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Abstract: In this paper, we introduce the concept of l_{(m.
n)} - J- closed sets in bigeneralized topological spaces and study some of their properties. The notion of Jg_{(m, n)}- continuous functions and strongly l_{(m.
n)} - J- closed set is also defined on bigeneralized topological spaces and investigate some of their characterizations.

Mathematics Subject Classification: 54A05, 54A10

Keywords: bigeneralized topological spaces, l_(m.n)-J-closed sets,Jg_(m,n)- continuous functions, l_{(m,
n)} ï¿½ J ^s ï¿½ closed set.

1. Introduction

ï¿½.Csï¿½szï¿½r [3] introduced the concepts of generalized neighborhood systems and he also introduced the concepts of continuous functions and associated interior and closure operators on generalized neighborhood systems and generalized topological spaces. In particular, he investigated characterizations for generalized continuous functions (= (g,g^ï¿½)- continuous functions). In [4], he introduced and studied the notions of g- a - open sets, g - semi-open sets, g ï¿½ pre open sets and g - b open sets in generalized topological spaces.

After the introduction of the concept of generalized closed set by Levine[9] in a topological spaces. Several other authors gave their ideas to the generalizations of various concepts in topology. Kelly[8] introduced the concept of bitopological spaces .Since many mathematicians generalized the topological concepts into bitopological setting. In 1986, Fukutake [7] generalized the notion to bitopological spaces and he defined a set ij-generalized closed set. Some new types of generalized closed sets in bitopological spaces were introduced. In this paper, we introduce the notion of l_{m,
n}- J - closed sets in bigeneralized topological spaces and study some of their properties. We also introduce the concept of strongly l_{m,
n}- J - closed sets and strongly l_{m,
n}- J -continuous functions and investigate some of their characterizations.

2. Preliminaries

Let X be a non ï¿½ empty set and l be a collection of subsets of X. Then l is called a generalized topology (briefly GT) on X iff f ï¿½ l and G_iï¿½ l for i ï¿½ I ï¿½ f implies G = .We call the pair (X,l), a generalized topological space(briefly GTS)on X. The elements of l are called l- open sets and the complements are called l- closed sets. The generalized closure of a subset S of X, denoted by c_l(S), is the intersection of generalized closed sets including S and the interior of S, denoted by i_l(S), is the union of generalized open sets contained in S.

Definitions 2.1. Let (X, l) be a generalized topological space and A ï¿½ X, then A is said to be

(1) l - semi open if A ï¿½ c_l(i_l(A))

(2) l - pre open if A ï¿½ i_l(c_l(A))

(3) l - a- open if A ï¿½ i_l(c_l(i_l(A)))

(4) l - b open if A ï¿½ c_l(i_l(c_l(A)))

The complement of l - semi open (resp l - pre open , l - a- open, l - b open) is said to be l - semi closed (resp l - pre closed , l - a- closed, l - b closed). The class of all l - semi open sets on X is denoted by s(l_x)(briefly s_x or s). The class of l - pre open (l - a- open and l - b open) sets on X as [p(gx), a(gx), b(gx) ] or briefly [[p, a, b].

Definition 2.3. [2] Let X be non-empty set and let l₁ and l₂ be generalized topologies on X. A triple (X, l₁, l₂) is said to be a bigeneralized topological space (briefly BGTS).

Let (X, l₁, l₂) be a bigeneralized topological space and A be a subset of X. The closure of A and the interior of A with respect to l_m are denoted by c_lm(A) and i_lm(A), respectively, for m = 1,2.

Definition 2.4. [10]Let (X, l) be a generalized topological space and A ï¿½ X, then A is said to be l - J- open if A ï¿½ i_l(c_π(A)). The complement of l - J- open is said to be l - J- closed.

The class of all l - J - open sets on X is denoted by λ ï¿½ JO(X). The class of all l - J - closed sets on X is denoted by λ ï¿½ JC(X).The l - J- closure of a subset S of X, denoted by c_J(S) is the intersection of l - J- closed sets including S. The l - J- interior of a subset S of X, denoted by i_J(S) is the union of l - J- open sets contained in S.

The class of all l - J - open sets is properly placed between l - open sets and l - pre ï¿½ open sets.

3. l_{(m,
n)} - J- closed sets

Definition 3.1. A Subset A of a bigeneralized topological space (X, l₁, l₂) is said to be l_{(m,
n)} ï¿½ J - closed set if c_ln(A) ï¿½ U, whenever A ï¿½ U and U is l_m ï¿½ J- open, where m, n = 1,2 and mï¿½n. The complement of l_{(m,
n)} ï¿½ J - closed set is said to be l_{(m,
n)} ï¿½ J-open set.

Remark 3.2. Every (m, n) - closed set is l_{(m,
n)} ï¿½ J - closed.

The converse is not true as can be seen from the following example.

Example 3.3. Let X = {a, b, c}. Consider two generalized topologies l₁ = {f,{a},{a, b}, X}, and l₂ = {f,{c},{b, c}},.Let A = {a, b}, then A is l_{(m,
n)} ï¿½ J- closed but not (m, n) - closed.

Proposition 3.4. Every l_{(m,
n)} ï¿½ closed set is l_{(m,
n)} ï¿½ J- closed.

Proof: Let U be a l_m ï¿½ J- open set such that A ï¿½ U. Since A is l_{(m,
n)} ï¿½ closed set, c_ln(A) ï¿½ U. Therefore A is l_{(m,
n)} ï¿½ J- closed.

Proposition 3.5. Let (X, l₁, l₂) be a bigeneralized topological space and A be a subset of X. If A is l_n ï¿½ closed, then A is l_{(m,
n)} ï¿½ J- closed, where m, n = 1, 2 and m ï¿½ n.

Remark 3.6.(i) The union of two l_{(m,
n)} ï¿½ J - closed need not be l_{(m,
n)} ï¿½ J- closed.

(ii) The intersection of two l_{(m,
n)} ï¿½ J- closed sets need not be l_{(m, n)} ï¿½ J ï¿½ closed.

Proposition 3.7. Let (X, l₁, l₂) be a bigeneralized topological space. If A is l_{(m,
n)} ï¿½ J - closed and F is (m, n) ï¿½ J- closed, then A ï¿½ F is l_{(m,
n)} ï¿½ J ï¿½ closed, where m, n = 1, 2 and m ï¿½ n.

Proof: Let U be a l_m ï¿½ J- open set such that A ï¿½ F ï¿½ U. Then A ï¿½ U ï¿½ (X - F) and so c_ln(A) ï¿½ U ï¿½ (X - F).Therefore c_ln(A) ï¿½ F ï¿½ U .Since F is (m, n) ï¿½ J- closed, c_ln (A ï¿½ F) ï¿½ c_ln (A) ï¿½ c_ln (F) ï¿½ U . Hence A ï¿½ F is l_{(m,
n)} ï¿½ J ï¿½ closed.

Proposition 3.8. For each element x of a bigeneralized topological space (X, l₁, l₂), {x} is l_m - J- closed or X ï¿½ {x} is l_{(m,
n)} - J - closed, where m, n = 1, 2 and m ï¿½ n.

Proof: Let x ï¿½ X and the singleton {x} be not l_m ï¿½ J- closed. Then X ï¿½ {x} is not l_m ï¿½J- open, if X ï¿½ l_m, then X is only l_m ï¿½ J- open set which contains X ï¿½ {x}, hence X ï¿½ {x} is l_{(m,
n)} ï¿½ J- closed and if X ï¿½ l_m, then X ï¿½ {x} is l_{(m,
n)} ï¿½ Jï¿½ closed.

Proposition 3.9. Let (X, l₁, l₂) be a bigeneralized topological space. Let A ï¿½ X be a l_{(m,
n)} ï¿½ J - closed subset of X, then c_ln(A) \ A does not contain any non- empty l_m- J- closed set, where m, n = 1, 2 and m ï¿½ n.

Proof: Let A be a l_{(m,
n)} ï¿½ J - closed set and F ï¿½ f is l_m ï¿½ J- closed such that F ï¿½ c_ln (A) \ A. Then F ï¿½ X \ A and hence A ï¿½ X \ F. Since A is l_{(m,
n)} ï¿½ J- closed, c_ln (A) ï¿½ X \ F and hence F ï¿½ X \ c_ln (A) .So, F ï¿½ c_ln(A) ï¿½ (X \ c_ln(A)) = f.Therefore c_ln (A) \ A does not contain any non- empty l_m- J- closed set.

Proposition 3.10. Let l₁ and l₂ be generalized topologies on X. If A is l_{(m,
n)} - J- closed set, then c_lm({x}) ï¿½ A ï¿½ f holds for each x ï¿½ c_ln(A), where m, n = 1, 2 and m ï¿½ n.

Proof: Let x ï¿½ c_ln(A). Suppose that c_lm({x}) ï¿½ A = f. Then A ï¿½ X - c_lm({x}). Since A is l_{(m,
n)} ï¿½ Jï¿½ closed and X - c_lm({x}) is l_m ï¿½ J- open. Thus c_ln(A) ï¿½ X - c_lm({x}). Hence c_ln(A) ï¿½ c_lm({x}) = f. This is a contradiction.

Proposition 3.11. If A is a l_{(m,
n)} ï¿½ J - closed set of (X, l₁, l₂) such that A ï¿½ B ï¿½ c_ln(A) then B is l_{(m,
n)} ï¿½ J - closed set, where m, n = 1, 2 and m ï¿½ n.

Proof: Let A be a l_{(m,
n)} ï¿½ J- closed set and A ï¿½ B ï¿½ c_ln(A). Let B ï¿½ U and U is l_m ï¿½ J-open. Then A ï¿½ U. Since A is l_{(m,
n)} ï¿½ J ï¿½ closed, we have c_ln(A) ï¿½ U. Since B ï¿½ c_ln(A), then c_ln(B) ï¿½ c_ln(A) ï¿½ U. Hence B is l_{(m,
n)} ï¿½ J ï¿½ closed.

Remark.3.12. l_(1,2) ï¿½ JC(X) is generally not equal to l_(2,1) ï¿½ JC(X) as can be seen from the following example.

Example 3.13. Let X = {a, b, c}. Consider two generalized topologies l₁ = {f, {a}, {a, b}, X} and l₂ = {f, {c}, {b, c}}.Then l_(1,2) ï¿½ JC(X) = {{a}, {b},{a, b}, {b, c},X} and l_(2,1) ï¿½ JC(X) = {{b}, {c},{a, b}, {b, c},f, X }. Thus l_(1,2) ï¿½ JC(X) ï¿½ l_(2,1) ï¿½ JC(X).

Proposition 3.14. Let l₁ and l₂ be generalized topologies on X. if l₁ ï¿½ l₂, then l_(2,1) ï¿½ JC(X) ï¿½ l_(1,2) ï¿½ JC(X)

Proposition 3.15. For a bigeneralized topological space (X, l₁, l₂), l_m- JO(X) ï¿½ l_n- JC(X)

if and only if every subset of X is l_{(m,
n)} - J - closed set, where m, n = 1, 2 and m ï¿½ n.

Proof: Suppose that l_m- JO(X) ï¿½ l_n- JC(X). Let A be a subset of X such that A ï¿½ U, where U ï¿½ l_m- JO(X). Then c_ln(U) ï¿½ c_ln(A) = U and hence A is l_{(m,
n)} ï¿½ J ï¿½ closed set.

Conversely, suppose that every subset of X is l_{(m,
n)} ï¿½ J ï¿½ closed. Let U ï¿½ l_m- JO(X).Since U is l_{(m,
n)} ï¿½ J ï¿½ closed, we have c_ln(U) ï¿½ U. Therefore, U ï¿½ l_n- JC(X) and hence l_m- jO(X) ï¿½ l_n- JC(X).

Proposition 3.16. Let (X, l₁, l₂) be a bigeneralized topological space. Let A ï¿½ X be a l_{(m,
n)} ï¿½ J - closed subset of (X, l₁, l₂). Then A is l_m- J- closed if and only if c_ln(A) \ A is l_m- J- closed, where m, n = 1, 2 and m ï¿½ n.

Proof: Let A be a l_{(m,
n)} ï¿½ J- closed set. If A is l_m ï¿½ J- closed, then c_ln(A) \ A = f, but f is l_m ï¿½ J- closed . Therefore c_ln (A) \ A is l_m- J- closed.

Conversely, Suppose that c_ln (A) \ A is l_m- J- closed, As A is l_{(m,
n)} ï¿½ Jï¿½ closed, c_ln(A) \ A = f, Consequently c_ln(A) = A.

Proposition 3.17. Let (X, l₁, l₂) be a bigeneralized topological space. If A is l_{(m,
n)} ï¿½ J - closed and A ï¿½ B ï¿½ c_ln(A), then c_ln (B) \ B has no non empty l_m- J- closed subset.

Proof: A ï¿½ B implies X ï¿½ B ï¿½ X ï¿½ A and B ï¿½ c_ln(B) implies c_ln((c_ln(A)) = c_ln(A).Thus c_ln(B) ï¿½ (X - B) ï¿½ c_ln(A) ï¿½ (X -A) which yields c_ln(B) / B ï¿½ c_ln(A) / A. As A is l_{(m,
n)} ï¿½ J ï¿½ closed, c_ln (A) \ A has no non empty l_mï¿½ J- closed subset. Therefore c_ln (B) \ B has no non empty l_m- J- closed subset.

Remark 3.18.(i) The intersection of two l_{(m,
n)} ï¿½ J- open sets need not be l_{(m,
n)} ï¿½ J- open

(ii) The union of two l_{(m,
n)} ï¿½ J- open sets need not be l_{(m,
n)} ï¿½ J ï¿½ open.

Proposition 3.19. A subset of a bigeneralized topological space (X, l₁, l₂) is l_{(m,
n)} - J - open iff every subset of F of X, F ï¿½ i_ln (A) whenever F is l_m ï¿½ J- closed and F ï¿½ A, where m, n = 1, 2 and m ï¿½ n.

Proof: Let A be a l_{m,
n} ï¿½ J- open. Let F ï¿½ A and F is l_m ï¿½ J-closed. Then X ï¿½ A ï¿½ X ï¿½ F and X ï¿½ F is l_m ï¿½ J- open, we have X ï¿½ A is l_{(m,
n)} ï¿½ Jï¿½ closed, then c_ln(X - A) ï¿½ X - F. Thus X - i_ln (A) ï¿½ X ï¿½ F and hence F ï¿½ i_ln (A).

Conversely, let F ï¿½ i_ln (A) where F is a l_m ï¿½ J- closed set such that F ï¿½ A. Let X - A ï¿½ U where U is a l_mï¿½ J- open. Then X - U ï¿½ A and X - U is l_m ï¿½ J- closed. By the assumption, X - U ï¿½ i_ln (A), then X - i_ln (A) ï¿½ U. Therefore, c_ln(X - A) ï¿½ U. Hence X - A is l_{(m,
n)} - J - closed and hence A is l_{(m,
n)} -J - open.

Proposition 3.20. Let A and B be subsets of a bigeneralized topological space(X, l₁, l₂) such that i_ln (A) ï¿½ B ï¿½ A. If A is l_{(m,
n)} - J - open then B is also l_{(m,
n)} - J- open, where m, n = 1, 2 and m ï¿½ n.

Proof: Suppose that i_ln (A) ï¿½ B ï¿½ A. Let F be a l_m ï¿½ J- closed set such that F ï¿½ B. Since A is l_{(m,
n)} - J - open, F ï¿½ i_ln. Since i_lnï¿½ B, we have i_ln(i_ln(A)) ï¿½ i_ln(B). Consequently i_ln(A) ï¿½ i_ln(B). Hence F ï¿½ i_ln(B). Therefore, B is l_{(m,
n)} - J -open.

Proposition 3.21. If A be a subset of a bigeneralized topological space (X, l₁, l₂) is l_{(m
,n)} - J- closed , then c_ln(A) - A is l_{m,
n} ï¿½ J ï¿½ open, where m, n = 1, 2 and m ï¿½ n.

Proof: Suppose that A is l_{m,
n} ï¿½ J ï¿½ closed. Let X ï¿½ (c_ln(A) - A) ï¿½ U and U is l_mï¿½ J- open. Then X ï¿½ U ï¿½ X ï¿½ (c_ln(A) - A) and X ï¿½ U is l_mï¿½ J- closed. Thus we have c_ln(A) - A does not contain non ï¿½ empty l_mï¿½ J- closed by Proposition 3.10 . Consequently, X ï¿½ U = f, then U = X. Therefore, c_ln(X- ( c_ln(A) ï¿½ A)) ï¿½ U, so we obtain X ï¿½ (c_ln(A) ï¿½ A) is l_{m,
n} ï¿½ J- closed. Hence c_ln(A) - A is l_{m,
n} - J - open.

4. Jg_{(m, n)}- continuous functions

Definition 4.1. Let (X,l_X¹, l_X²) and (Y,l_Y¹, l_Y²) be generalized topological spaces. A function f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is said to be (m, n) - J- generalized continuous (briefly Jg_{(m,
n)}- continuous) if f^-1(F) is l_{(m,
n)}- J-closed in X for every l_n- closed F of Y, where m, n = 1,2 and mï¿½ n.

A function f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is said to be pairwise J-generalized continuous(briefly pairwise Jg-continuous) if f is Jg_(1,2)-continuous and Jg_(2,1)-continuous.

Proposition 4.2. For an injective function f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²), the following properties are equivalent:

(1) f is Jg_{(m, n)}- continuous,

(2) For each x ï¿½ X and for every l_n- open set V containing f(x), there exists a l_{(m,
n)}-J-open set U containing x such that f(U) ï¿½ V:

(3) f((A)) ï¿½ (f(A)) for every subset A of X;

(4) (f^-1(B)) ï¿½ f^-1((B) for every subset B of Y.

Proof: (1) ï¿½(2): Let x ï¿½ X and V be a l_n-open subset of Y containing f(x). Then by (1), f ^-1(V) is l_{(m,
n)}-J-open of X containing x. If U = f ^-1(V), then f (U) ï¿½ V.

(2)ï¿½(3): Let A be a subset of X and f(x) ï¿½ ((f(A)). Then, there exists a l_n ï¿½ open subset V of Y containing f(x) such that V ï¿½ f(A) = f.Then by(2), there exist a l_{(m,
n)}-J-open set such that f(x) ï¿½ f(U) ï¿½ V. Hence, f(U) ï¿½ f(A) = f implies U ï¿½ A = f. Consequently, x ï¿½ and f(x) ï¿½ f ((A)).

(3) ï¿½(4): Let B be a subset of Y. By (3) we obtain f((f ^-1(B)) ï¿½ (f(f ^-1(B)). Thus (f ^-1(B)) ï¿½ f ^-1 ( (B)).

(4) ï¿½(1): Let F be a l_n ï¿½ closed subset of Y. Let U be a l_m ï¿½ J- open subset of X such that f^-1(F) ï¿½ U. Since (F) = F and by (4), (f ^-1(F)) ï¿½ U. Hence f is Jg_{(m, n)} continuous.

Definition 4.3. Let (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) be generalized topological spaces. A function f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is said to be Jg_m- continuous if f ^-1(F) is l_m ï¿½J-closed in X for every l_m ï¿½ closed F of Y, for m = 1,2.

Definition 4.4. Let (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) be generalized topological spaces. A function f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is said to be Jg_m- closed(resp. Jg_m- open) if f(F) is l_m ï¿½ J-closed(resp l_m ï¿½J- open) of Y for every l_m ï¿½ closed (resp. l_m ï¿½ open) F of X, for m = 1,2.

Proposition 4.5. If f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is Jg_m- continuous and Jg_m- closed , then f(A) is l _{(m, n)}- J-closed subset of Y for every l _{(m, n)}- J-closed subset A of X, where m, n = 1,2 and mï¿½ n.

Proof: Let U be l_m ï¿½ J- open subset of Y such that f (A) ï¿½ U. Then A ï¿½ f^-1(U) and f^-1(U) is l_m ï¿½ J-open subset of X. Since A is l _{(m, n)}- J-closed, (f(A)) ï¿½ f^-1(U) and hence f( (A)) ï¿½ U. Therefore we have (f(A)) ï¿½ (f((A))) = f((A)) ï¿½ U. Therefore, f(A) is l _{(m, n)}- J-closed subset of Y.

Lemma 4.6. If f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is Jg_{m ï¿½}closed, then for each subset S of Y and each l_m ï¿½ J- open subset U of X containing f ^-1(S), there exists a l_m ï¿½ J- open subset V of Y such that f^-1(V) ï¿½ U.

Proposition 4.7. If f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is injective, Jg_m-closed and Jg_{(m ,n)}-continuous, then f^-1(B) is l _{(m, n)}- J-closed subset of X for every l _{(m, n)}- J-closed subset of B of Y, where m, n = 1,2 and mï¿½ n.

Proof: Let B be a l _{(m, n)}- J-closed subset of Y. Let U be a l_m ï¿½ J- open subset of X such that f^-1(B) ï¿½ U. Since f is Jg_m-closed and by lemma 4.6, there exists a l_m ï¿½ J- open subset of Y such that B ï¿½ V and f^-1(V) ï¿½ U. Since B is l _{(m, n)}- J-closed set and B ï¿½ V, then (B) ï¿½ V. Consequently, f^-1((B)) ï¿½ f^-1(V) ï¿½ U. By theorem 4.2, (f^-1(B)) ï¿½ f^-1((B) ï¿½U and hence f ^-1(B) is l _{(m, n)}- J-closed subset of X.

Definition 4.8. A bigeneralized topological space (X,l_X¹, l_X²) is said to be l _{(m, n)} ï¿½ JT_1/2 ï¿½ space if, for every l _{(m, n)}- J-closed set is l_n ï¿½ closed, where m, n = 1,2 and mï¿½ n.

Definition 4.9. A bigeneralized topological space (X,l_X¹, l_X²) is said to be pairwise l ï¿½ JT_1/2 ï¿½ space if it is both l _{(1, 2)} ï¿½JT_1/2 ï¿½ space and l _{(2, 1)} ï¿½ J_1/2 ï¿½ space.

Proposition 4.10. A bigeneralized topological space is a l _{(m, n)}- JT_1/2-space if and only if {x} is l_n - open or l_m ï¿½J-closed for each x ï¿½ X, where m, n = 1,2 and mï¿½ n.

Proof: Suppose that {x} is not l_mï¿½J- closed. Then X ï¿½ {x} is l_{(m,
n)} -J-closed by proposition 3.9. Since X is l _{(1, 2)} -T_1/2 -space, X ï¿½ {x} is l_n -closed. Hence, {x} is l_n - open.

Conversely, let F be a l _{(m, n)}- J-closed set. By assumption , {x} is l_n - open or l_m ï¿½J- closed for any x ï¿½ c_ln(F).Case (i) Suppose that {x}is l_n - open. Since {x} ï¿½ Fï¿½ f, we have x ï¿½ F. Case (ii) Suppose that {x} is l_m ï¿½ J-closed. If x ï¿½ F, then {x} ï¿½ c_ln(F) ï¿½ F, which is a contradiction to proposition 3.9, Therefore, x ï¿½ F. Thus in both case, we conclude that F is l_n - closed. Hence, (X,l_X¹, l_X²) is a l _{(m, n)} -JT_1/2 -space.

Definition 4.11. Let (X,l_X¹, l_X²) and (Y,l_Y¹, l_Y²) be generalized topological spaces. A function f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is said to be Jg _{(m, n)}- irresolute if f ^-1(F) is l_{(m,
n)}-J- closed in X for every l_{(m,
n)} ï¿½J-closed F of Y, where m, n = 1,2 and mï¿½ n.

Proposition 4.12. Let f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) and g: (Y,l_Y¹, l_Y²) ï¿½ (Z,l_Z¹, l_Z²) be functions, the following properties hold:

(i) If f is Jg _{(m, n)}- irresolute and Jg _{(m, n)} -continuous, then g ï¿½ f is Jg _{(m, n) -}continuous.

(ii) If f and g are Jg _{(m, n)}- irresolute , then g ï¿½ f is Jg _{(m, n)}- irresolute;

(iii) Let (Y,l_Y¹, l_Y²) be a l _{(m, n)} -JT_1/2 -space. If f and g are Jg _{(m,
n)}-continuous, then g ï¿½ f is Jg _{(m, n)}-continuous.

Proof: (i) Let F be a l_n -closed subset of Z. Since g is Jg _{(m, n)}- continuous, then g ^-1(F) is l_{(m,
n)} -J-closed subset of Y. Since f is Jg _{(m, n)}- irresolute, then (g ï¿½ f)^-1(F) = f ^-1(g ^-1(F)) is l_{(m,
n)} -J-closed subset of X. Hence g ï¿½ f is Jg _{(m, n) -}continuous.

(ii) Let F be a l_{(m,
n)} -J- closed subset of Z. Since g is Jg _{(m, n)}- irresolute, then g ^-1(F) is l_{(m,
n)} -J-closed subset of Y. Since f is Jg _{(m, n)}- irresolute, then (g ï¿½ f)^-1(F) = f ^-1(g ^-1(F)) is l_{(m,
n)} ï¿½J- closed subset of X. Hence, g ï¿½ f is Jg _{(m, n)}- irresolute.

(iii) Let F be a l_n -closed subset of Z. Since g is Jg _{(m, n)}-continuous, then g ^-1(F) is l_{(m,
n)} -J- closed subset of Y. Since (Y,l_Y¹, l_Y²) is a l _{(m, n)} -JT_1/2 -space, then g ^-1(F) is l_n -closed subset of Y. Since f is Jg _{(m, n)}-continuous, then (g ï¿½ f)^-1(F) = f ^-1(g ^-1(F)) is l_{(m,
n)} ï¿½J-closed subset of X. Hence then g ï¿½ f is Jg _{(m, n)}-continuous.

Proposition 4.13. Let: (X,l_X¹, l_X²) be a l _{(m, n)} -JT_1/2 -space. If f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is surjective, g_n -closed and Jg _{(m, n)} -irresolute, then (Y,l_Y¹, l_Y²) is a l _{(m, n)} -JT_1/2 -space, where m, n = 1,2 and mï¿½ n.

Proof: Let F be a l _{(m, n)} -J-closed subset of Y. Since f is Jg _{(m, n)}- irresolute, we have f ^-1(F) is a l _{( m, n)} -J-closed subset of X. Since (X,l_X¹, l_X²) is a l _{(m, n)} -JT_1/2 -space, f ^-1(F) is a l_n -closed subset of X. It follows by assumption that F is a l_n -closed subset of Y. Hence (Y,l_Y¹, l_Y²) is a l _{(m, n)} -JT_1/2 space.

Proposition 4.14. Let: (X,l_X¹, l_X²) be a l _{(m, n)} - JT_1/2 space. If f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) is bijective, g_n -open and Jg _{(m, n)}ï¿½ irresolute, then (Y,l_Y¹, l_Y²) is a l _{(m, n)} -JT_1/2space, where m, n = 1,2 and mï¿½ n.

5. Strongly-λ_{(m, n)} - J - closed sets

Definition 5.1. A subset A of a bigeneralized topological space (X, l₁, l₂) is said to be strongly l_{(m,
n)} ï¿½ J ^s - closed set (briefly l_{(m,
n)} ï¿½ J ^s - closed) if c_ln (A) ï¿½ U, whenever A ï¿½ U and U is l_{(m,
n)} ï¿½ J - open, where m, n = 1,2 and mï¿½n.

Definition 5.2. A subset A of a bigeneralized topological space (X, l₁, l₂) is said to be pairwise strongly l_{(m,
n)} ï¿½ P- J ^s closed (briefly l_{(m,
n)} ï¿½ P- J ^s closed)if A is l_(1,2) -J^s ï¿½ closed and l_(2,1) - J^s ï¿½ closed..

Proposition 5.3. Every λ_n- closed set is l_{(m,
n)} ï¿½ J ^s ï¿½ closed.

Proof: Let U be a l_{(m,
n)} ï¿½ J ï¿½ open set such that A ï¿½ U and A be λ_n- closed set. Then c_ln (A) = A and so c_ln (A) ï¿½ U. Thus A is l_{(m,
n)} ï¿½ J ^s ï¿½ closed.

Proposition 5.4. Let (X, l₁, l₂) be a bigeneralized topological space. Let A ï¿½ X be a l_{(m,
n)} ï¿½ J^s - closed if and only if , then c_ln (A) \ A does not contain any non- empty l _(m,n)- J- closed set, where m, n = 1, 2 and m ï¿½ n.

Proof: Let F be a l_{(m,
n)} ï¿½ J - closed subset of c_ln (A) \ A . Now, F ï¿½ c_ln (A) \ A and A ï¿½ X \ F where A is l_{(m,
n)} ï¿½ J ^s ï¿½ closed and X \ F is l_{(m,
n)} ï¿½ J ï¿½ open. Thus c_ln (A) ï¿½ X \ F and hence F ï¿½ X \ c_ln (A) .So, F ï¿½ c_ln (A) ï¿½ (X \ c_ln (A)) = f.Therefore F = j.

Conversely, assume c_ln (A) \ A contains no non- empty l _(m,n)- J- closed set. Let U be l _(m,n)- J- open such that A ï¿½ U. Suppose that c_ln (A) is not contained in U, then c_ln (A) ï¿½ U^c is a nonempty l _{(m, n)}- J- closed set of c_ln (A) \ A, which is a contradiction. Therefore c_ln (A) ï¿½ U and hence A is l_{(m,
n)} ï¿½ J ^s ï¿½ closed.

Proposition 5.5. Let (X, l₁, l₂) be a bigeneralized topological space. Let A ï¿½ X be a l_{(m,
n)} ï¿½ J^s - closed and A ï¿½ B ï¿½ c_ln (A) then B is l_{(m,
n)} ï¿½ J^s - closed set, where m, n = 1, 2 and m ï¿½ n.

Proof: Let A be a l_{(m,
n)} ï¿½ J^s- closed set and A ï¿½ B ï¿½ c_ln (A). Let B ï¿½ U and U is l_{(m,
n)} ï¿½ J - open. Then A ï¿½ U. Since A is l_{(m,
n)} ï¿½ J^s ï¿½ closed, we have c_ln (A) ï¿½ U. Since B ï¿½ c_ln (A), then c_ln (B) ï¿½ c_ln (A) ï¿½ U. Hence B is l_{(m,
n)} ï¿½ J^s ï¿½ closed.

Definition 5.6. A subset A of a bigeneralized space X is called strongly l_{(m,
n)} ï¿½ J ^s ï¿½ open if A^c is l_{(m,
n)} ï¿½ J ^s ï¿½ closed.

Proposition 5.7. Let (X, l₁, l₂) be a bigeneralized topological space. Let A ï¿½ X be a l_{(m,
n)} ï¿½ J^s ï¿½ open if and only if F ï¿½ i_ln (A) whenever F ï¿½ A and F is l_{(m,
n)} ï¿½ Jï¿½ closed.

Proof: Let A be l_{(m,
n)} ï¿½ J^s- open and suppose F ï¿½ A where F is a l_{(m,
n)} ï¿½ Jï¿½ closed. Then X \ A is l_{(m,
n)} ï¿½ J ^s ï¿½ closed and X \ A ï¿½ X \ F, where X \ F is l_{(m,
n)} ï¿½ Jï¿½ open set. This implies that c_ln (X \ A) ï¿½ X \ F. Now c_ln (X \ A) = X \ i_ln (A). Hence X \ i_ln (A) ï¿½ X \ F and F ï¿½ i_ln (A).

Conversely. If F is an l_{(m,
n)} ï¿½ Jï¿½ closed with F ï¿½ i_ln (A) whenever F ï¿½ A. Then X \ A ï¿½ X \F and X \ i_ln (A) ï¿½ X \ F. Thus c_ln (X \ A) ï¿½ X \ F. Hence X \ A is l_{(m,
n)} ï¿½ J^s- closed and A is l_{(m,
n)} ï¿½ J^s- open.

Proposition 5.8. Let (X, l₁, l₂) be a bigeneralized topological space. For each x ï¿½ X.{x} is l_{(m,
n)} ï¿½ Jï¿½ closed or {x}^c is l_{(m,
n)} ï¿½ J^s- closed.

Proof: If {x} is not l_{(m,
n)} ï¿½ Jï¿½ closed, then the only l_{(m,
n)} ï¿½ Jï¿½ open containing {x}^c is X. Thus c_ln ({x}^c) ï¿½ X and {x}^c is l_{(m,
n)} ï¿½ J^s- closed.

6. On Strongly - λ_{(m, n)} - J - continuous and irresolute functions

Definition 6.1. Let (X,l_X¹, l_X²) and (Y,l_Y¹, l_Y²) be generalized topological spaces. A function f: (X,l_X¹, l_X²) ï¿½(Y,l_Y¹, l_Y²) is said to be l_{(m,
n)} -strongly J - continuous(briefly λ_{(m, n)}-J^s- continuous) if f^-1(F) is l_{(m,
n)}- J^s-closed in X for every l_n- closed F of Y, where m, n = 1,2 and m ï¿½ n.

Definition 6.2. Let (X,l_X¹, l_X²) and (Y,l_Y¹, l_Y²) be generalized topological spaces. A function f: (X,l_X¹, l_X²) ï¿½(Y,l_Y¹, l_Y²) is said to be l_{(m,
n)} -strongly J ï¿½ irresolute (briefly λ_{(m, n)}-J^s- irresolute) if f^-1(F) is l_{(m,
n)}- J^s-closed in X for every l_{(m,
n)}- J^s-closed F of Y, where m, n = 1,2 and m ï¿½ n.

Proposition 6.3. Let f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²), then, (1) ï¿½(2): (2)ï¿½(3): (3) ï¿½(4)

(1) f is λ_{(m, n)}-J^s- continuous,

(2) For each x ï¿½ X and for every l_n- open set V containing f(x), there exists a l_{(m,
n)}-J^s-open set U containing x such that f(U) ï¿½ V:

(3) f (λ_{(m, n)}-J^s c_λ(A)) ï¿½ λ_n-c_λ(f(A)) for every subset A of X;

(4) λ_{(m, n)}-J^s c_λ (f^-1(B)) ï¿½ f^-1(λ_n-c_λ (B)) for every subset B of Y.

Proof: (1) ï¿½(2): Let x ï¿½ X and V be a l_n-open subset of Y containing f(x). Then by (1), f ^-1(V) is l_{(m,
n)}-J^s-open of X containing x. If U = f ^-1(V), then f (U) ï¿½ V.

(2)ï¿½(3): Let A be a subset of X and f(x) ï¿½(λ_n-c_λ (f(A)). Then, there exists a l_n ï¿½ open subset V of Y containing f(x) such that V ï¿½ f(A) = f.Then by(2), there exist a l_{(m,
n)}-J^s-open set such that f(x) ï¿½ f(U) ï¿½ V. Hence, f(U) ï¿½ f(A) = f and U ï¿½ A = f. Consequently, x ï¿½ λ_{(m, n)}-J^s c_λ(A) and f(x) ï¿½ f (λ_{(m, n)}-J^s c_λ(A)).

(3) ï¿½(4): Let B be a subset of Y and A = f^-1(B). By (3) we obtain f( λ_{(m, n)}- J^s c_λ(f ^-1(B))) ï¿½ l_n - c_λ (f(f ^-1(B))) ï¿½ l_n - c_λ (B).Thus λ_{(m, n)}-J^s c_λ (f^-1(B)) ï¿½ f^-1(λ_n-c_λ (B)).

Proposition 6.4. If f: (X,l_X¹, l_X²) ï¿½ (Y,l_Y¹, l_Y²) and g: (Y,l_Y¹, l_Y²) ï¿½ (Z,l_Z¹, l_Z²) be two functions, then the following will hold

(i) If g is l_n ï¿½ continuous and f is λ_{(m, n)}-J^s ï¿½ continuous, then gï¿½f is λ_{(m, n)}-J^s ï¿½ continuous.

(ii) If g is λ_{(m, n)}- J^s ï¿½irresolute and f is λ_{(m,
n)}-J^s ï¿½irresolute, then gï¿½f is λ_{(m, n)}-J^s ï¿½irresolute.

(iii) If g is λ_{(m, n)}- J^s ï¿½continuous and f is λ_{(m,
n)}-J^sï¿½irresolute, then gï¿½f is λ_{(m, n)}-J^s ï¿½continuous.

Proof: obvious

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