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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 12, no. 2, 2011

pp. 163-173

On a type of generalized open sets

Bishwambhar Roy
∗

Abstract

In this paper, a new class of sets called µ-generalized closed (briefly
µg-closed) sets in generalized topological spaces are introduced and
studied. The class of all µg-closed sets is strictly larger than the class

of all µ-closed sets (in the sense of Á. Császár). Furthermore, g-closed
sets (in the sense of N. Levine) is a special type of µg-closed sets in
a topological space. Some of their properties are investigated here.
Finally, some characterizations of µ-regular and µ-normal spaces have
been given.

2010 MSC: 54D10, 54D15, 54C08, 54C10.

Keywords: µ-open set, µg-closed set, µ-regular space, µ-normal space.

1. Introduction

In the past few years, different forms of open sets have been studied. Re-
cently, a significant contribution to the theory of generalized open sets, was
extended by A. Császár. Especially, the author defined some basic operators
on generalized topological spaces.

It is observed that a large number of papers is devoted to the study of
generalized open like sets of a topological space, containing the class of open
sets and possessing properties more or less similar to those of open sets. For
example, [22] has introduced g-open sets, [4, 30, 2] sg-open sets, [25] pg-open
sets, [27, 28] gα-open sets, [13] δg∗-open sets, [21, 17] bg-open sets.

Owing to the fact that corresponding definitions have many features in
common, it is quite natural to conjecture that they can be obtained and a
considerable part of the properties of generalized open sets can be deduced
from suitable more general definitions. The purpose of this paper is to point

∗The author acknowledges the financial support from UGC, New Delhi.



164 B. Roy

out extremely elementary character of the proofs and to get many unknown
results by special choice of the generalized topology.

We recall some notions defined in [9]. Let X be a non-empty set, expX
denotes the power set of X. We call a class µ ⊆ expX a generalized topology
[9], (briefly, GT) if ∅ ∈ µ and union of elements of µ belongs to µ. A set X,
with a GT µ on it is said to be a generalized topological space (briefly, GTS)
and is denoted by (X, µ). The θ-closure [35] (resp. δ-closure [35]) of a subset
A of a topological space (X, τ) is defined by {x ∈ X : clU ∩ A 6= ∅ for all
U ∈ τ with x ∈ U} (resp. {x ∈ X : A ∩ U 6= ∅ for all regular open sets U
containing x}, where a subset A is called regular open if A = int(cl(A))). A is
called δ-closed [35] (resp. θ-closed [35]) if A = clδA (resp. A = clθA) and the
complement of a δ-closed set (resp. θ-closed) set is known as a δ-open (resp. θ-
open) set. A subset A of a topological space (X, τ) is called preopen [29] (resp.
semiopen [23], δ-preopen [33], δ-semiopen [32], α-open [27], β-open [1], b-open
[21]) if A ⊆ int(cl(A)) (resp. A ⊆ cl(int(A)),A ⊆ int(clδA), A ⊆ cl(intδA),
A ⊆ int(cl(int(A))), A ⊆ cl(int(clA)), A ⊆ cl(int(A)) ∪ int(cl(A))). We note
that for any topological space (X, τ), the collection of all open sets denoted
by τ (preopen sets denoted by PO(X), semi-open sets denoted by SO(X), δ-
open sets denoted by δO(X), δ-preopen sets denoted by δ-PO(X), δ-semiopen
sets denoted by δ-SO(X), α-open sets denoted by αO(X), β-open sets denoted
by βO(X), θ-open sets denoted by θO(X), b-open sets denoted by BO(X) or
γO(X)) forms a GT.

For a GTS (X, µ), the elements of µ are called µ-open sets and the comple-
ment of µ-open sets are called µ-closed sets. For A ⊆ X, we denote by cµ(A)
the intersection of all µ-closed sets containing A, i.e., the smallest µ-closed set
containing A; and by iµ(A) the union of all µ-open sets contained in A, i.e.,
the largest µ-open set contained in A (see [9, 10]). Obviously in a topological
space (X, τ), if one takes τ as the GT, then cµ becomes equivalent to the usual
closure operator. Similarly, cµ becomes pcl, scl, clδ, pclδ, sclδ, clα, clβ, bcl
if µ stands for PO(X) (resp. SO(X), δO(X), δ-PO(X), δ-SO(X), αO(X),
βO(X), BO(X) or γO(X)).

It is easy to observe that iµ and cµ are idempotent and monotonic, where
γ : expX → expX is said to be idempotent iff A ⊆ B ⊆ X implies γ(γ(A))
= γ(A) and monotonic iff γ(A) ⊆ γ(B). It is also well known from [10, 11]
that if g is a GT on X and A ⊆ X, x ∈ X, then x ∈ cµ(A) iff x ∈ M ∈ µ ⇒
M ∩ A 6= ∅ and cµ(X \ A) = X \ iµ(A).

In this paper we introduce the concepts of µg-closed sets and µg-open sets.
It is shown that many results in previous papers can be considered as special
cases of our results.

2. Properties of µg-closed sets

Definition 2.1. Let (X, µ) be a GTS. Then a subset A of X is called a µ-
generalized closed set (or in short, µg-closed set) iff cµ(A) ⊆ U whenever A ⊆ U
where U is µ-open in X. The complement of a µg-closed set is called a µg-open
set.



On a type of generalized open sets 165

Remark 2.2.

(i) If (X, τ) is a topological space, the definition of g-open set [22] (resp.
sg-open set [4, 2], pg-open set [25], gα-open set [27], δg∗-open set [13],
bg-open set [21] or γg-open set [17]) can be obtained by taking µ = τ
(resp. SO(X), PO(X), αO(X), δO(X), γO(X)).

(ii) Every µ-open set in a GTS (X, µ) is µg-open. In fact, if A is a µ-open
set in (X, µ), then X \ A is a µ-closed set. Let X \ A ⊆ U ∈ µ. Then
cµ(X \ A) = X \ A ⊆ U. Thus X \ A is a µg-closed set and hence A is
a µg-open set.

The converse of Remark 2.2(ii) is not true as seen from the next example :

Example 2.3. Let X = {a, b, c} and µ = {∅, X, {a}, {b, c}, {a, c}}. Then
(X, µ) is a GTS. It is easy to verify that {c} is µg-open in (X, µ) but not
µ-open.

The next two examples show that the union (intersection) of two µg-open
sets is not in general µg-open.

Example 2.4.

(a) Let X = {a, b, c} and µ = {∅, X, {a}}. Then (X, µ) is a GTS. It can
be shown that if A = {b} and B = {c}, then A and B are two µg-open
sets but A ∪ B = {b, c} is not a µg-open set.

(b) Let X = {a, b, c, d} and µ = {∅, X, {a, b}, {a, c, d}, {a, b, d}, {b, c, d}}.
Then (X, µ) is a GTS. It follows from Remark 2.2(ii) that {a, b} and
{a, c, d} are two µg-open sets but it is easy to check that their intersec-
tion {a} is not µg-open.

Theorem 2.5. A subset A of a GTS (X, µ) is µg-closed iff cµ(A)\ A contains
no non-empty µ-closed set.

Proof. Let F be a µ-closed subset of cµ(A) \ A. Then A ⊆ F
c (where F c

denotes as usual the complement of F). Hence by µg-closedness of A, we have
cµ(A) ⊆ F

c or F ⊆ (cµ(A))
c. Thus F ⊆ cµ(A) ∩ (cµ(A))

c = ∅, i.e., F = ∅.
Conversely, suppose that A ⊆ U where U is µ-open. If cµ(A) * U, then

cµ(A) ∩ U
c (6= ∅) is a µ-closed subset of cµ(A) \ A - a contradiction. Hence

cµ(A) ⊆ U. �

Theorem 2.6. If a µg-closed subset A of a GTS (X, µ) be such that cµ(A)\ A
is µ-closed, then A is µ-closed.

Proof. Let A be a µg-closed subset such that cµ(A) \ A is µ-closed. Then
cµ(A) \ A is a µ-closed subset of itself. Then by Theorem 2.5, cµ(A) \ A = ∅
and hence cµ(A) = A, showing A to be a µ-closed set. �

That the converse is false follows from the following example.

Example 2.7. Let X = {a, b, c} and µ = {∅, {a}, {a, b}}. Then (X, µ) is a
GTS. It is easy to observe that {b, c} is µ-closed and hence a µg-closed set (by
Remark 2.2), but cµ(A) \ A = ∅, which is not µ-closed.



166 B. Roy

Theorem 2.8. Let A be a µg-closed set in a GTS (X, µ) and A ⊆ B ⊆ cµ(A).
Then B is µg-closed.

Proof. Let B ⊆ U, where U is µ-open in (X, µ). Since A is µg-closed and
A ⊆ U, cµ(A) ⊆ U. Now, B ⊆ cµ(A) ⇒ cµ(B) ⊆ cµ(A). So cµ(B) ⊆ U. �

Theorem 2.9. In a GTS (X, µ), µ = Ω (the collection of all µ-closed sets) iff
every subset of X is µg-closed.

Proof. Suppose µ = Ω and A (⊆ X) be such that A ⊆ U ∈ µ. Then cµ(A) ⊆
cµ(U) = U and hence A is µg-closed.

Conversely, suppose that every subset of X is µg-closed. Let U ∈ µ. Then
U ⊆ U and by µg-closedness of U, we have cµ(U) ⊆ U, i.e., U ∈ Ω. Thus
µ ⊆ Ω.
Now, if F ∈ Ω then F c ∈ µ, so F c ∈ Ω (as µ ⊆ Ω), i.e., F ∈ µ. �

Theorem 2.10. A subset A of a GTS (X, µ) is µg-open iff F ⊆ iµ(A), when-
ever F is µ-closed and F ⊆ A.

Proof. Obvious and hence omitted. �

Theorem 2.11. A set A is µg-open in a GTS (X, µ) iff U = X whenever U
is µ-open and iµ(A) ∪ A

c ⊆ U.

Proof. Suppose U is µ-open and iµ(A) ∪ A
c ⊆ U. Now, Uc ⊆ (iµ(A))

c ∩ A =
cµ(X \ A) \ (X \ A). Since U

c is µ-closed and X \ A is µg-closed, by Theorem
2.5, Uc = ∅, i.e., U = X.

Conversely, let F be a µ-closed set and F ⊆ A. Then by Theorem 2.10,
it is enough to show that F ⊆ iµ(A). Now, iµ(A) ∪ A

c ⊆ iµ(A) ∪ F
c, where

iµ(A) ∪ F
c is µ-open. Hence by the given condition, iµ(A) ∪ F

c = X, i.e.,
F ⊆ iµ(A). �

Theorem 2.12. A subset A of a GTS (X, µ) is µg-closed iff cµ(A) \ A is
µg-open.

Proof. Suppose A is µg-closed and F ⊆ cµ(A)\A, where F is a µ-closed subset
of X. Then by Theorem 2.5, F = ∅ and hence F ⊆ iµ[cµ(A) \ A]. Then by
Theorem 2.10, cµ(A) \ A is µg-open.

Conversely, suppose that A ⊆ U where U is µ-open. Now, cµ(A) ∩ U
c ⊆

cµ(A) ∩ A
c = cµ(A) \ A. Since cµ(A) ∩ U

c is µ-closed and cµ(A) \ A is µg-open,
cµ(A) ∩ U

c = ∅ (by Theorem 2.5). Thus cµ(A) ⊆ U, i.e., A is µg-closed. �

Definition 2.13. A GTS (X, µ) is said to be

(i) µ-T0 [34] iff x, y ∈ X, x 6= y implies the existence of K ∈ µ containing
precisely one of x and y.

(ii) µ-T1 [34] iff x, y ∈ X, x 6= y implies the existence of K, K
1 ∈ µ such

that x ∈ K, y 6∈ K and x 6∈ K1, y ∈ K1.
(iii) µ-T1/2 iff every µg-closed set is µ-closed.



On a type of generalized open sets 167

Remark 2.14. A topological space (X, τ) is Ti [16] (resp. semi-Ti [4], pre-Ti
[25], α-Ti [28], δ-Ti [13], b-Ti [21]) for i = 0, 1/2, 1 by taking µ = τ (resp.
SO(X), PO(X), αO(X), δO(X), BO(X) or γO(X)).

Theorem 2.15. If a GTS (X, µ) is µ-T1/2 then it is µ-T0.

Proof. Suppose that (X, µ) is not a µ-T0 space. Then there exist distinct points
x and y in X such that cµ({x}) = cµ({y}). Let A = cµ({x}) ∩ {x}

c. We shall
show that A is µg-closed but not µ-closed. Suppose that A ⊆ V ∈ µ. We have
to show that cµ(A) ⊆ V . Thus it is enough to show that cµ({x}) ⊆ V (as
A ⊆ cµ({x})). Again, since cµ({x}) ∩ {x}

c = A ⊆ V , we need only to show
that x ∈ V . In fact, if x 6∈ V , then y ∈ cµ({x}) ⊆ V

c (as V c is µ-closed). So
y ∈ A ⊆ V c and hence y ∈ V ∩ V c - a contradiction.
If x ∈ U ∈ µ, then U ∩ A ⊇ {y} 6= ∅, and hence x ∈ cµ(A). Clearly, x 6∈ A and
thus A is not µ-closed. �

Example 2.16. Let X = {a, b, c, d} and µ = {∅, X, {a, b}, {a, c, d}, {a, b, d},
{b, c, d}}. Then (X, µ) is a GTS. Clearly, this GTS is µ-T0 and it can be shown
that the collection of all µg-open sets are {∅, X, {d}, {a, b}, {a, c, d}, {a, b, d},
{b, c, d}}. Thus this space is not µ-T1/2.

Theorem 2.17. If a GTS (X, µ) is µ-T1 then it is µ-T1/2.

Proof. Suppose that A is a subset of X which is not µ-closed. Take x ∈
cµ(A) \ A. Then {x} ⊆ cµ(A) \ A and {x} is µ-closed (as (X, µ) is µ-T1). Thus
by Theorem 2.5, A is not µg-closed. �

Example 2.18. Let X = {a, b, c, d} and µ = {∅, X, {d}, {a, b}, {b, c, d}, {a, c, d},
{a, b, d}}. Then (X, µ) is a GTS. It is easy to verify that (X, µ) is µ-T1/2 but
not µ-T1.

Definition 2.19. A GTS (X, µ) is said to be µ-symmetric iff for each x, y ∈ X,
x ∈ cµ({y}) ⇒ y ∈ cµ({x}).

Remark 2.20. It is easy to check that the above definition of a µ-symmetric
space GT unifies the existing definitions of δ-symmetric space [8], (δ, p)-symmetric
space [5], α-symmetric [6], δ-semi symmetric space [7] if (X, τ) is a topological
space and µ = δO(X), δ-PO(X), αO(X), δ-SO(X) respectively.

Theorem 2.21. A GTS (X, µ) is µ-symmetric iff {x} is µg-closed for each
x ∈ X.

Proof. Let {x} ⊆ U ∈ µ and (X, µ) be µ-symmetric but cµ({x}) * U. Then
cµ({x}) ∩ U

c 6= ∅. Let y ∈ cµ({x}) ∩ U
c. Then x ∈ cµ({y}) ⊆ U

c ⇒ x 6∈ U - a
contradiction.

Conversely, let for each x ∈ X, {x} is µg-closed and x ∈ cµ({y}) ⊆ (cµ({x}))
c

(as {y} is µg-closed). Thus x ∈ (cµ({x}))
c - a contradiction. �

Corollary 2.22. If a GTS (X, µ) is µ-T1 then it is µ-symmetric.

Example 2.23. Let X = {a, b} and µ = {∅, X}. Then (X, µ) is a µ-
symmetric space which is not µ-T1.



168 B. Roy

Theorem 2.24. A GTS (X, µ) is µ-symmetric and µ-T0 iff (X, µ) is µ-T1.

Proof. If (X, µ) is µ-T1 then it is µ-symmetric (by Corollary 2.22) and µ-T0
(by Definition 2.13).

Conversely, let (X, µ) be µ-symmetric and µ-T0. We shall show that (X, µ)
is µ-T1. Let x, y ∈ X and x 6= y. Then by µ-T0-ness of (X, µ), there exists
U ∈ µ such that x ∈ U ⊆ {y}c. Then x 6∈ cµ({y}) and hence y 6∈ cµ({x}).
Thus there exists V ∈ µ such that y ∈ V and x 6∈ V . Thus (X, µ) is µ-T1. �

Theorem 2.25. If (X, µ) is µ-symmetric, then (X, µ) is µ-T0 iff (X, µ) is
µ-T1/2 iff (X, µ) is µ-T1.

Proof. Follows from Theorem 2.24 and the fact that µ-T1 ⇒ µ-T1/2 ⇒ µ-T0.
�

3. Preservation of µg-closed sets

Definition 3.1. Let (X, µ
1
) and (Y, µ

2
) be two GTS’s. A mapping f :

(X, µ
1
) → (Y, µ

2
) is said to be

(i) (µ
1
, µ

2
) continuous [9] iff f−1(G2) ∈ µ1 for each G2 ∈ µ2;

(ii) (µ
1
, µ

2
)-closed iff for any µ

1
-closed subset A of X, f(A) is µ

2
-closed in

Y .

Theorem 3.2. Let (X, µ
1
) and (Y, µ

2
) be two GTS’s and f : (X, µ

1
) → (Y, µ

2
)

be (µ
1
, µ

2
)-continuous and (µ

1
, µ

2
)-closed mapping. If A is µ

1
g-closed in X

then f(A) is µ
2
g-closed in Y .

Proof. Let f(A) ⊆ G2, where G2 is a µ2-open set in Y . Then A ⊆ f
−1(G2),

where f−1(G2) is a µ1-open set in X. Thus by µ1g-closedness of A, cµ1 (A) ⊆

f−1(G2). Thus f(cµ1(A)) ⊆ G2 and f(cµ
1
(A)) is µ

2
-closed in Y . It thus follows

that cµ
2
(f(A)) ⊆ cµ

2
(f(cµ1(A))) = f(cµ

1
(A)) ⊆ G2. Thus f(A) is µ2g-closed

in Y . �

Theorem 3.3. Let (X, µ
1
) and (Y, µ

2
) be two GTS’s and f : (X, µ

1
) → (Y, µ

2
)

be a (µ1, µ2)-continuous and (µ1, µ2)-closed mapping. If B is a µ2g-closed set
in Y , then f−1(B) is µ

1
g-closed in X.

Proof. Suppose that B is a µ
2
g-closed set in Y and f−1(B) ⊆ G1, where G1 is

µ
1
-open in X. We shall show that cµ1(f

−1(B)) ⊆ G1. Now f[cµ
1
(f−1(B)) ∩

Gc1] ⊆ cµ
2
(B) \ B and by Theorem 2.5, f[cµ

1
(f−1(B)) ∩ Gc1] = ∅. Thus

cµ
1
(f−1(B)) ∩ Gc1 = ∅. Thus cµ

1
(f−1(B)) ⊆ G1 and hence f

−1(B) is µ
1
g-

closed in X. �

Next two examples show that (µ
1
, µ

2
)-continuity and (µ

1
, µ

2
)-closedness in

both of the above theorems are essential.

Example 3.4. Let X = {a, b, c, d}, µ
1
= {∅, X, {a, b}, {c, d}, {a, c, d}, {a, b, d}}

and µ
2
= {∅, X, {a, b}, {c, d}, {a, c, d}}. Then (X, µ

1
) and (X, µ

2
) are two

GTS’s. Consider the identity mapping f : (X, µ
1
) → (X, µ

2
). It is easy to see



On a type of generalized open sets 169

that f is a (µ
1
, µ

2
)-continuous mapping which is not (µ

1
, µ

2
)-closed. The fami-

lies of µ
1
g-open and µ

2
g-open sets are respectively {∅, X, {a}, {d}, {c, d}, {a, d},

{a, b}, {a, c, d}, {a, b, d}} and {∅, X, {a}, {c}, {d}, {c, d}, {a, d}, {a, b}, {a, c},
{a, c, d}, {a, b, d}, {a, b, c}}. We note that {d} is gµ

2
-closed but f−1({d}) is not

gµ
1
-closed.
Again, the identity map h defined by h : (X, µ

2
) → (X, µ

1
) is not a (µ2, µ1)-

continuous mapping but it is (µ
2
, µ

1
)-closed. Clearly, {d} is a µ

2
g-closed set

but h({d}) is not a µ
1
g-closed set.

Example 3.5. Let X = {a, b, c, d} , µ
1
= {∅, X, {a, b}, {c, d}, {a, c, d}, {a, b, d}}

and µ
2
= {∅, X, {a, b}, {a, b, d}, {a, c, d}}. Then (X, µ

1
) and (X, µ

2
) are GTS’s.

Now, consider the identity map f : (X, µ
1
) → (X, µ

2
). It is easy to verify that

f is a (µ
1
, µ

2
)-continuous mapping which is not (µ

1
, µ

2
)-closed. The family of

gµ
1
-open and gµ

2
-open sets are respectively {∅, X, {a}, {d}, {c, d}, {a, d}, {a, b},

{a, c, d}, {a, b, d}} and {∅, X, {a}, {d}, {a, b}, {a, d}, {a, b, d}, {a, c, d}}. We note
that {a, b} is µ

1
g-closed but f({a, b}) is not µ

2
g-closed.

Again, consider the identity map h : (X, µ
2
) → (X, µ

1
). Then, clearly h

is a (µ
2
, µ

1
)-closed map which is not (µ

2
, µ

1
)-continuous. Clearly, {a, b} is

µ
1
g-closed but h−1({a, b}) is not a µ

2
g-closed set.

4. Properties of µ-regular and µ-normal spaces

Definition 4.1. A GTS (X, µ) is said to be µ-regular if for each µ-closed set
F of X not containing x, there exist disjoint µ-open set U and V such that
x ∈ U and F ⊆ V .

Remark 4.2. Regular space, pre-regular space, semi-regular space, β-regular
space, α-regular space are defined and studied in [16, 31, 15, 19, 20] respectively.
The above definition gives a unified version of all these definitions if µ takes
the role of τ, PO(X), SO(X), βO(X), αO(X) respectively.

Theorem 4.3. For a GTS (X, µ) the followings are equivalent:

(a) X is µ-regular.
(b) For each x ∈ X and each U ∈ µ containing x, there exists V ∈ µ such

that x ∈ V ⊆ cµ(V ) ⊆ U.
(c) For each µ-closed set F of X, ∩{cµ(V ) : F ⊆ V ∈ µ} = F.
(d) For each subset A of X and each U ∈ µ with A ∩ U 6= ∅, there exists

a V ∈ µ such that A ∩ V 6= ∅ and cµ(V ) ⊆ U.
(e) For each non-empty subset A of X and each µ-closed subset F of X

with A ∩ F = ∅, there exist U, V ∈ µ such that A ∩ V 6= ∅, F ⊆ W
and W ∩ V = ∅.

(f) For each µ-closed set F with x 6∈ F there exist U ∈ µ and a µg-open
set V such that x ∈ U, F ⊆ V and U ∩ V = ∅.

(g) For each A ⊆ X and each µ-closed set F with A ∩ F = ∅ there exist
a U ∈ µ and a µg-open set V such that A ∩ U 6= ∅, F ⊆ V and
U ∩ V = ∅.

(h) For each µ-closed set F of X, F = ∩{cµ(V ) : F ⊆ V, V is µg-open}.



170 B. Roy

Proof. (a) ⇒ (b) : Let U be a µ-open set containing x. Then x 6∈ X\U, where
X \ U is µ-closed. Then by (a) there exist G, V ∈ µ such that X \ U ⊆ G and
x ∈ V and G ∩ V = ∅. Thus V ⊆ X \ G and so x ∈ V ⊆ cµ(V ) ⊆ X \ G ⊆ U.

(b) ⇒ (c) : Let X \ F ∈ µ be such that x 6∈ F . Then by (b) there exists
U ∈ µ such that x ∈ U ⊆ cµ(U) ⊆ X \ F . So, F ⊆ X \ cµ(U) = V (say)∈ µ
and U ∩ V = ∅. Thus x 6∈ cµ(V ). Hence F ⊇ ∩{cµ(V ) : F ⊆ V ∈ µ}.

(c) ⇒ (d) : Let U ∈ µ with x ∈ U ∩ A. Then x 6∈ X \ U and hence by (c)
there exists a µ-open set W such that X \ U ⊆ W and x 6∈ cµ(W). We put
V = X \ cµ(W), which is a µ-open set containing x and hence A ∩ V 6= ∅ (as
x ∈ A ∩ V ). Now V ⊆ X \ W and so cµ(V ) ⊆ X \ W ⊆ U.

(d) ⇒ (e) : Let F be a µ-closed set as in the hypothesis of (e). Then X \ F
is a µ-open set and (X \ F) ∩ A 6= ∅. Then there exists V ∈ µ such that
A ∩ V 6= ∅ and cµ(V ) ⊆ X \ F . If we put W = X \ cµ(V ), then F ⊆ W and
W ∩ V = ∅.

(e) ⇒ (a) : Let F be a µ-closed set not containing x. Then by (e), there
exist W, V ∈ µ such that F ⊆ W and x ∈ V and W ∩ V = ∅.

(a) ⇒ (f) : Obvious as every µ-open set is µg-open (by Remark 2.2).

(f) ⇒ (g) : Let F be a µ-closed set such that A ∩ F = ∅ for any subset
A of X. Thus for a ∈ A, a 6∈ F and hence by (f), there exist a U ∈ µ and a
µg-open set V such that a ∈ U, F ⊆ V and U ∩ V = ∅. So A ∩ U 6= ∅.

(g) ⇒ (a) : Let x 6∈ F , where F is µ-closed. Since {x}∩F = ∅, by (g) there
exist a U ∈ µ and a µg-open set W such that x ∈ U, F ⊆ W and U ∩ W = ∅.
Now put V = iµ(W). Then F ⊆ V (by Theorem 2.10) and U ∩ V = ∅.

(c) ⇒ (h) : We have F ⊆ ∩{cµ(V ) : F ⊆ V and V is µg-open} ⊆ ∩{cµ(V ) :
F ⊆ V and V is µ-open} = F .

(h) ⇒ (a) : Let F be a µ-closed set in X not containing x. Then by (h)
there exists a µg-open set W such that F ⊆ W and x ∈ X \ cµ(W). Since F is
µ-closed and W is µg-open, F ⊆ iµ(W) (by Theorem 2.10). Take V = iµ(W).
Then F ⊆ V , x ∈ X\cµ(V ) = U (say) (as (X\F)∩V = ∅) and U ∩V = ∅. �

Definition 4.4. A GTS (X, µ) is µ-normal [12] if for any pair of disjoint µ-
closed subsets A and B of X, there exist disjoint µ-open sets U and V such
that A ⊆ U and B ⊆ V .

Remark 4.5. Normal space, pre-normal space, semi-normal space, α-normal
space, β-normal space, γ-normal space are defined and studied in [16, 31, 2,



On a type of generalized open sets 171

20, 19, 17] respectively. The above definition gives a unified version of all these
definitions if µ takes the role of τ, PO(X), SO(X), αO(X), βO(X) respectively.

Theorem 4.6. For a GTS (X, µ) the followings are equivalent:

(a) X is µ-normal;
(b) For any pair of disjoint µ-closed sets A and B, there exist disjoint

µg-open sets U and V such that A ⊆ U and B ⊆ V ;
(c) For every µ-closed set A and µ-open set B containing A, there exists

a µg-open set U such that A ⊆ U ⊆ cµ(U) ⊆ B;
(d) For every µ-closed set A and every µg-open set B containing A, there

exists a µ-open set U such that A ⊆ U ⊆ cµ(U) ⊆ iµ(B);
(e) For every µg-closed set A and every µ-open set B containing A, there

exists a µ-open set U such that A ⊆ cµ(A) ⊆ U ⊆ cµ(U) ⊆ B.

Proof. (a) ⇒ (b) : Let A and B be two disjoint µ-closed subsets of X. Then
by µ-normality of X, there exist disjoint µ-open sets U and V such that A ⊆ U
and B ⊆ V . Then U and V are µg-open by Remark 2.2.

(b) ⇒ (c) : Let A be a µ-closed set and B be a µ-open set containing
A. Then A and Bc are two disjoint µ-closed sets in X. Then by (b), there
exist disjoint µg-open sets U and V such that A ⊆ U and Bc ⊆ V . Thus
A ⊆ U ⊆ X \ V ⊆ B. Again, since B is µ-open and X \ V is µg-closed,
cµ(X \ V ) ⊆ B. Hence A ⊆ U ⊆ cµ(U) ⊆ B.

(c) ⇒ (d) : Let A be a µ-closed subset of X and B be a µg-open set
containing A. Since B is a µg-open set containing A and A is µ-closed, by
Theorem 2.10, A ⊆ iµ(B). Thus by (c) there exists a µg-open set U such that
A ⊆ U ⊆ cµ(U) ⊆ iµ(B).

(d) ⇒ (e) : Let A be a µg-closed set and B be a µ-open set in X containing
A. A ⊆ B implies cµ(A) ⊆ B, where cµ(A) is µ-closed and B is µg-open (as B
is µ-open). Then by (d), there exists a µ-open set U such that A ⊆ cµ(A) ⊆
U ⊆ cµ(U) ⊆ iµ(B). Thus A ⊆ cµ(A) ⊆ U ⊆ cµ(U) ⊆ B.

(e) ⇒ (a) : Let A and B be two disjoint µ-closed subsets of X. Then A
is µg-closed and A ⊆ X \ B, where X \ B is µ-open. Thus by (e), there exists
a µ-open set U such that A ⊆ cµ(A) ⊆ U ⊆ cµ(U) ⊆ X \ B. Thus A ⊆ U,
B ⊆ X \ cµ(U) and U ∩ (X \ cµ(U)) = ∅. Hence X is µ-normal. �

Remark 4.7. (a) By using µ = τ [22] (resp. PO(X) [25], SO(X) [4], αO(X)
[27], δO(X) [13], BO(X) [17, 21]) on a topological space (X, τ) several modifi-
cations of g-closed sets (resp. sg-closed sets, gα-closed sets, δg∗-closed sets, bg-
closed sets) are introduced and investigated. Since each of τ, PO(X), SO(X),
αO(X), δO(X), BO(X) forms a GT on X, the characterizations of each of the
families are obtained from µg-open set.



172 B. Roy

(b) The definition of many other similar types of generalized closed sets can
be defined on a topological space (X, τ) from the definition of µg-closed set by
replacing µ by the corresponding GT on X.

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(Received December 2010 – Accepted July 2011)

Bishwambhar Roy (bishwambhar roy@yahoo.co.in)
Department of Mathematics, Women’s Christian College, 6 Greek Church Row,
Kolkata-700026, India.


	On a type of generalized open sets. By B. Roy