Ai)| Ai œ i; i I } So by the above theorem it suffices to show that no φ with FUP (finite union property) is a cover. Let φ with F U P be given: and for iI let = { A œi | Pi-1 (A) }. Then each i has FUP; i 2251 | P a g e
2.
Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2251-2255 For if nj=1 Aj = 1xi; First Vn Ain = (ai)x. since Where Pi-1 (Aj) Vn(ai)[N] = (ai)ℕ in <xi:œi ;> Then n and Pi-1 preserves suprema so that Vj=1 P-1i (Aj) = 1x VinAin= Vi (Vn Ain) = Vi (ai)x = l Since Pi (1xi) = 1x and P-1i preserves -1 Second no finite sub-collection Therefore is a non-cover for i {Ain | < i;n> J } ; |J| < can each i I and there exits xi Xi such that Cover; since V<i,n> J Ain (x) = O (V) (xi) = ai <1 i For any x with Now let x = < xi > X and Xi ≥ V {n| <i; n> J } For any i ’ = { P-1i (A)| A œi } ∩ I and infinitely many such x always exist. Then φ implies = I ’ and I 1.8 Initial and Final Topologies- In this paper we ( V ) (xi) = ai implies i introduce the notations of initial and final ( V) (x) = ai i topologies. For this firstly we defined initial Since ( V ’ ) (x) = V { (Pi-1 (A)) (x) | A I and final topologies then their theorems. Weœi show that from a categorical point of view and Pi-1 (A) } they are the right concept of generalize the topological ones. We show that with our notion of= V { A (Pi(x))| Pi-1 (A) fuzzy compactness the Tychonoff product and Aœ i } = (V )a (xi). i theorem is safeguarded. We also show that ’ this is not the case for weak fuzzy Therefore (V (x) = V ((VI) (x)) ) iI compactness. = V ai < 1 1.9 Initial Fuzzy Topologies- If we consider a set E; i a fuzzy topological space (f:y) and a function since I is α- isolated in I; each ƒ : E → F then we can a i < 1 and |I| ≤ α . define ƒ-1 (Y) = { ƒ-1 ( ג : )ג Y }1.7 Theorem- If 1 is not α –isolated in L; then there It is easily seen that ƒ-1 (Y) is the smallest is a collection <xi: œ i> for i I and |I| = α fuzzy topology making ƒ fuzzy continuous. of compact L-spaces such that the product More generally: consider a family of fuzzy <x:œ > is non-compact. topological space Proof- non α- isolation given I with |I|= α (Fj : Yj) jJ and for each j J a function and ai <1 for i I such that Vi ai = 1. ƒj : E → fj then it is easily seen that the union Let Xi = ℕ for iI and Let œ i = { 0, (ai)[n]. 1 }*; where As ; for a L and S Xi; denote the ƒ-1j(Yj) function equal to a on S and O or Xi-S; j J where [n] = { 0,1 ……………. N-1} and where φ indicates the L- topology generated by φ as a sub-basis. is a sub-base for a fuzzy topology on E making every ƒi fuzzy continues . we denote Then < ℕ, œi> is a compact. As every A it by œi except 1 is contained in (ai)N; and (ai)N <1; that œi is a cover iff 1 therefore every cover has {1} as a sub-cover. ƒ-1j (Yj) or sup But <x : œ > = π iI <Xi : œi > is non -1 ƒ j (Yj) compact. Let Ain denote the open L-set Pi-1 ((ai) [n]) œ : for i I j J jJ at is given for x X by the formula Ain (x) = a i if xi<n; 1.10 Definition- The initial fuzzy topology on E for and Ain (x) = O otherwise the family of fuzzy topological space (Fj: { Ain} i I; n ℕ } Yj) j J and the family of functions is an open cover with no finite sub-cover. ƒj : E → (Fj : yj) 2252 | P a g e
3.
Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2251-2255 is the smallest fuzzy topology on E making ƒ E : i (ƒ-1 ()) → F. each function ƒj fuzzy continuous. i () is continuous and this implies Within the notations of the definition let (G:) be a fuzzy topological i (ƒ-1()) ƒ-1 (i()). space and let ƒ be a function from G to E then ƒ will be fuzzy continuous iff all the 1.15 Theorem- If E is a set (Fj: Tj)jJ is a family of compositions ƒjoƒ are fuzzy continues. topological space and Moreover the initial fuzzy topology on E is the finest one possessing this property; it is ƒj : E → Ej is a family of equally clear that the transitivity property for functions then we have initial topologies holds for initial fuzzy topologies. Sup ƒ-1j (w(Tj))= w (supƒj- 1 (Tj))1.11 Definition- If (Ej, j) j J is a family of fuzzytopological space; then the fuzzy product topology on jJ jJπjJ Ej is defined as the initial fuzzy topology on πjJ Proof- we can prove this theorem By the followingEj for the family of spaces (Ej. j) j J and function lemma:-ƒi: πjJ Ej → Ei Lemma 1:-if ƒ: E → F : then Where i J i (ƒ-1 ( )) = ƒ-1(i()) = i (ƒ-1()). ƒ i = Pri, Lemma 2:-if ƒ: E → F: W (T) then the protection on the ith coordinate. W (ƒ-1(T)) = ƒ-1 (W(T)).1.12 Definition- if (Ej, j)jJ is a family of fuzzy topological space then the fuzzy product Consider now a set E; a family of fuzzy topology, on πjJ Ej is defined as the initial topological spaces (Fj: j)jJ and a family of fuzzy topology on πjJ Ej for the family of functions ƒj : E → Fj spaces (Ej,j)jJ and functions ƒi: πjJ πjJ Ej → Ei where i J fi = pri the projection Then on the one hand: we can consider the on ith coordinate we denote this fuzzy initial fuzzy topology as defined and take topology πjJ j. the associated topology1.13 Theorems of Initial Topologies- The i.e. i (sup) ƒ-1 (j)) Theorems related to initial fuzzy topologies is as follows- jJ1.14 Lemma- if ƒ: E → F, then on the other hand we can consider the family of topological spaces (Fj; i (j))jJ and take i (ƒ-1 ( )) = ƒ-1 (i( )) = i (ƒ-1()). the initial topology i.e. Proof - ƒ:E; ƒ-1 (i ()) → F. Sup ƒ-1j (i(j)). i () is continuous: thus ƒ: E; W (ƒ- jJ1 (i( )))→ F; Showing that these two topologies are equal is fuzzy continuous. is trivial. This implies W (ƒ-1 (i( ))) ƒ-1 () ƒ-1 () 1.16 Theorem- if E is a set (Fj. j)jJ is a family of ƒ-1 (i()) i (ƒ-1 ( ))) i (ƒ-1 ()). fuzzy topological spaces: and Further: ƒ: E ; ƒ-1 ( ) → F ƒj : E → Fj is a family of function: then we have is fuzzy continuous thus 2253 | P a g e
4.
Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2251-2255 i (sup ƒ- ƒj: (Ej:j) → F1 j (j)) = sup ƒj-1(i(j)). is the finest fuzzy topology on F making jJ each functions ƒj: fuzzy continuous.jJ And j J :Proof- we can prove this theorem by the help oflemma 1 Let j = (Tj) if ƒ: E → F: then Then on the other hand we have i (ƒ-1()) = ƒ-1(i()) = i (ƒ- ∩ ƒj ( (Tj)). 1 ()). jJ It follows that jJ and on the other hand we have ƒj-1 (i(j)) = i ƒ-ij (j). W ( ∩ ƒj (Tj)) and the result then follows analogously by the lemma 2 jJ again we can show the two fuzzyLemma 3:- if E is a set and (Tj)jJ is a family of topological to be the same. topologies on E then 1.19 Lemma 1- if ƒ:E → F Sup (Tj) = (sup Tj). Then we have i (ƒ()) = ƒ(i()) i (ƒ()). jJ jJ Proof- to show the inclusions1.17 Final Fuzzy Topologies- Consider a set F a i (ƒ( )) ƒ (i()) i (ƒ())fuzzy topological space (E:): and a function we proceed as we proceeded in lemma:- ƒ: E → F if ƒ: E → F: then Then we can define i (ƒ-1()) = ƒ-1 (i()) = i (ƒ-1 ( )). ƒ () = <{ :ƒ-1 ( ) }> The remaining inclusion is shown using the latter one upon . more generally consider a family of fuzzy topological space (Ej: j)jJ and for each jJ i.e i (ƒ()) ƒ(i()) = ƒ(io a function (i()))=ƒ (i()). ƒj : Ej → F. 1.20 Lemma 2- if ƒ:E: (T) → F then we have It is easily seen that the intersection ƒ ((T)) = (ƒ (T)). ∩ ƒj (j) Proof- from lemma as proved above. at follows with jJ = (T) then is the finest fuzzy topology on F making all the functions fj fuzzy continuous. ƒ ( (T)) = W (ƒ (T))1.18 Definition- the final fuzzy topology on F: for the However: ƒ ( (T) is topologically family of fuzzy topological spaces (Ej:j)jJ generated indeed it is the finest fuzzy and the family of functions topology making 2254 | P a g e
5.
Dr. H.S.Nayal, Manoj Kabadwal / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue4, July-August 2012, pp.2251-2255 ƒ: E : (T) → F: ƒ (W(T)) fuzzy continuous But since (T) is topologicallygenerated ƒ: E: (T) → F: ƒ ((T)) still is fuzzy continuous and thus ƒ ((T) = ƒ (w(T)).1.21 Conclusion- In this section we discussed about the fuzzy Tychonoff theorems and initial and final fuzzy topology fuzzy Tychonoff theorems is describes as “ arbitrary products of compact spaces are compact”. We show that with our notion of fuzzy compactness the Tychonoff product theorem is safeguarded.REFERENCES: [1] E. Bishop, “Foundations of Constructive Analysis,” McGraw-Hill, New York, 1967. [2] J.A. Goguen; Catogories of fuzzy sets; Ph. D. Thesis; university of cal. At Berkeley; Dept. of Math. 1968. (available through University microfilm). [3] J.A. Goguen; L-fuzzy sets; J.math Anal. Appl. 18 (1967); 145 -174. [4] J.L. Kelley “ General Topology” Van- Nostrand: Princetor: new Jersey : 1955. [5] L.A. Zadeh; fuzzy sets inform and control 8 (1965); 338-358 [6] S. Mac Lane, “Categories for the Working Mathematician,” Spriner-Verlag New York, 1971 [7] P.M. Cohn, “universal Algebra,” Harper and Row, New York, 1965. 2255 | P a g e
Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.
Be the first to comment