Survey
* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project
Download FULL TEXT - RS Publication
Document related concepts
no text concepts found
Transcript
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
STRONGLY *- CLOSED SETS IN TOPOLOGICAL SPACES
Veronica Vijayan , Associate Professor, Nirmala College for Women , Coimbatore.
Akilandeswari. K, PG student, Nirmala College for Women, Coimbatore.
ABSTRACT
In this paper we introduce a new class of sets namely, strongly
*- closed sets which settled in between
the class of closed sets and the class of strongly g-closed sets .Applying these sets we introduce three new
classes of spaces namely T s * spaces ,
Keywords: Strongly
*T s * spaces and s * T sg spaces .
*-closed sets, T s * spaces , *T s * spaces and s * T sg spaces .
1. INTRODUCTION
Levine.N[5] introduced the class of g-closed sets, a super class of closed sets in 1970. S.P Arya and Tour
[1] defined gs-closed sets in 1990. . Levine.N[4] introduced semi-open sets and pre-open sets
respectively. Maki.et.al[6] defined
g-closed sets in 1994. M.K.R.S Veerakumar[12] introduced the
generalized g*-closed sets in 2000. Pauline Mary Helen, Ponnuthai Selvarani ,Veronica Vijayan , Punitha
Tharani [8]introduced strongly
g-closed sets and strongly g**-closed sets in 2012.
The purpose of this paper is to introduce the concept of strongly
*-closed sets, T s * spaces , *T s *
spaces and s * T sg spaces and investigate some of their properties.
2.PRELIMINARIES
Throughout this paper(X, ) represents a non-empty topological spaces on which no separation axioms
are assumed unless otherwise mentioned . For a subset A of a space (X, ), Cl(A) ,and int(A) denote the
closure and the interior of A respectively.
Definition 2-1: A subset A of a topological space (X, ) is called a
1) - open set [7] if A int(Cl(int(A)) and an - closed set if Cl(int(Cl(A))) A.
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 12
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
2) Semi-open set [4] if A Cl(int(A)) and a semi-closed set if int(Cl(A)) A.
3) g-closed set [5] if Cl(A) U whenever A U and U is open in (X, ).
4) Strongly g-closed set [8] if Cl(int(A)) U whenever A U and U is open in (X, ).
5) * -closed set [13] if Cl(A) U whenever A U and U is - open in (X, ).
6) gs-closed set [1] if Scl(A) U whenever A U and U is open in (X, ).
7) g-closed set [6] if cl(A) U whenever A U and U is open in (X, ).
8) -closed set [11] if Cl(A) U whenever A U and U is semi-open in (X, ).
9) g*-closed set [12] if Cl(A) U whenever A U and U is g-open in (X, ).
10) g*s-closed set [10] if Scl(A) U whenever A U and U is gs-open in (X, ).
11) g**-closed set [9] if Cl(A) U whenever A U and U is g*-open in (X, ).
12) Sg-closed set [2]if Scl(A) U whenever A U and U is semi-open in (X, ).
13) -closed set [14] if Cl(A) U whenever A U and U is sg-open in (X, ).
Definition 2-2: A topological space (X, ) is said to be a
1) T * space[13] if every
*-closed set in it is closed.
2)
T 1/ 2S space[15] if every strongly g -closed set in it is closed.
3)
Td space[3] if every gs-closed set in it is g-closed.
3. BASIC PROPERTIES OF STRONGLY *- CLOSED SETS
We introduce the following definition.
Definition 3.1:
A subset A of a topological space (X, ) is said to be a strongly
whenever A U and U is
*-closed set if Cl(int(A)) U
- open in X.
Proposition 3.2: Every closed set is a strongly
*- closed set.
Proof follows from the definition.
The following example supports that a strongly
*- closed set need not be closed in general.
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 13
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
Example 3.3: Let X={a,b,c} , = { ,X,{a},{a,b}}. Then A= {b} is strongly
*- closed but not closed
in (X, ).
Proposition 3.4: Every strongly *- closed set is strongly g-closed.
Proof: Let A be strongly
* - closed . Let A U and U be open. Then A U and U is -open and
Cl(int (A)) U since A is strongly * -closed. Hence A is strongly g -closed .
The converse of the above proposition need not be true in general as seen in the following example.
Example 3.5: Let X= {a,b,c} ,
= { ,X,{b}}.Then A= {a,b} is strongly g - closed but not strongly
*- closed in (X, ).
Proposition 3.6: Every
*- closed set is strongly *- closed .
*- closed set. Let A U and U be -open. Then Cl(A) U since A is *-closed
and Cl(int(A)) Cl(A ) U. Hence A is strongly *-closed.
Proof: Let A be a
The following example supports that a strongly
* closed need not be *- closed in general.
Example 3.7: Let X={a,b,c} , ={ ,X,{b,c}}.Then A={b} is strongly
*- closed but not a *-
closed set in (X, ).
Remark 3.8: Strongly
*- closedness is independent from g-closedness , gs-closedness and g –
closedness .
Example 3.9: Let X={a,b,c} , ={ ,X,{b,c}}.Then A={b} is strongly
*- closed but not g-closed in
(X, ).
Let X={a,b,c}, ={ ,X,{b}}.Then A={a,b} is g-closed but not a strongly
Hence strongly
*- closed set in (X, ).
*- closedness is independent of g-closedness .
Example 3.10: Let X={a,b,c} , ={ ,X,{c}}.Then A={a,c} is gs-closed but not a strongly
set in (X, ).Let X={a,b,c} ,
*- closed
={ ,X,{a,c}}. Then A={a} is strongly *- closed but not a gs-closed
set in (X, ).
Hence strongly
*- closedness is independent of gs-closedness.
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 14
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
Example 3.11: Let X={a,b,c} , ={ ,X,{a,c}}.Then A={a} is strongly
* closed but not g-closed
in (X, ).
Let X={a,b,c}, ={ ,X,{b},{b,c}}.Then A={a,b} is
g -closed but not a strongly *- closed set in
(X, ).
Hence strongly
*- closedness is independent of g -closedness.
Proposition 3.12: Every
-closed set is strongly *- closed .
Proof follows from the definitions.
The reverse implication does not hold as it can be seen from the following example.
Example 3.13: Let X= {a,b,c} , = { ,X,{a}}.Then A={b}is strongly
*- closed but not a -closed
set in (X, ).
Remark 3.14: Strongly *- closedness is independent from g*-closedness , g*s- closedness and g**closedness.
Example 3.15: Let X={a,b,c} , ={ ,X,{a},{a,c}}.Then A={c} is strongly
closed set in (X, ). B={a,b} is g*-closed but not strongly
*- closed but not a g*-
*- closed in (X, ).Hence strongly *-
closedness is independent of g*-closedness.
Example 3.16: Let X={a,b,c} , ={ ,X,{a},{b},{a,b}}.Then A= {b} is g*s- closed but not a strongly
* -closed set in (X, ). Let X={a,b,c} , ={ ,X,{a,b},{c}}.Then A={b} is strongly *- closed but
not a g*s-closed set in (X, ).Hence strongly
*- closedness is independent of g*s-closedness.
Example 3.17: Let X={a,b,c} , ={ ,X,{a},{a,c}}.Then A={c} is strongly
closed set in (X, ). B={a,b} is g**-closed but not strongly
*- closed but not a g**-
*- closed in (X, ).Hence strongly *-
closedness is independent of g**-closedness.
Proposition 3.18: Every - closed set is strongly
*- closed but not conversely.
Example 3.19: Let X={a,b,c} , ={ ,X,{b,c}} . Then A={b} is strongly
*- closed but not a -
closed set in(X, ).
Theorem 3.20: If A is strongly
*- closed and A B Cl(int(A)), then B is strongly *- closed .
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 15
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
- open. Then A U where U is - open. Cl(int(A)) U,since A is
*-closed. Cl(int(B)) Cl(B) Cl(int(A)) U. B is strongly *-closed.
Proof: Let B U where U is
strongly
Theorem 3.21: If a subset of a topological space (x, ) is both open and strongly
*- closed, then it is
closed.
Proof: Suppose A is both open and strongly
*- closed. Cl(int(A)) A and Cl(A) = Cl(int(A)) A .
Hence A is closed.
Corollary 3.22: If A is both open and strongly
*- closed in X, then it is both regular open and regular
closed .
Corollary 3.23: If a subset A of a topological space (x, ) is both strongly
is
*- closed and open , then it
*-closed.
The above results can be represented in the following figure.
Strongly g-closed
* closed
closed
g-closed
g**-closed
strongly *- closed
gs-closed
- closed
g -closed
g*s-closed
g*-closed
*
where A
B(resp. A
- closed
B) represents A implies B but not conversely(resp. A and B are
independent).
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 16
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
4. APPLICATIONS OF STRONGLY *-CLOSED SETS
As applications of strongly
sg
*-closed sets, new spaces, namely, T s * space , * T s * space and s * T
space are introduced.
We introduce the following definitions.
Definition 4.1: A space (X, ) is called a T s * space if every strongly
Definition 4.2: A space (X, ) is called a
*-closed set is closed.
* T s * space if every strongly *-closed set is *- closed.
Definition 4.3: A space (X, ) is called a s * T sg space if every strongly g-closed set is strongly
*-
closed.
Theorem 4.4: Every T s * space is a T * space .
Proof: Let (X, ) be a T s * space. Let A be
*-closed .Then A is a strongly *-closed set, and since
(X, ) is a T s * space, A is a closed set. Hence the space (X, ) is a T * space .
The reverse implication does not hold as it can be seen from the following example.
Example 4.5: Let X= {a,b,c} , = { ,X,{b}}.Then (X, ) is a T * space. A={a} is strongly
*-
closed but not a closed set. The space (X, ) is not a T s * space.
Theorem 4.6: Every T s * space is a
* T s * space .
Proof: Let (X, ) be a T s * space. Let A be strongly *-closed. Then A is closed since (X, ) is a T s *
space. Since every closed set is a * - closed set, A is
*-closed . Hence the space (X, ) is a * T s *
space .
The converse of the above theorem is not true in general as it can be seen from the following example.
Example 4.7: Let X={a,b,c} , ={ ,X,{a},{b,c}}.Then (X, ) is a
* T s * space .A={b}is strongly
*- closed but not a closed set . The space (X, ) is not a T s * space.
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 17
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
Theorem 4.8: Every T 1/ 2 S space is a s * T sg space.
The converse of the above theorem need not be true in general as seen in the following example.
Example 4.9: Let X={a,b,c} , ={ ,X,{b,c}}. Then (X, ) is a s * T sg space. A={b} is a strongly g
– closed set but not a closed set . The space (X, ) is not a T 1/ 2 S space .
Theorem 4.10: A space which is both
Proof: Let A be strongly
* T s * and T * space is a T s * space .
*- closed . Then A is *- closed, since the space is a * T s * space. Now A
is closed, since the space is a T * space . The space (X, ) is a T s * space .
Theorem 4.11: A space which is both s * T sg and T s * space is a T 1/ 2 S space .
Proof follows from the definitions.
Remark 4.12: T s * ness ,
* T s * ness and s * T sg ness are independent from T d ness.
Example 4.13: Let X={a,b,c} , ={ ,X,{b,c}}. Then (X, ) is a T d space . A={b} is strongly *closed but not closed. Hence the space (X, ) is not a T s * space but it is a T d space .
Let X={a,b,c} , ={ ,X,{a},{b},{a,b}} .Then (X, ) is a T s * space . A={a}is gs-closed but not a gclosed set. Hence the space (X, ) is not a T d space .Hence T s * ness is independent from T d ness.
Example 4.14: Let X={a,b,c} , ={ ,X,{b,c}}.Then (X, ) is a T d space .A={b} is strongly *closed but not a
*-closed set .Hence (X, ) is not a * T s * space .Let X={a,b,c} , ={
,X,{a},{b},{a,b}}. Then(X, ) is a
* T s * space. A={a} is gs-closed but not a g-closed set. The
space (X, ) is not a T d space .
Hence
* T s * ness is independent from T d ness.
Example 4.15: Let X={a,b,c} , ={ ,X,{b}}.Then (X, ) is a T d space .A= {a,b} is strongly g closed but not strongly
*- closed. The space (X, ) is not a s * T sg space .
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 18
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
Let X={a,b,c} , ={ ,X,{a},{b},{a,b}}.Then (X, ) is a s * T sg space. A={a} is gs-closed but not a
g-closed set. The space (X, ) is not a T d space .Hence s * T sg ness is independent from T d ness.
Remark 4.16: s * T sg ness is independent from T * ness.
Example 4.17: Let X={a,b,c} , ={ ,X,{b}}.Then (X, ) is a T * space . A= {a,b} is strongly g closed but not strongly
*- closed. The space (X, ) is not a s * T sg space .
Example 4.18: Let X={a,b,c} , ={ ,X,{b,c}}. Then space (X, ) is a s * T sg space. A= {a,b} is
*-closed but not closed. The space (X, ) is not a T * space .Hence s * T sg ness is independent
from T * ness.
The above results can be represented in the following figure.
* T s *
T s *
Td
s *
T *
where A B(resp. A
T sg
T 1/ 2 S
B ) represents A implies B(resp. A and B are independent).
REFERENCES
[1] Arya.S.P and Nour.T, Characterizations of s-normal spaces , Indian.J. Pure. Appl. Math.,
21(8)(1990),717-719.
[2] Bhattacharya.P and Lahiri.B.K, Semi-generalized closed sets in topology , Indian J. Math.,
29(3)(1987),
375-382.
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 19
International Journal of Computer Application
Available online on http://www.rspublication.com/ijca/ijca_index.htm
Issue 4, Volume 1 (February 2014)
ISSN: 2250-1797
[3]Devi.R, Balachandran.K and Maki.H, Semi-generalized closed maps and generalized closed maps,
Mem. Fac. Sci. Kochi Univ. Ser.A. Math., 14(1993), 41-54.
[4] Levine.N, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly,70(1963),
36-41.
[5]Levine.N, Generalized closed sets in topology , Rend. Circ. Math. Palermo, 19(2)(1970),89-96.
[6] Maki.H, Devi.R and Balachandran.K, Associated topologies of generalized
-closed maps and -
generalized closed sets, Mem. Fac. Sci. Kochi Univ. Ser.A, Math., 15(1994), 51-63.
[7]Njastad.O, On some classes of nearly open sets, Pacific J.Math., 15(1965), 961-970.
[8]Pauline Mary Helen, Veronica Vijayan , Ponnuthai Selvarani , Punitha Tharani, Strongly g-closed sets
and Strongly g**-closed sets , IJMA , 3(8)(2012), 3108-3112.
[9] Pauline Mary Helen.M, Veronica Vijayan, Ponnuthai Selvarani.S, g**-closed sets in topological
spaces,IJMA,3(5)(2012),1-15.
[10]Pushpalatha.P and Anitha .K , g*s-closed sets in topological space, Int.J. contemp.Math.Sciences,
Vol.6., March 2011, no 19, 917-929.
[11]Sundaram.P and Shiek john.M, on
-closed sets in topological spaces , Actaciencia Indica 4(2000),
389-392.
[12] Veerakumar M.K.R.S , Between closed sets and g-closed sets , Mem. Fac. Sci. Kochi Univ. Ser.A,
Math.,17(1996),33-42
[13]Veronica Vijayan, Priya.F, *-closed sets in topological spaces, IJCA, Issue 3, Volume 4 (July –
August2013), ISSN:2250-1797,49-60.
[14] Veronica Vijayan, Selvapriya.K , A study on
*, , * -closed sets in topological spaces , IJCA,
Issue 3,Volume 2(April 2013) ISSN:2250-1797, 105-115.
[15] Veronica Vijayan, Thenmozhi.B, Strongly g, g*,g** - closed sets in bitopological spaces , IJCA,
Issue 3, Volume 3 (May-June 2013) ISSN:2250-1797, 28-35.
R S. Publication (rspublication.com), rspublicationhouse@gmail.com
Page 20
Related documents