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Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
STRONGLY ψ
͞ –CLOSED AND STRONGLY ͞ψ* CLOSED SETS IN TOPOLOGICAL
SPACES
Veronica Vijayan
Associate Professor, Nirmala college for women, Coimbatore, Tamil Nadu
Nasreen Fathima.S
PG student, Nirmala college for women, Coimbatore, Tamil Nadu
____________________________________________________________________________________
ABSTRACT
In this paper we introduce strongly ͞ψ and strongly ψ
͞ * closed sets in topological spaces.
Properties of these sets are investigated and applying these sets, we introduce three new spaces namely
Ts ψ͞ space, Ts ψ͞ * space and T*s ψ͞ space.
Key words: Strongly ͞ψ- closed sets and Strongly ͞ψ*- closed sets
____________________________________________________________________________________
1. INTRODUCTION
N.Levine[3] introduced the class of generalized closed sets in the year 1970.Veronica Vijayan,
K.Selvapriya[8] introduced ͞ψ,͞ψ* closed sets in topological spaces in the year 2013. In this paper strongly
͞ψ and strongly ψ
͞ * closed sets are defined and their properties are investigated.
2. PRELIMINARIES
Throughout this paper (X,τ) represents a non-empty topological space on which no separation axioms are
assumed unless otherwise mentioned. For a subset A of a topological space (X,τ), cl(A) and int(A) denote
the closure and the interior of the subset A respectively.
DEFINITIONS 2.1: A subset A of a topological space (X,τ) is called
1. a semi-open set [2] if A Int(cl(A)) and a semi-closed set if Int(cl(A))  A.
2. a semi-generalized set(briefly sg-closed)[1] if Scl(A)  U whenever A  U and U is semi-open in (X,τ).
3. a  -closed set[8] if Cl(A)  U whenever A  U and U is sg-open in (X,τ).
4. a

*
-closed set[8] if Cl(A)  U whenever A  U and U is  -open in (X,τ).
5. a generalized closed set (briefly g-closed)[3] if Cl(A)⊆U whenever A⊆U and U is open in (X,τ).
6. a g star closed set(briefly g*-closed)[7] if Cl(A)⊆U whenever A⊆U and U is g-open in (X,τ).
7. a strongly generalized-closed set(briefly strongly g-closed)[6] if Cl(int(A))⊆U whenever A⊆U and U is
open in (X,τ).
8. a strongly g star closed set(briefly strongly g*-closed)[4] if Cl(int(A))⊆U whenever A⊆U and U is g-open
in (X,τ).
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Issue 4, Volume 1 (February 2014)
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9. a g star star closed set(briefly g**-closed)[5] if Cl(A)⊆U whenever A⊆U and U is g* open in (X,τ).
DEFINITIONS 2.2: A topological space (X,τ) is called a
T  space[8] if every  -closed set is closed.
aT
space[8] if every  -closed set is closed.

1. a
*
2.
*
T  space[8] if every  -closed set is  -closed.
a T space[9] if every strongly g closed set is closed.
a T space[9] if every strongly g closed set is closed.
a T space[9] if every strongly g closed is strongly g
a T space[7] if every g closed is closed.
a T space[5] if every g -closed is closed.
a T space[5] if every g -closed is g closed.
4.
5.
6.
7.
8.
9.
*
*
3. a
1
s
2
*
1
s
2
*
**
*
*
1
s
2
*
1
2
*
**
1
2
**
1
2
**
closed.
*
3. Basic Properties of Strongly  - closed sets and Strongly

*
- closed sets
We introduce the following definitions.
Definition 3.1: A subset A of a topological space (X,τ) is said to be strongly  -closed if Cl(int(A))⊆U
whenever A⊆U and U is sg-open.
Definition 3.2: A subset A of a topological space (X,τ) is said to be strongly

*
-closed if
Cl(int(A))⊆U whenever A⊆U and U is  -open.
Proposition 3.3: Every closed set is strongly  -closed.
Proof follows from the definitions.
The converse of the above proposition is not true.
Example 3.4: Let X={a,b,c} τ ={ Φ,{a},{a,c},X}. A={c} is strongly  -closed but not a closed set of (X,
τ).
Proposition 3.5: Every closed set is strongly

*
-closed.
Proof follows from the definitions.
The following example shows that the converse of the above proposition is not true in general.
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Example 3.6: Let X={a,b,c} τ ={ Φ,{a},X}. A={a,b} is strongly
Issue 4, Volume 1 (February 2014)
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
*
-closed but not a closed set of (X,
τ).
Proposition 3.7: Every strongly  -closed set is strongly

*
closed.
Proof: Let A be a strongly  -closed set. Let A⊆U and U be  -open. Then A⊆U and U is sg-open.
Therefore, Cl(int(A))⊆U since A is strongly  -closed.... A is strongly

*
closed.
The converse of the above proposition is not true in general.
Example 3.8: Let X={a,b,c} τ ={ Φ,{a},X}. A={a,b} is a strongly

*
-closed but not a strongly  -
closed set of (X,τ).
Proposition 3.9: Every strongly  -closed set is strongly g-closed but not conversely.
Example 3.10: Let X={a,b,c} τ ={ Φ,{a},{a,c},X}. A={a,b} is a strongly g-closed set but not strongly
 - closed.
Proposition 3.11: Every  -closed set is strongly  -closed
Proof: Let A be a  -closed set. Let A⊆U and U be sg-open. Then Cl(A)⊆U since A is  -closed.
Cl(int(A))⊆Cl(A)⊆U. ... A is strongly  -closed.
The converse of the above proposition need not be true.
Example 3.12: Let X={a,b,c} τ ={ Φ,{a},X}. A={b} is strongly  - closed but not a  -closed set of
(X,τ).
Proposition 3.13: Every

*
-closed set is strongly

*
-closed.
The converse of the above proposition is not true in general as we see the following example.
Example 3.14: Let X={a,b,c} τ ={ Φ,{a},{a,c},X}. A={c} is strongly
Proposition 3.15: Every  -closed set is strongly


*
-closed but not

*
- closed.
*
-closed
The following example shows that the converse of the above proposition is not true.
Example 3.16: Let X={a,b,c} τ ={ Φ,{a},{a,c},X}. A={c} is strongly

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*
- closed but not  -closed
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Remark 3.17: Strongly  -closedness is independent of

Example: Let X={a,b,c} τ ={ Φ,{a},{a,c},X}. A={a,b} is
B={c} is strongly  -closed but not

Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
*
-closedness.

*
- closed but not strongly  -closed and
*
- closed.
Hence strongly  -closedness is independent of

*
-closedness.
Remark 3.18: g-closedness is independent of strongly  -closedness.
Example: Let X={a,b,c} τ ={ Φ,{a},{a,c},X}. A={a,b} is g-closed but not strongly  -closed and
B={c} is a strongly  -closed set but not g-closed.
Hence g-closedness is independent of strongly  -closedness.
Proposition 3.19: Every g*-closed set is strongly

*
-closed.
Proof follows from the definitions.
The converse of the above proposition is not true in general as it can be seen from the following
example.
Example 3.20: Let X={a,b,c} τ ={ Φ,{a},{a,c},X}. A={c} is a strongly

*
- closed set but not g*-
closed.
Proposition 3.21: Every strongly g*-closed set is strongly

*
-closed. But the converse is not true.
Example 3.22: Let X={a,b,c} τ ={ Φ,{a},X}. A={a,b} is strongly

*
- closed but not strongly g*-
closed.
Remark 3.23: g*-closedness is independent of strongly  -closedness.
Example: Let X={a,b,c} τ ={ Φ,{b},{a,b},X}. A= {b,c} is g*-closed but not strongly  -closed and
B= {a} is a strongly  -closed set but not g*-closed.
Theorem 3.24: Let A be an open subset of X which is strongly

*
-closed then A is closed.
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Proof: A is both open and strongly
is strongly

*

*
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-closed. Since A is open, A is  -open. ... Cl(int(A)) ⊆A, since A
-closed. Now, cl(int(A))=Cl(A), since int(A)=A.... Cl(A) ⊆A. Hence A is closed.
Corollary 3.25: If A is both open and strongly

*
-closed in ( X , ) then it is both regular open and
regular closed in (X,τ).
Proof: As A is open and closed, A=Int(A)= Int(Cl(A)). Hence A is regular open. Cl(Int(A))=Cl(A)=A
since A is both open and closed. Therefore, A is regular closed.
Corollary 3.26: If a subset A of a topological space (X,τ) is both open and strongly
is


*
-closed, then it
*
-closed.

*
Proof: Let A be open and strongly

*
Theorem 3.27: A set A is strongly
-closed. By theorem(3.24) A is closed and hence A is

*
-closed.
-closed if and only if Cl(Int(A)) –A contains no non empty  -
closed set.
Proof: Necessity: Let F be a non-empty  -closed subset of Cl(Int(A)) –A.
Now F⊆Cl(Int(A)) –A implies F⊆Cl(Int(A))  Ac. Thus F⊆Cl(Int(A)).Now, F⊆Ac implies A⊆Fc. Here

Fc is  -open and A is strongly
*
-closed, we have Cl(Int(A)) ⊆Fc. Thus, F⊆ (Cl(Int(A)))c. Hence F⊆
(Cl(Int(A)))  (Cl(Int(A)))c = Φ.
Sufficiency: Let A⊆U and U is  -open. Suppose that Cl(Int(A)) is not contained in U then Cl(Int(A)) 
Uc= Φ and Cl(Int(A))  Uc is  -closed. ... Φ ≠ Cl(Int(A))  Uc⊆ Cl(Int(A)) –A which is a contradiction
and hence Cl(Int(A))⊆U. ... A is a strongly
Corollary 3.28: A strongly

*

*
-closed set.
-closed set A is regular closed if and only if cl(int(A)) –A is  -closed.
Proof: Assume that A is regular closed. Then Cl(int(A))=A, Cl(int(A)) –A= Φ which is  -closed.
Conversely, assume that cl(int(A)) –A is  -closed. By theorem 3.27, cl(int(A)) –A contains no non
empty  -closed set. Therefore, cl(int(A)) –A= Φ and hence cl(int(A))=A. Hence A is regular closed.
Theorem 3.29: If A is strongly

*
-closed and A⊆B⊆Cl(Int(A)) then B is strongly
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
*
-closed.
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Proof: Let B⊆U where U is  -open. A⊆B⊆U and A is strongly
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
*
-closed. Therefore, Cl(Int(A)) ⊆U.
Int(B) ⊆B⊆Cl(Int(A)) Then Cl(Int(B)) ⊆Cl(B) ⊆Cl(Int(A)) ⊆U. Hence, B is strongly

*
-closed.
Theorem 3.30: If A is a subset of a topological space (X,τ) is open and strongly  -closed, then A is
closed.
Corollary 3.31: If A is both open and strongly  -closed in (X,τ) then it is both regular open and
regular closed in (X,τ).
Corollary 3.32: If a subset A of a topological space X is both open and strongly  -closed, then it is  closed.
Theorem 3.33: A set A is strongly  -closed if and only if Cl(Int(A)) –A contains no non empty sgclosed set.
Proof: Necessity: Let F be a non-empty sg-closed subset of Cl(Int(A)) –A. Now F⊆Cl(Int(A)) –A
implies F⊆Cl(Int(A))  Ac. Thus F⊆Cl(Int(A)). Now, F⊆Ac implies A⊆Fc. Here Fc is sg-open and A is
strongly  -closed, we have Cl(Int(A)) ⊆Fc. Thus, F⊆ (Cl(Int(A)))c. Hence F⊆(Cl(Int(A))) 
(Cl(Int(A)))c = Φ.
Sufficiency: Let A⊆U, U is sg-open. Suppose that Cl(Int(A)) is not contained in U, then Cl(Int(A))  Uc=
Φ and Cl(Int(A))  Uc is sg-closed. ... Φ ≠ Cl(Int(A))  Uc⊆ Cl(Int(A)) –A which is a contradiction and
hence Cl(Int(A)) ⊆U. ... A is a strongly  -closed set.
Corollary 3.34: A strongly  -closed set A is regular closed if and only if Cl(Int(A)) –A is sg-closed.
Theorem 3.35: If A is strongly  -closed and A⊆B⊆Cl(Int(A)) then B is strongly  -closed.
Remark 3.36: g**-closedness is independent of strongly  -closedness.
Example: Let X={a,b,c} τ ={ Φ ,{b},{a,b},X}. A={b,c} is g**-closed but not strongly  -closed and
B={a} is a strongly  -closed set but not g**-closed.
Proposition 3.37: Every g**-closed set is strongly

*
-closed but not conversely.
Example 3.38: Let X={a,b,c} τ ={ Φ,{b},{a,b},X}. A={a} is a strongly

*
-closed set but not g**-
closed.
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g** closed
g closed
Strongly g- closed
Closed
Strongly  - closed
 -closed
Strongly

*
- closed
g* closed
Strongly g* -closed
Where A
are independent).
B (resp.A

*
- closed
B) represents A implies B but not conversely (resp. A and B
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
*

*
4. Applications of Strongly  -closed and Strongly
As application of strongly  -closed and strongly
space,
T
*
s
space,
Issue 4, Volume 1 (February 2014)
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-closed sets
-closed sets, 3 new spaces namely
T
s
*
T  space are introduced.
s
space if every strongly  -closed set is closed.
Definition 4.1: A space (X,τ) is called a
T
Definition 4.2: A space (X,τ) is called a
T
Definition 4.3: A space (X,τ) is called a
s
s
*
T
s
*
space if every strongly
space if every strongly


*
*
-closed set is closed.
-closed set is strongly  -
closed.
Proposition 4.4: Every
Proof: Let (X,τ) be a
(X,τ) is a T
s
*
T s
T s
*
space is
*
*
T
s
space.

*
space. Let A be a strongly
-closed set of (X,τ). Then A is closed, since
space and hence A is strongly  -closed. Hence (X,τ) is a
*
T
s
space.
But the converse is not true.
Example 4.5: Let X={a,b,c} τ ={ Φ,{a,c},X}. (X,τ) is a
is strongly

*
T
space but not a
T
is a
s
T s
*
space since A= {a}
*
-closed set but not closed.
Proposition 4.6: A space (X,τ) which is both
Proof: Let A be a strongly

*
T
s
s
and
s
T s
*
space.
-closed set. Then A is strongly  -closed since the space is a
Now A is closed since the space is a
Theorem 4.7: Every
*
T
T
space is a
s
T
space. Hence the space is a
T s
*
*
T
s
space.
space.
space.
Proof: Let A be a  -closed set. Then A is strongly  -closed. Now A is closed since the space is a
space. Hence the space is a
T
T
s
space.
The converse of the above theorem is not true in general as seen in the following example.
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Example 4.8: X={a,b,c}, τ ={ Φ,{a},X}. We have already proved in example (3.6) that the  -closed
sets are Φ,{b,c},X and in example (3.8) that the strongly -closed sets are Φ,{b},{c},{b,c},X. Hence
every  -closed is closed and (X,τ) is a
hence (X,τ) is not a
T
Theorem 4.9: Every
space. Hence every
s
T s

space. A={b} is a strongly  -closed but not closed and
T
space is a
*
*
T
T
space need not be a
s
space.
space.

is closed and hence A is  -closed. Hence the space is a T
Proof: Let A be a
T
*
*
-closed set. Then A is a strongly
-closed set. Since (X,τ) is a
*

T s
*
space. A
space.
The converse need not be true in general as seen in from the following example.
Example 4.10: X= {a,b,c}, } τ ={ Φ,{a,c},X}. In example (4.5) we have already proved that (X,τ) is a
*
T
space since every

*
T
-closed is closed. But it is not a
*
s
space since A= {c} is strongly

*
-
closed set but not  -closed.
Theorem 4.11: Every
T
*
s
space is a
T
*
1
2
space.
Proof: Let A be g* closed set. Then A is strongly
space. Hence the space is
T
*
1
2

*
-closed. Now A is closed since the space is
T
*
s
space. But the converse is not true.
Example 4.12: X={ a,b,c}. τ ={Φ,{a},X}. In example (3.6) and (3.8) we have proved that every g* closed
set is closed and hence (X,τ) is a
T
*
1
2
space but not a
T
*
s
space since A={b} is strongly

*
-closed
set but not closed.
Theorem 4.13: A space (X,τ) which is both
*
T
and
s
T
s
Proof: Let A be a strongly g*- closed set then A is strongly
since the space is a
T
*
1
s
2
*
T
s
is a

*
T
*
1
s
2
space.
-closed. Then A is strongly  -closed
space. Now A is closed since the space is a
T
space. Hence the space is a
T
space.
s
space.
Theorem 4.14: A space (X,τ) which is both
T
s
*
and
*
T
1
s
2
is a
1
s
2
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Proof: Let A be a strongly g closed set then A is strongly g*- closed since the space is a
Then A is strongly
T
1
s
2

*
T
-closed. Now A is closed since the space is a
*
T
1
s
2
space.
space. Hence the space is a
*
s
space.
Theorem 4.15: A space (X,τ) which is both
T
*
and
*
s
T
1
s
2
T
is a
space.
s
Proof: Let A be strongly  closed set. Then A is strongly g closed. A is strongly g*- closed since the
space is
*
T
1
s
2

space. Then A is strongly
Hence the space is a
T
s
*
-closed. Now A is closed since the space is a
T
*
s
space.
space.
Theorem 4.16: A space (X,τ) which is both
T
and
*
s
*
T
1
s
2
is a
T
space.
Proof: Let A be  closed. Then A is strongly  -closed and hence A is strongly g closed set. Now A is
strongly g*- closed since the space is a *
since the space is a
T
s
*
T
1
s
2
space. Hence the space is a
Theorem 4.17: A space (X,τ) which is both
Proof: Let A be
space is a
*
T
s

space. Then A is strongly
T
closed. Then A is strongly
space. Since the space is a

T
T
s
Proof: Let A be g**- closed set. Then A is strongly
space is a
*
T
s
space. Since the space is a
T
Theorem 4.19: A space (X,τ) which is both
*
-closed. Now A is closed
space.
T  is a T
space.
*
s
-closed. Now, A is strongly  -closed since the
*
**
1
2
T
s
*
-closed. Now A is closed since the space is a
*
space.
T

-closed. Now, A is strongly  -closed since the
s
*
is a
T
space.
space, A is closed and hence the space is
s
T
and
*
and
**
T
1
2
is a
T
**
1
2
Proof: Let A be g**- closed set. Then A is g*- closed since the space is a **

*
space. A is closed and hence the space is a
s
Theorem 4.18: A space (X,τ) which is both
and
*
s
*
T

T
s
*
**
1 space.
2
space.
T
1
2
space. Then A is strongly
space. Hence the space is a
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T
T
**
1
2
space.
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Theorem 4.20: A space (X,τ) which is both
T
and
s
Proof: Let A be a g*- closed set. Then A is strongly
is a
*
T
s
space. Since the space is a
T
s

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ISSN: 2250-1797
*
T  space is a T
s
*
*
1
2
space.
-closed. A is strongly  -closed since the space
space, A is closed and hence the space is a
*
Theorem 4.21: A space (X,τ) which is both
T
Theorem 4.22: A space (X,τ) which is both
T
s
and
*
T  space is a T
and
s
T
1
s
2
space is a
*
T
*
1
2
space.
space.
T
*
space.
References
1. P.Bhattacharya and B.K.Lahiri, semi-generalized closed sets in Topology, Indian J.Math.,
29(3)(1987),375-382.
2. N.Levine, semi-open sets and semi continuity in topological spaces, Amer.Math.monthly,70(1963),
36-41.
3. N.Levine, generalized closed sets in topological spaces, Rend.Circ.Math.Palermo 19(2)(1970),89-96.
4. R.Parimelazhagan and V.Subramaniapillai, Strongly g* closed sets in topological spaces, Int.Journal of
Math Analysis, Vol 6.,(30)(2012),1481-1489.
5. Pauline Mary Helen.M, Veronica Vijayan, Ponnuthai Selvarani, g** closed sets in topological spaces,
IJMA 3(5),(2012),1-15.
6. Sr.Pauline Mary Helen.M, Mrs.Ponnuthai Selvarani, Mrs.Veronica Vijayan, strongly g closed sets and
strongly g** closed sets in topological spaces, IJMA 3(8),(2012),3108-3112.
7. MKRS. Veerakumar, Between closed sets and g closed sets, Mem.Fac.Sci.Koch.Univ.Ser. A. Math.,
17(1996), 33-42.
8. Veronica Vijayan, K.Selvapriya, A Study on ψ*, ͞ψ and ͞ψ* closed sets in Topological spaces, IJCA,
Issume 3, Volume 2 (April 2013) ISSN: 2250-1797, 105-115.
9. Veronica Vijayan, B.Thenmozhi, strongly g,g* ,g** closed sets in Bitopological spaces, IJCA, Issume 3,
Volume 3(May- june 2013) ISSN: 2250-1797.
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