Fuzzy set as proposed by Zadeh [12] in 1965 is a framework to encounter uncertainty, vagueness and partial truth and it represents a degree of membership for each member of the universe of discourse to a subset of it. After the introduction of fuzzy topology by Chang [2] in 1968 there have been several generalizations of notions of fuzzy sets and fuzzy topology. Different aspects of such spaces have been developed, by several investigators. This paper is also devoted to the development of the theory of fuzzy topological spaces. K.K.Azad [1] introduced fuzzy continuity, fuzzy almost continuity and fuzzy weakly continuity in 1981. which appeals more accurately to uncertainty quantification and provides the opportunity to precisely model the problem, based on the existing knowledge and observations. In 2007 .Dr.Sadananda Patil [7] [8] [9] introduced the new class of fuzzy sets in fuzzy topological space. We introduce fuzzy weakly g**-Lindelof maps and fuzzy weakly g**-regular and also fuzzy weakly g**-normal spaces are also introduced, studied and some of their properties are obtained.

Throughout this paper (X, T), (Y,σ) & (Z,ƞ) or (simply X, Y & Z) represents non-empty fuzzy topological spaces on which no separation axiom is assumed unless explicitly stated. For a subset A of a space (X, T). cl (A) , int(A) & C(A) denotes the closure, interior and the compliment of A respectively.

Definition 2.01: A fuzzy set A of a fts (X, T) is called: 1) a generalized closed (g-closed) fuzzy set, if cl(A) ≤ U ,whenever A ≤ U and U is open fuzzy Set in (X, T). [7] [8] [9] 2) a weakly-generalized-closed (wg-closed) fuzzy Set, if cl(A) ≤ U , whenever A ≤ U and U is open fuzzy set in (X, T). [7] [8] [9] 3) a weakly-generalized* closed (wg*-closed) fuzzy set, if cl(A) ≤ U, whenever A ≤ U and U is open fuzzy set in (X,T). [7] [8] [9] Complement of g-closed fuzzy (respectively wg-closed fuzzy set and wg*-closed fuzzy set) sets are called g-open (respectively wg-open fuzzy set and wg*-open fuzzy set) sets. Definition 2.02: Let X, Y be two fuzzy topological spaces. A function f: X→Y is called 1) Fuzzy continuous (f-continuous) [7] [8] [9] if f-1(B) is open fuzzy set in X , for every open fuzzy set B of Y 2) Fuzzy generalized- continuous (fg-continuous) function [7] [8] [9] if f-1(A) is g-closed fuzzy set in X , for every closed fuzzy set A of Y 3) Fuzzy g*-continuous (fg*-continuous) function [7] [8] [9] if f-1(A) is g*-closed fuzzy set in X , for every closed fuzzy set A of Y

9

1) Fuzzy -open (f-open) [7] [8] [9] iff f (V) is open fuzzy set in Y, for every open fuzzy set in X. 2) Fuzzy g-open (fg-open) [7] [8] [9] iff f (V) is g-open- fuzzy set in Y, for every open fuzzy set in X. 3) Fuzzy g*-open (fg*-open) [7] [8] [9] iff f(V) is g-open- fuzzy set in Y, for every open fuzzy set in X. Definition 2.04: A fuzzy set A of fuzzy topological space in (X, T) is called weakly g**-closed fuzzy sets if cl(int(A)) ≤ U whenever A ≤ U and U is g*- open fuzzy set in (X,T). Definition 2.05: If A is any fuzzy set in a fts, then wg** cl(A)= ˄{U:U is wg**-closed fuzzy set and A ≤ U} wg**int(A)= ˅{V:V is wg**-open fuzzy set and A ≥ V} We introduced the following Definition 2.06: Let X and Y be two fts. A function f: X→Y is said to be fuzzy wg**-continuous (briefly fwg**-continuous) if the inverse image of every open fuzzy set in Y is wg**-open fuzzy set in X.[10] Definition 2.07: A function f: X→Y is said to be fuzzy wg**-irresolute (briefly fwg**-irresolute) if the inverse image of every wg**-closed fuzzy set in Y is wg**-closed fuzzy set in X.[10] Definition 2.08: A function f: X→Y is said to be fuzzy wg**-open (briefly fwg**-open) if the image of every open fuzzy set in X is wg**-open fuzzy set in Y.[10] Definition 2.09: A function f: X→Y is said to be fuzzy wg**-closed (briefly fwg**-closed) if the image of every closed fuzzy set in X is wg**-closed fuzzy set in Y.[10] Definition 2.10: Let X and Y be two fts. A bijective map f: X→Y is called fuzzy-homeomorphism (briefly f-homeomorphism) if f and f-1 are fuzzy-continuous.[10] Definition 2.11: A function f: X→Y is called fuzzy wg**-homeomorphism (briefly fwg**- homeomorphism) if f and f-1 are wg**- continuous.[10] Definition 2.12: Let X and Ybe two fts. A bijective map f: X→Y is called fuzzy fwg**- c-homeomorphism (briefly fwg**- c-homeomorphism) if f and f-1 are fuzzy wg**- irresolute.[10]

fuzzy set B in X if B ≤ ∨ A λ.

Definition 3.2: A fts X is called fuzzy wg**-compact if every fwg**-open cover of X has a finite sub cover. Definition 3.3: A fuzzy set A in X is said to be fwg**- compact relative to X if for every collection {Aλ: λ ∈ A} of wg**- open fuzzy sets of X such that A ≤ ∨ A λ , there exists a finite sub A0 of A such that A ≤ ∨ A λ Definition 3.4: A fuzzy set A of X is said to be fwg**- compact if A is fwg**-compact relative to X. Theorem 3.5: A wg**- closed crisp sub set of a fwg**- compact fts X is fwg**-compact as a sub space.

Proof: Let Y be a wg** - closed crisp subspace of fwg** - compact fts X.To prove that Y is fwg** - compact. Let u={Uλ: λ∈A} be any fwg** - open cover of Y.Then {Uλ: λ∈A}∨{1-Y} is a fwg**- open cover of X. Let x∈X then x∈Y or xY.If xY, Then x ∈ 1-Y.That is (1-Y)(x)=1where 1-Y∈{Uλ: λ∈A} ∨ {1-Y}. Suppose x∈Y Since {Uλ:λ∈A} is fwg**-open cover of Y,there exists Uλ0 such that Uλ0(x)=1. Thus v={Uλ: λ∈A}∨ {1-Y} is fwg**- open cover of X. Since X is wg**-compact fts ,v has a finite sub cover say v={Uλ1, Uλ2, Uλ3,………. Uλk}∨{1-Y} for X. Then the family u={Uλ1, Uλ2, Uλ3,………. Uλk} is a finite sub cover of u for Y. Let x∈Y,then x∈X and x1-Y. Since v is a finite sub cover of X,there exists Uλi {i=1,2,……..k}such that Uλi(x)=1. It follows that u is a fwg**- open cover u of Y has a finite sub cover u. Hence Y is fwg**-compact fts. Theorem 3.6: The image of a fwg**- compact fts under a fwg**- continuous map is fuzzy compact. Proof: Let f:X→Y be fwg**-continuous map from a fwg**-compact fts X onto fts Y. Let u={Aλ:λ∈A} be fuzzy open cover of Y by open fuzzy sets in Y .Then the collection v={f-1(Aλ):λ∈A} is a fwg**-open cover of X, since f is fwg**- continuous .Since X is fwg**- compact, v has a finite subcover {f-1(Aλ i):i=1,2,….,n}. Then {Aλ i: i=1,2,….,n}is a finite sub cover of u for Y and hence Y is fuzzy compact. Theorem 3.7: If a map f:X→Y is fwg**- irresolute and crisp subset B is fwg**-compact relative to X, then the image f(B) is fwg**-compact relative to Y. Proof: Let {Aλ: λ∈A} be any collection of wg** - open fuzzy sets of Y such that f(B)≤ ∨ Aλ , Then B ≤ ∨ f-1(Aλ) by hypothesis,there exists a finite subset A0 of A such that B≤∨ f-1(Aλ) Therefore,we have f(B)≤ ∨ Aλ. Which shows that f(B) is fwg**-compact relative to Y.

Satyamurthy V. Parvatkar1*, Sadanand N. Patil2 9

Proof: Follows from the previous result. Theorem 3.9: Every fwg**-compact fts is fuzzy compact. Proof: Let X be a fwg**-compact fts Let u={Aλ:λ∈A} be an f-open cover of X . And therefore u={Aλ:λ∈A} is a fwg**-open cover of X . Since X is a fwg**-compact, u has a finite sub cover for X. Hence X is fuzzy compact. Theorem 3.10: Every fwg**-compact fts is fg**- compact fts. Proof:Let X be a fwg**-compact fts.Let {Aλ:λ∈A} be a fg**-open cover of X .Since every g**-open fuzzy set is wg**-open, the cover {Aλ:λ∈A} is fwg**-open cover of X.Since X is fwg**-compact, {Aλ:λ∈A} has a finite sub cover and hence X is fg**-compact. Theorem 3.11: If f:X→Y is a strongly fwg**-continuous map fro a fuzzy compact fts X onto a fts Y.Then Y is fwg**-compact. Proof: Proof is omitted. Theorem 3.12: The image of a fwg**-compact fts under a strongly fwg**-continuous function is fwg**-compact fts. Proof: Proof is omitted. Theorem 3.13: If f:X→Y is completely fwg**-continuous map from a nearly fuzzy compact fts X onto ftsY.Then Y is fwg**-compact fts. Proof: Let u={Aλ:λ∈A} be any fwg**-open cover of Y. Since f is completely fwg**- continuous, then v={f-1(Aλ):λ∈A} is a fuzzy regular open cover of X .Also since X is fuzzy nearly compact, v has a finite sub cover v’={f-1(Aλ i):i=1,2,….,n}. Then u’={(Aλ i):i=1,2,….,n} is a finite sub cover of u for Y.Hence Y is fwg**-compact fts. Theorem 3.14: A fts X is fwg**-compact iff every family of wg**-closed fuzzy sets of X having f.i.p has a non-empty intersection. Proof: Proof is omitted. Definition 3.15: A fts X is said to be countably wg**-compact if every countable fwg**-open cover of X has a finite subcover. Theorem 3.16: Every countably wg**-compact fts is countably compact fts. a countable wg*-open cover of X .Since X is countably wg**-compact .u has a finite sub cover for X .Hence X is countably compact. Theorem 3.17:Every wg**-compact fts is countably wg**-compact. Proof:The proof is omitted. Theorem 3.18: A wg**-closed crisp subset of a countably fwg**-compact fts is countably wg**-compact. Proof: Proof is omitted. Theorem 3.19: The image of a countably wg**-compact fts under a fwg**-continuous map is countably compact. Proof: Let f:X→Y be fwg**-continuous map from a countably wg**-compact fts X onto fts Y. Let u={Uλ:λ∈A} be countable open cover of Y by open fuzzy sets in Y .Then the collection v={f-1(Uλ):λ∈A} is a countable wg*- open cover of X, since f is fwg**-continuous .Since X is countably wg**-compact, v has a finite subcover say v’={f-1(Uλ1),…. f-1(Uλn)}. Then u’= {Uλ1,…., Uλn}is a finite sub cover of Y and hence Y is countably compact fts. Theorem 3.20: The image of a countably wg**-compact space under a fwg**-irresolute map is countably wg**-compact. Proof: Proof is omitted. Theorem 3.21: If f:X→Y is a strongly fwg**-continuous map from a countably compact fts X onto a fts Y.Then Y is countably wg**-compact. Proof: The routine proof is omitted. Theorem 3.22: If f:X→Y is completely fwg**-continuous map from a nearly fuzzy compact fts X onto ftsY.Then Y is countably wg**-compact fts. Proof: Let u={Aλ:λ∈A} be any countable wg**-open cover of Y. Since f is completely fwg**- continuous, then v={f-1(Aλ):λ∈A} is a fuzzy regular open cover of X .Also since X is fuzzy nearly compact, v has a finite sub cover v’={f-1(Aλ i):i=1,2,….,n}. Then u’={(Aλ i):i=1,2,….,n} is a finite sub cover of u for Y.Hence Y is countably wg**-compact fts. Theorem 3.23: The image of a countably wg**-compact fts under a strongly fwg**-continuous function is countably wg**-compact fts.

1

Let u={Aλ:λ∈A}be a countable wg**-open cover of Y .Then the collection u’={f-1(Aλ):λ∈A} is a countable open cover of X ,since f is strongly fwg**-continuous function .And so u’={f-1(Aλ):λ∈A} is a countable wg**-open cover of X .Since X is countably fwg**-compact, u’ has a finite sub cover say {f-1(Aλ i):i=1,2,….,n}. Then {(Aλ i):i=1,2,….,n} is a finite sub cover of Y.And hence Y is countably fwg**-compact. Theorem 3.24: A fts X is countably wg**-compact iff every family of wg**-closed fuzzy sets of X having f.i.p has a non-empty intersection. Proof: The routine proof is omitted. Definition 3.25: A fts X is said to be wg**-Lindelof if every wg**-open cover of X has a countable sub cover. Theorem 3.26: Every wg**-Lindelof fts is Lindelof Proof: Follows from the two definitions. Theorem 3.27: Every wg**- Lindelof fts is g**- Lindelof fts Proof: Let X be wg**-Londelof fts .Let u={Uλ: λ∈A} be a g**-Open cover of X by g**-open fuzzy sets in X .Then u is a wg**-open cover of X by wg**-open fuzzy sets as every g**-open fuzzy set is wg**-open . since X is wg**-Lindelof fts, wg**-open cover u has a countable sub cover .Hence X is g**-Lindelof. Theorem 3.28: Every wg**-compact fts is wg*-Lindelof Proof: Follows from the two definitions. Theorem 3.29: If X is wg**-Lindelof and countably wg**-compact fts then X is wg**-compact. Proof:Suppose X is countably wg**-compact and wg**-Lindelof Let u be a wg**-open cover of X .since X is wg**-Lindelof, u has a countable sub cover say v.Therefore v is a countable cover of X and v≤u and so v is countable wg**-open cover of X . Again since X is countably wg**-compact ,v has finite sub cover say V.Therefore V≤v implies V≤u. Therefore V is a finite sub cover of u. Hence X is wg**-compact. Theorem 3.30:A wg**-closed crisp subset of a wg**-Lindelof fts is wg**-Lindelof as a subspace. Proof: Proof is omitted. Theorem 3.31: The image of a wg* Lindelof fts under a fwg**-continuous map is Lindelof fts. open cover of Y by open fuzzy sets in Y .Then the collection u*={f-1(Uλ):λ∈A} is a wg**-open cover of X by wg**-open fuzzysets in X ,since f is fwg**-continuous .Since X is wg**-Lindelof , u* has a countable subcover say v={f-1(Uλ1),…. f-1(Uλn)}. Therefore X= ∨ f-1(Uλn),implies f(X)=f(∨ f-1(Uλn)) Y=∨ Uλn Therefore { Uλn:n∈N} is a countable sub cover of u for Y .Hence Y is Lindelof. Theorem 3.32: The image of a wg**-Lindelof fts under a fwg**-irresolute map is wg**-Lindelof Proof: The routine proof is omitted. Theorem 3.33: Let f:X→Y be strongly fwg**-continuous map from a Lindelof fts X onto Y.Then Y is wg**-Lindelof . Proof: The easy proof is omitted. Theorem 3.34: The image of a wg**-Lindelof fts under a strongly fwg**-continuous map is wg**-Lindelof Proof: The easy proof is omitted Definition 3.35:A fts (X,T) is said to be wg**-regular fts if for each x∈X and wg**-closed fuzzy set A with A(x)=0, there exist open fuzzy sets G,H such that G(x)=1,AH and G1-H Theorem 3.36: Every wg**-regular fts is regular fts. Proof: Follows from the two concepts. Example 3.37: Let X={ a,b,c} fuzzy sets A,B,C and D be defined as follows: A={(a,1),(b,0),(c,0)},B={(a,0),(b,1),(c,1)}, C={(a,0),(b,1),(c,0)} and D={(a,0),(b,0),(c,1)}.(X,T) is fts with topology T={0,1,A,B}.Then (X,T) is wg**-regular fts, a∈X and B is wg**-closed with B(a)=0, then A and B are open fuzzy sets that A(a)= 1, B B and A1-B Theorem 3.38:Every g**-regular fts is wg**-regular fts.

Satyamurthy V. Parvatkar1*, Sadanand N. Patil2 1

fuzzy set in X. Since X is g** –regular fts ,therer exists open fuzzy sets G. H such that G(x)=1, A H and G1-H. Hence X is wg**-regular fts. The coverse of the above theorem need not be true be true as seen from the following example. Example 3.39:In the 2.37, the fts (X,T) is wg**-regular fts but not g**-regular fts as therer does not exist open fuzzy sets say G. H with G 1-H which contain c∈X and a g**-closed fuzzy setC. Theorem 3.40:The following three properties are equivalent 1) X is wg**-regular fts. 2) For each x∈X and a wg**-open fuzzy set U with U(x)=1, there exists an open fuzzy set V with V(x)=1, such that V U 3) For each x∈X and a wg**-closed fuzzy set A with A(x)=0.There is an open fuzzy set V with V(x)=1, such that A1- or 1-A Proof: Proof is omitted. Theorem 3.41:A fuzzy subspace of a wg**-regular fts is wg**-regular. Proof: Let Y be a fuzzy subspace of wg**-regular fts X.Let x∈Y and A be wg**-closed fuzzy set in Y with A(x)=0.Then there is a closed fuzzy set and so wg**- closed fuzzy set B of X such that A =BY and B(x)=0.Since X is wg**-regular fts,therer exist open fuzzy sets G,H such that G(x)=1,BH and G1-H. then GY and HY are open fuzzy sets such that (G Y)(x)=1,A HY and GY 1- HY. Hence Y is wg**-regular. Theorem 3.42:If f:XY is an open .fwg**-irresolute bijection and X is wg**-regular fts then Y is wg**-regular fts. Proof: Let y∈Y and A be wg**-closed fuzzy set of Y with A(y)=0. Since f is fwg**-irresolute, f-1(A) is wg**-closed fuzzy set in X .put f(x)=y,then (1- f-1(A))(x)=1. Since X is wg**-regular fts, there exists open fuzzy sets G ,H such that G(x)=1, f-1(A)H and G1-H. Since f is open and bijective,we have (f(G))(y)=1,Af(H) and f(G)1-f(H).This shows that Y is wg**-regular fts. Definition 3.43: A fts X is said to be wg**-normal ,if for every wg**-closed fuzzy set K and wg**-open fuzzy set Theorem 3.44:For a fts X the following statements are equivalent 1) X is a wg**-normal fts. 2) For any two wg**-closed fuzzy sets A and B in X such that A1-B, there exist open fuzzy sets C,D such that AC,BD and C1-D. 3) For any two wg**-closed fuzzy sets A and B in X such that A1-B there is an open fuzzy set C such that AC and 1-B. 4) For any two wg**-closed fuzzy sets A and B in X such that A1-B,there are open fuzzy sets C ,D such that AC, BD and 1-D. Proof: Proof is omitted. Example 3.45:In the example 2.37. X is wg**-normal fts ,as the fuzzy set A and B are wg**-closed fuzzy sets with AB. The fuzzy sets A and B are open fuzzy sets such that AA,B B,A1-B. Hence (X,T) is wg**-normal fts. Theorem 3.46:Every g**-normal fts is wg**-normal fts. Proof: Let X be a g**-normal fts . Let K be a wg**-closed and B be wg**-open fuzzy sets in X such that K B,then K is g**-closed and B is g**-open fuzzy sets and KB.Since X is g**-normal fts ,there exista fuzzy set A such that KAoB. Hence X is wg**-normal fts. The converse of the above theorem need not be true as seen from the following example. Example 3.47:in the example2.37, X is wg**-normal fts . but X is not g**-normal fts, as the fuzzy sets C,D are g**-closed fuzzy sets with C 1-D, but fuzzy sets say A and B such that CA,DB and A1-B.

REFERENCE:

K.K.Azad,On fuzzy continuity, fuzzy almost continuity and fuzzy weaklycontinuity, J.Math.Anal. Appl.82,14-32,(1981). C. L. Chang, Fuzzy topological spaces, J Math Anal Appl 24,182-190 (1968).

1

systems,13,187- 192(1984). A.Hayder.Es, Almost compactness and near compactness in fuzzy topological space,Fuzzy sets and systems,22,289-295(1987). R. Lowen, Fuzzy topological spaces & fuzzy compactness, J Math Anal Appl 56,621- 633 (1976). P.Sundaram and N.Nagaveni,Weakly generalised closed sets in topological spaces,Proc 84th Ind.sci.Cong.Delhi 18(1997). Sadanand.N.Patil, On g#- closed fuzzy set & fuzzy g#-continuous maps in fuzzy Topological spaces, proc of the KMANational seminar on Fuzzy Math & Appl, Kothamangalam (53-79). Sadanand.N.Patil, On g#-semi closed fuzzy sets and g#-semi contours maps in Fuzzy topological spaces. IMS conference Roorkey (UP) (26-30) Dec2005. Sadanand.N.Patil, On some Recent Advances in Topology. Ph.D Theses, Karnataka University, Dharwad 2008. Sadanand.N. Patil and Satyamurthy.V.Parvatkar, On Fuzzy weakly g**-Continuous Maps and Fuzzy weakly g**-Irresolute Mappings in Fuzzy Topological spaces,intl. conf on math sci and magnt.(ICMSM-16) held at Goa.11/09/2016. L.A.Zadeh, Fuzzy sets, Information and control,8(1965)338-353.

Assistant Prof., Department of Mathematics, KLE Institute of Technology, Hubballi, Karnataka (India)