3_300_hussain.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 2, No. 3, 2009, (338-351)

ISSN 1307-5543 – www.ejpam.com

On Minimal γ-open Sets

Sabir Hussain1∗ and Bashir Ahmad2

1 Department of Mathematics, Islamia University, Bahawalpur, Pakistan.

Present Address: Department of Mathematics, Yanbu University, P. O. Box 31387, Yanbu

Alsinaiyah, Saudi Arabia.

2 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya

University, Multan, Pakistan.

Present Address: Department of Mathematics, King Abdul Aziz University P. O. Box 80203,

Jeddah 21589, Saudi Arabia.

Abstract. In this paper, we introduce and discuss minimal γ-open sets in topological spaces.

We establish some basic properties of minimal γ-open sets and provide an example to illustrate

that minimal γ-open sets are independent of minimal open sets introduced and discussed in

[3]. We obtain some properties of pre γ-open sets using properties of minimal γ-open sets.

As an application of a theory of minimal γ-open sets, we obtain a sufficient condition for a

γ-locally finite space to be a pre γ-T2 space.

2000 Mathematics Subject Classifications: 54A05, 54A10, 54D10, 54D99.

Key Words and Phrases: γ-closed (open), γ-closure , minimal γ-open , pre γ-open, finite

γ-open , γ-locally finite, pre γ-T2 space.

∗Corresponding author.

Email addresses: sabiriub�yahoo.
om (S. Hussain), drbashir9�gmail.
om (B. Ahmad)

http://www.ejpam.com 338 c© 2009 EJPAM All rights reserved.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 339

1. Introduction

Several known characterizations of compact spaces, nearly compact spaces and H-

closed spaces are unified by generalizing the notion of compactness with the help of

a certain operation γ of a topology τ into the power set P(X) of a space X introduced

and discussed by S. Kasahara [2]. By using operation γ, H. Ogata [4], introduced

the concept of γ-open sets and investigated the related topological properties of the

associated topology τγ and τ. He introduced the notions of γ-Ti (i = 0, 1/2, 1, 2)

spaces which generalize Ti - spaces (i = 0, 1/2, 1, 2) respectively. Moreover, he

investigated general operator approaches of the closed graph mappings.

In 2003, B. Ahmad and S. Hussain [1] continued studying the properties of γ-

operations on topological spaces and investigated many interesting results.

In this paper, we introduce and discuss minimal γ-open sets in topological spaces.

We establish some basic properties of minimal γ-open sets and provide an example to

illustrate that minimal γ-open sets are independent of minimal open sets introduced

and investigated in [3]. We obtain some properties of pre γ-open sets using properties

of minimal γ-open sets. As an application of a theory of minimal γ-open sets, we

obtain a sufficient condition for a γ-locally finite space to be a pre γ-T2 space.

First, we recall some definitions and results used in this paper. Hereafter, we shall

write a space in place of a topological space.

2. Preliminaries

Definition 2.1. [2] Let (X,τ) be a space. An operation γ : τ→ P(X) is a function from

τ to the power set of X such that V ⊆ V γ , for each V ∈ τ, where V γ denotes the value

of γ at V. The operations defined by γ(G) = G, γ(G) = cl(G) and γ(G) = intcl(G) are

examples of operation γ.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 340

Definition 2.2. [4] Let A⊆ X. A point x∈ A is said to be γ-interior point of A if there

exists an open nbd N of x such that Nγ ⊆ A and we denote the set of all such points by

intγ(A). Thus

intγ (A) = {x ∈ A : x ∈ N ∈ τ and Nγ ⊆ A} ⊆ A.

Note that A is γ-open [1] iff A = intγ(A). A set A is called γ- closed [1] iff X-A is γ-open.

Definition 2.3. [4] A point x∈ X is called a γ-closure point of A⊆ X, if Uγ ∩ A 6= φ, for

each open nbd U of x. The set of all γ-closure points of A is called γ-closure of A and is

denoted by clγ(A). A subset A of X is called γ-closed, if clγ(A) ⊆ A. Note that clγ(A) is

contained in every γ-closed superset of A.

Definition 2.4. [4] An operation γ on τ is said be regular, if for any open nbds U,V of

x ∈ X, there exists an open nbd W of x such that Uγ ∩ V γ ⊇W γ.

Definition 2.5. [4] An operation γ on τ is said to be open, if for any open nbd U of each

x ∈ X , there exists γ-open set B such that x ∈ B and Uγ ⊇ B.

3. Minimal γ-open Sets

In view of the definition of minimal open sets [3], we define minimal γ-open sets

as:

Definition 3.1. Let X be a space and A ⊆ X a γ-open set. Then A is called a minimal

γ-open set if φ and A are the only γ-open subsets of A.

The following Example shows that minimal γ-open sets and minimal open sets are

independent of each other.

Example 3.1. Let X= {a, b, c}, τ = {φ, X , {a}, {b}, {a, b}, {a, c}}. For b ∈ X , define an

operation γ : τ→ P(X ) by



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 341

γ(A) = Aγ =







A, if b ∈ A

cl(A), if b 6∈ A.

The γ-open sets are φ, X, {b},{a, b}, {a, c} [4]. Here {a} is a minimal open set which is

not minimal γ-open. Also {a, c} is minimal γ-open set which is not minimal open.

The following is immediate:

Proposition 3.1. Let X is a space. Then

(1) Let A be a minimal γ-open set and B a γ-open set. Then A∩ B = φ or A⊆ B, where

γ is regular.

(2) Let B and C be minimal γ-open sets. Then B ∩ C = φ or B = C , where γ is regular.

Proposition 3.2. Let X be a space and A a minimal γ-open set. If a ∈ A, then for any

γ-open nbd B of a, A⊆ B , where γ is regular.

Proof. Suppose on the contrary that B is a γ-open nbd B of a ∈ A such that A* B

. Since γ is a regular operation, therefore A∩ B is a γ-open set [4] with A∩ B ⊆ A and

A∩ B 6= φ. This is a contradiction to our supposition that A is a minimal γ-open set.

Hence the proof.

The following example shows that the condition that γ is regular is necessary for

the above Proposition.

Example 3.2. Let X= {a, b, c}, τ = {φ, X , {a}, {b}, {a, b}, {a, c}}. For b ∈ X , define an

operation γ : τ→ P(X ) by

γ(A) = Aγ =







A, if b ∈ A

cl(A), if b 6∈ A .

Then calculations show that the operation γ is not regular [4]. The γ-open sets are φ

, X, {b}, {a, b}, {a, c}[4]. Clearly A = {a, c} is a minimal γ-open set. Thus for a ∈ A,

there does not exist γ-open nbd B of a such that A⊆ B.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 342

The following proposition easily follows from proposition 3.1.

Proposition 3.3. Let X be a space and A a minimal γ-open set. Then for any y ∈ A,

A= ∩{B : B is γ-open nbd of y}, where γ is regular.

Similarly we have:

Proposition 3.4. Let A be a minimal γ-open set in X and x ∈ X such that x /∈ A . Then

for any γ-open nbd C of x, C ∩ A= φ or A⊆ C.

Corollary 3.1. Let A be a minimal γ-open set in X and x ∈ X such that x /∈ A . If

Ax = {B : B is a γ-open nbd of x }. Then Ax ∩ A= φ or A⊆ Ax .

If Γ(X ) denotes the class of monotone operators, then we have:

Corollary 3.2. Let X be a space and γ ∈ Γ(X ). If A is a nonempty minimal γ-open set of

X, then for a nonempty subset C of A, A⊆ clγ(C) , where γ is regular.

Proof. Let C be any nonempty subset of A. Let y ∈ A and B be any γ-open nbd B of

y. By Proposition 3.3, we have A⊆ B. Also since γ is monotone, C = Aγ ∩ C ⊆ Bγ ∩ C .

Thus we have Bγ∩C 6= φ and hence y ∈ clγ(C)[4]. This implies that A⊆ clγ(C). This

completes the proof.

Proposition 3.5. Let A be a nonempty γ-open subset of a space X . If A⊆ clγ(C), then

clγ(A) = clγ(C), for any nonempty subset C of A, where γ is open.

Proof. Since for any nonempty C such that C ⊆ A implies clγ(C) ⊆ clγ(A). On the

other hand, by supposition we have A⊆ clγ(C). Since γ is open, clγ(A)⊆ clγ(clγ(C)) =

clγ(C)[4] implies clγ(A) ⊆ clγ(C). Hence the proof.

The following example shows that the condition that γ is open is necessary for the

above Proposition.

Example 3.3. Let X= {a, b, c}, τ = {φ, X , {a}, {b}, {a, b}, {a, c}}. For b ∈ X , define an

operation γ : τ→ P(X ) by



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 343

γ(A) = Aγ =







cl(A), if b ∈ A

int(cl(A)), if b 6∈ A.

Then the operation γ is not open. The γ-open sets are φ, X, {b}, {a, c}. Let A= {b} and

C = {a, b}, then clγ(A) = {b} 6= X = clγ(C).

Proposition 3.6. Let A be a nonempty γ-open subset of a space X. If clγ(A) = clγ(C), for

any nonempty subset C of A, then A is a minimal γ-open set.

Proof. We suppose on the contrary that A is not a minimal γ-open set. Then there

exists a nonempty γ-open set D such that D ⊆ A and hence there exists an element

x ∈ A such that x /∈ D. Then we have clγ({x}) ⊆ Dγ implies that clγ({x}) 6= clγ(A).

This contradiction proves the proposition.

Combining Propositions 3.4, 3.5 and 3.6, we have:

Theorem 3.1. Let A be a nonempty γ-open subset of space X and γ ∈ Γ(X ). Then the

following are equivalent:

(1) A is minimal γ-open set, where γ is regular.

(2) For any nonempty subset C of A, A⊆ clγ(A) , where γ is open.

(3) For any nonempty subset C of A , clγ(A) = clγ(C).

Definition 3.2. Let X be a space and A ⊆ X . Then A is called a pre-γ-open set, if

A⊆ intγ(clγ(A)) . The family of all pre-γ-open sets of X will be denoted by POγ(X ).

In view of the definition of a pre-Hausdorff space [3], we define a γ-T2 space as:

Definition 3.3. A space X is called a pre γ-T2 space, if for any x , y ∈ X ,x 6= y, there

exist subsets U and V of POγ(X ) such that x ∈ U, y ∈ V and U ∩ V = φ.

Proposition 3.7. Let X be a space and γ ∈ Γ(X ). If A⊆ X is a minimal γ-open set, then

φ 6= C ⊆ A is a pre-γ-open set, where γ is regular.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 344

Proof. Let A be a minimal γ-open set and φ 6= C ⊆ A. By Proposition 3.6, we

have A ⊆ clγ(C) implies intγ(A) ⊆ intγ(clγ(C)). Since A is a γ-open set, therefore

C ⊆ A = intγ(A) ⊆ intγ(clγ(C)) or C ⊆ intγ(clγ(C)), that is, C is pre-γ-open. Hence

the proof.

We use Theorem 3.1(3) and prove the following:

Theorem 3.2. Let B be a nonempty subset of a space X. Let A be a minimal γ-open set

in X and γ ∈ Γ(X ). If there exists a γ-open set C containing B such that C ⊆ clγ(B ∪ A)

, then for any nonempty subset D of A, B ∪ D is a pre-γ-open set, where γ is regular and

open.

Proof. Suppose A is a minimal γ-open set in X. Since γ is regular, therefore for

any nonempty subset D of A, we have

clγ(B ∪ D) = clγ(B)∪ clγ(D) = clγ(B)∪ clγ(A) = clγ(B ∪ A).

By supposition, we have C ⊆ clγ(B∪A) = clγ(B∪D) implies intγ(C) ⊆ intγ(clγ(B∪D))

, C being γ-open set such that B ⊆ C . It follows that

B ⊆ C = intγ(C)⊆ intγ(clγ(B ∪ D)) or B ⊆ intγ(clγ(B ∪ D)) .....(1)

and

intγ(A) = A⊆ clγ(A) ⊆ clγ(B)∪ clγ(A) = clγ(B ∪ A) implies intγ(A) ⊆ intγ(clγ(B ∪ A))

.....(2)

Since A is a γ-open set, therefore

D ⊆ A= intγ(A)⊆ intγ(clγ(B ∪ A)) =⊆ intγ(clγ(B ∪ D)) .....(3)

From (1) and (3),

B ∪ D ⊆ intγ(clγ(B ∪ D)) implies B ∪ D is a pre-γ-open set. This completes the proof.

Corollary 3.3. Let X be a space, φ 6= B ⊆ X , A a minimal γ-open set of a space X and

γ ∈ Γ(X ). If there exists a γ-open set C containing B such that C ⊆ clγ(A), then for any

nonempty subset D of A, B ∪ D is a pre γ-open set, where γ is regular and open.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 345

Proof. Let A be a minimal γ-open set and B ⊆ X . Suppose there exists a γ-open set

C containing B such that C ⊆ clγ(A). Then we have C ⊆ clγ(B)∪clγ(A) = clγ(A∪B)[4].

By Theorem 3.2, it follows that for any nonempty subset D of A, B∪ D is a pre γ-open

set. This completes the proof.

4. Finite γ-open Sets

Proposition 4.1. Let X be a space and φ 6= B a finite γ-open set in X. Then there exists

at least one (finite) minimal γ-open set A such that A⊆ B.

Proof. Suppose that B is a finite γ-open set in X. Then we have the following two

possibilities:

(1) B is a minimal γ-open set.

(2) B is not a minimal γ-open set.

In case (1), if we choose B = A, then the theorem is proved. If the case (2) is

true, then there exists a nonempty (finite) γ-open set B1 which is properly contained

in B. If B1 is minimal γ-open, we take A= B1. If B1 is not a minimal γ-open set, then

there exists a nonempty (finite) γ-open set B2 such that B2 ⊂ B1 ⊂ B. We continue

this process and have a sequence of γ-open sets ... ⊂ Bm ⊂ ... ⊂ B2 ⊂ B1 ⊂ B. Since B

is a finite, this process will end in a finite number of steps. That is, for some natural

number k, we have a minimal γ-open set Bk such that Bk = A. This completes the

proof.

In view of the Definition of locally finite space [3], we define γ-locally finite space

as:

Definition 4.1. A space X is said to be a γ-locally finite space, if for each x ∈ X there

exists a finite γ-open set A in X such that x ∈ A.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 346

Example 4.1. Let X= {a, b, c}, τ= {φ, X , {a}, {b}, {a, b}, {a, c}} [4]. For b ∈ X , define

an operation γ : τ→ P(X ) by

γ(A) = Aγ =







A, if b ∈ A

cl(A), if b 6∈ A.

Then calculations show that the γ-open sets are φ , X, {b}, {a, b}, {a, c}[4]. Clearly X

is γ-locally finite space.

Proposition 4.2. Letφ 6= B be a γ-open set in a γ-locally finite space X. Then there exists

at least one (finite) minimal γ-open set A which is contained in B, where γ is regular.

Proof. Let y ∈ B. Since X is a γ-locally finite space, then there exists a finite

γ-open set By such that y ∈ By . Since B ∩ By is a finite γ-open set [4], therefore by

proposition 4.1 there exists a minimal γ-open set A such that A ⊆ B ∩ By ⊆ B. This

completes the proof.

Proposition 4.3. Let X be a γ-locally finite space and for any α ∈ I , Bα a γ-open set and

φ 6= A a finite γ-open set. Then A∩ (
⋂

α∈I
Bα) is a finite γ-open set, where γ is regular.

Proof. Since X is a γ-locally finite space, then there exists an integer k such that

A∩ (
⋂

α∈I
Bα) = A∩ (
⋂k

i=1
Bi) . Since γ is regular [4], A∩ (

⋂

α∈I
Bα) is a finite γ-open

set. This completes the proof.

Using Proposition 4.3, we can prove the following:

Theorem 4.1. Let X be a space and for any α ∈ I , Bα a γ-open set and for any β ∈ J ,

Aβ a nonempty finite γ-open set. Then (
⋃

β∈J
Aβ)∩ (
⋂

α∈I
Bα) is a γ-open set, where γ is

regular.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 347

5. Applications

Let A be a nonempty finite γ-open set. It is clear, by Proposition 3.1 and Proposi-

tion 4.3, that if γ is regular, then there exists a natural number m such that

{A1, A2, ..., Am} is the class of all minimal γ-open sets in A satisfying the following two

conditions:

(1) For any l, n with 1≤ l, n ≤ m and l 6= n, Al ∩ An = φ.

(2) If C is a minimal γ-open set in A, then there exists l with 1 ≤ l ≤ m such that

C = Al .

Theorem 5.1. Let X be a space and φ 6= A a finite γ-open set such that A is not a

minimal γ-open set. Let {A1, A2, ..., Am} be a class of all minimal γ-open sets in A and

y ∈ A− (A1 ∪ A2 ∪ ...∪ Am). If Ay = ∩{B : B is a γ-open nbd of y }. Then there exists a

natural number k ∈ {1, 2, ..., m} such that Ak is contained in Ay , where γ is regular.

Proof. Suppose on the contrary that for any natural number k ∈ {1, 2, ..., m} , Ak is

not contained in Ay . By Corollary 3.1, for any minimal γ-open set Ak in A, Ak∩Ay = φ

. By Proposition 4.3, φ 6= Ay is a finite γ-open set. Therefore by Proposition 4.1, there

exists a minimal γ-open set C such that C ⊆ Ay . Since C ⊆ Ay ⊆ A, then C is a

minimal γ-open set in A. By supposition, for any minimal γ-open set Ak, we have

Ak ∩ C ⊆ Ak ∩ Ay = φ. Therefore for any natural number k ∈ {1, 2, ..., m}, C 6= Ak.

This is a contradiction to our supposition. Hence the proof.

Proposition 5.1. Let X be a space and φ 6= A be a finite γ-open set which is not a

minimal γ-open set. Let {A1, A2, ..., Am} be a class of all minimal γ-open sets in A and

y ∈ A− (A1 ∪ A2 ∪ ... ∪ Am). Then there exists a natural number k ∈ {1, 2, ..., m} such

that for any γ-open nbd By of y, Ak is contained in By , where γ is regular.

Proof. This follows from Theorem 5.1 as ∩{B : B is a γ-open nbd of y } ⊆ By .

Hence the proof.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 348

Theorem 5.2. Let X be a space and φ 6= A be a finite γ-open set which is not a minimal

γ-open set. Let {A1, A2, ..., Am} be the class of all minimal γ-open sets in A and y ∈

A− (A1 ∪ A2 ∪ ... ∪ Am). Then there exists a natural number k ∈ {1, 2, ..., m} such that

y ∈ clγ(Ak) , where γ is regular.

Proof. It follows from Proposition 5.1 that there exists a natural number k ∈

{1, 2, ..., m} such that Ak ⊆ B for γ-open nbd B of x. Therefore φ 6= Ak ∩Ak ⊆ Ak∩B ⊆

Ak ∩ Bγ implies y ∈ clγ(Ak). This completes the proof.

Proposition 5.2. Let φ 6= A be a finite γ-open set in a space X, γ ∈ Γ(X ) and for each

k ∈ {1, 2, ..., m} , Ak a minimal γ-open set in A. If the class {A1, A2, ..., Am} contains all

minimal γ-open sets in A, then for any φ 6= Bk ⊆ Ak, A⊆ clγ(B1 ∪ B2 ∪ ...∪ Bm), where

γ is regular and open.

Proof. Let φ 6= A be a finite γ-open set. We consider the following two cases:

Case 1. If A is a minimal γ-open set, then this follows directly from Proposition 3.6.

Case 2. If A is not a minimal γ-open set. y ∈ A−(A1∪A2∪ ...∪Am). Then by Theorem

5.2, it follows that y ∈ clγ(A1)∪ clγ(A2)∪ ...∪ clγ(Am). Therefore by Proposition 3.6,

we have

A⊆ clγ(A1)∪clγ(A2)∪...∪clγ(Am) = clγ(B1)∪clγ(B2)∪...∪clγ(Bm) = clγ(B1∪B2∪...∪Bm).

This completes the proof.

Proposition 5.3. Let X be a space and φ 6= A a finite γ-open set and Ak a minimal γ-

open set in A, for each k ∈ {1, 2, ..., m}. If for any φ 6= Bk ⊆ Ak, A⊆ clγ(B1∪B2∪ ...∪Bm)

then clγ(A) = clγ(B1 ∪ B2 ∪ ...∪ Bm), where γ is open.

Proof. For anyφ 6= Bk ⊆ Ak, k ∈ {1, 2, ..., m}, we have clγ(B1∪B2∪...∪Bm)⊆ clγ(A).

Also, we have

clγ(A)⊆ clγ(clγ(B1 ∪ B2 ∪ ...∪ Bm)) = clγ(B1 ∪ B2 ∪ ...∪ Bm) .

This implies that for any φ 6= Bk ⊆ Ak, clγ(A) = clγ(B1∪B2∪ ...∪Bm). Hence the proof.



S. Hussain and B. Ahmad / Eur. J. Pure Appl. Math, 2 (2009), (338-351) 349

Proposition 5.4. Let X be a space and φ 6= A be a finite γ-open set and for each k ∈

{1, 2, ..., m}, Ak a minimal γ-open set in A. If for any φ 6= Bk ⊆ Ak, clγ(A) = clγ(B1 ∪

B2 ∪ ...∪ Bm), then the class {A1, A2, ..., Am} contains all minimal γ-open sets in A.

Proof. Suppose on the contrary that C is a minimal γ-open set in A and for

k ∈ {1, 2, ..., m}, C 6= Ai. Therefore, for each k ∈ {1, 2, ..., m}, C ∩ clγ(Ak) = φ. This

implies that any element of C is not contained in clγ(A1∪A2∪...∪Am). This is a contra-

diction to the fact that C ⊆ A⊆ clγ(A) = clγ(B1∪B2∪...∪Bm). This completes the proof.

Combining Propositions 5.2, 5.3 and 5.4, we have the following theorem:

Theorem 5.3. Let X be a space and φ 6= A be a finite γ-open set and for each k ∈

{1, 2, ..., m}, Ak a minimal γ-open set in A. Then the following three conditions are equiv-

alent:

(1) The class {A1, A2, ..., Am} contains all minimal γ-open sets in A.

(2) For any φ 6= Bk ⊆ Ak, clγ(A)⊆ clγ(B1 ∪ B2 ∪ ...∪ Bm).

(3) For any φ 6= Bk ⊆ Ak, clγ(A) = clγ(B1 ∪ B2 ∪ ...∪ Bm), where γ is regular and open.

Remark 5.1. Suppose that φ 6= A is a finite γ-open set and {A1, A2, ..., Am} is a class

of all minimal γ-open sets in A such that for each k ∈ {1, 2, ..., m}, yk ∈ Ak. Then by

Theorem 5.3, it is clear that {y1, y2, ..., ym} is a pre-γ-open set.

Theorem 5.4. Let X be a space. Suppose thatφ 6= A is a finite γ-open set and {A1, A2, ..., Am}

is a class of all minimal γ-open sets in A. If for any B ⊆ A−{A1, A2, ..., Am}and φ 6= Bk ⊆

Ak, for each k ∈ {1, 2, ..., m}, then B ∪ B1 ∪ B2 ∪ ... ∪ Bm is a pre-γ-open set, where γ is

regular and open.

Proof. Suppose that φ 6= A is a finite γ-open set and {A1, A2, ..., Am} is a class of all

minimal γ-open sets in A. Then by Proposition 5.2

A⊆ clγ(B1 ∪ B2 ∪ ...∪ Bm)⊆ clγ(B ∪ B1 ∪ B2 ∪ ...∪ Bm).



REFERENCES 350

Also, A is γ-open implies

B ∪ B1 ∪ B2 ∪ ...∪ Bm ⊆ A= intγ(A)⊆ intγ(clγ(B ∪ B1 ∪ B2 ∪ ...∪ Bm)).

This follows that B ∪ B1 ∪ B2 ∪ ...∪ Bm is a pre-γ-open set. This completes the proof.

Theorem 5.5. Let X be a γ-locally finite space and γ ∈ Γ(X ). If a minimal γ-open set

A ⊆ X has more than one element, then X is a pre γ-T2 space, where γ is regular and

open.

Proof. Let a, b ∈ X such that a 6= b. Since X is γ-locally finite, therefore there

exist finite γ-open sets V and W containing a and b respectively. Proposition 4.1

implies that there exist a class {V1, V2, ..., Vm} of all minimal γ-open sets in V and a

class {W1, W2, ..., Wl} of all minimal γ-open sets in W. We consider three possibilities:

1. Suppose there exist k ∈ {1, 2, ..., m} and i ∈ {1, 2, ..., l} such that a ∈ Vk and

b ∈ Wi. Then Proposition 3.7 implies that {a} and {b} are pre-γ-open sets such that

a ∈ {a}, b ∈ {b} and {a} ∩ {b} = φ .

2. Suppose there exist k ∈ {1, 2, ..., m} and i ∈ {1, 2, ..., l} such that a ∈ Vk and

b /∈ Wi . Then by supposition, proposition 3.7 and Theorem 5.4, we can find for

each i, bi ∈ Wi such that {a} and {b, b1, b2, ..., bl} are pre-γ-open sets and {a} ∩

{b, b1, b2, ..., bl}= φ.

3. Suppose that there exist k ∈ {1, 2, ..., m} and i ∈ {1, 2, ..., l} such that a /∈ Vk and

b /∈Wi. Then by supposition and Theorem 5.4, for each k and i, we can find elements

ak ∈ Vk and bi ∈ Wi such that {a, a1, a2, ..., am} and {b, b1, b2, ..., bl} are pre-γ-open

sets and {a, a1, a2, ..., am} ∩ {b, b1, b2, ..., bl} = φ. Hence X is a pre γ-T2 space. This

completes the proof.

References

[1] B. Ahmad and S. Hussain: Properties of γ-Operations on topological spaces, Aligarh Bull.

Math. 22(1) (2003), 45-51.



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