*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_{i }ï¿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 l_{1}
and l_{2}
be generalized topologies on X. A triple (X,
l_{1},
l_{2})
is said to be a bigeneralized topological space (briefly BGTS).

Let (X,
l_{1},
l_{2})
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,
l_{1},
l_{2})
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
l_{1}
= {f,{a},{a,
b}, X}, and l_{2}
= {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,
l_{1},
l_{2})
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, l_{1},
l_{2})
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,
l_{1},
l_{2}),
{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, l_{1},
l_{2})
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 l_{1}
and l_{2
} 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, l_{1},
l_{2})
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
l_{1}
= {f,
{a}, {a, b}, X} and l_{2}
= {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 l_{1}
and l_{2}
be generalized topologies on X. if
l_{1}
ï¿½
l_{2},
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,
l_{1},
l_{2}),
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, l_{1},
l_{2})
be a bigeneralized topological space. Let A
ï¿½
X be a l_{(m,
n)} ï¿½ J
- closed subset of (X, l_{1},
l_{2}).
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, l_{1},
l_{2})
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_{m
}ï¿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,
l_{1},
l_{2})
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_{m
}ï¿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,
l_{1},
l_{2})
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_{ln
}ï¿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,
l_{1},
l_{2})
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_{m
}ï¿½ J-
open. Then X ï¿½ U ï¿½
X ï¿½ (c_{ln
}(A) -
A) and X ï¿½ U is l_{m
}ï¿½ J-
closed. Thus we have c_{ln
}(A) -
A does not contain non ï¿½ empty
l_{m
}ï¿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_{X}^{1},
l_{X}^{2})
and (Y,l_{Y}^{1},
l_{Y}^{2})
be generalized topological spaces. A function f: (X,l_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2}),
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
be generalized topological spaces. A function f: (X,l_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
be generalized topological spaces. A function f: (X,l_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
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_{X}^{1},
l_{X}^{2})
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_{m
}ï¿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_{X}^{1},
l_{X}^{2})
is a l
_{(m, n)} -JT_{1/2} -space.

**
**

**
Definition 4.11.**
Let (X,l_{X}^{1},
l_{X}^{2})
and (Y,l_{Y}^{1},
l_{Y}^{2})
be generalized topological spaces. A function f: (X,l_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
and g: (Y,l_{Y}^{1},
l_{Y}^{2})
ï¿½
(Z,l_{Z}^{1},
l_{Z}^{2})
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_{Y}^{1},
l_{Y}^{2})
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_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
be a l
_{(m, n)} -JT_{1/2} -space. If f: (X,l_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
is surjective, g_{n} -closed and Jg _{(m, n) }
-irresolute, then (Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
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_{Y}^{1},
l_{Y}^{2})
is a l
_{(m, n)} -JT_{1/2} space.

**
**

**
Proposition 4.14.**
Let: (X,l_{X}^{1},
l_{X}^{2})
be a l
_{(m, n)} - JT_{1/2} space. If f: (X,l_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
is bijective, g_{n} -open and Jg _{(m, n) }ï¿½
irresolute, then (Y,l_{Y}^{1},
l_{Y}^{2})
is a l
_{(m, n)} -JT_{1/2}space, 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,
l_{1},
l_{2})
is said to be s*trongly **
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,
l_{1},
l_{2})
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, l_{1},
l_{2})
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, l_{1},
l_{2})
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, l_{1},
l_{2})
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 J^{ }ï¿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 J^{ }ï¿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 J^{ }ï¿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 J^{ }ï¿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, l_{1},
l_{2})
be a bigeneralized topological space. For each x
ï¿½
X.{x} is
l_{(m,
n)}
ï¿½ J^{ }ï¿½ closed or {x}^{c} is
l_{(m,
n)}
ï¿½ J^{s}- closed.

*
Proof: *
If {x} is not
l_{(m,
n)}
ï¿½ J^{ }ï¿½ closed, then the only
l_{(m,
n)}
ï¿½ J^{ }ï¿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_{X}^{1},
l_{X}^{2})
and (Y,l_{Y}^{1},
l_{Y}^{2})
be generalized topological spaces. A function f: (X,l_{X}^{1},
l_{X}^{2})
ï¿½(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
and (Y,l_{Y}^{1},
l_{Y}^{2})
be generalized topological spaces. A function f: (X,l_{X}^{1},
l_{X}^{2})
ï¿½(Y,l_{Y}^{1},
l_{Y}^{2})
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2}),
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_{X}^{1},
l_{X}^{2})
ï¿½
(Y,l_{Y}^{1},
l_{Y}^{2})
and g: (Y,l_{Y}^{1},
l_{Y}^{2})
ï¿½
(Z,l_{Z}^{1},
l_{Z}^{2})
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^{s}ï¿½irresolute, then gï¿½f
is λ_{(m, n)}-J^{s} ï¿½continuous.

*
Proof:*
obvious

**
**

**
References:**

**
**

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