Download g* λ* - closed sets in topological spaces

International Journal of Conceptions on Computing and Information Technology
Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808
g* λ* - closed sets in topological spaces
N Murugavalli
A Pushpalatha
Dept. of Mathematics,
Sri Eshwar College of Engineering,
Coimbatore, India.
[email protected]
Dept. of Mathematics,
Government Arts College,
Udumalpet, Tamilnadu, India.
[email protected]
Abstract— In this paper, we introduce the new notion g**-closed
sets and g**-open sets in Topological spaces. Also we introduce a
new space called and Tg** -space. We study some properties of
g**- closed sets and g**-open sets in Topological spaces.
Keywords- g**-closed sets ,g**-open sets and Tg* *- space
I. INTRODUCTION
N. Levine [12] introduced generalized closed sets in
general topology as a generalization of closed sets. This
concept was found to be useful and many results in general
topology were improved. Many researchers like Balachandran,
Sundaram and Maki [4], Bhattacharyya and Lahiri [5],
Arockiarani [1], Dunham [9], Gnanambal [10], Malghan [13],
Palaniappan and Rao [17], Park [18] Arya and Gupta [2] and
Devi [7] have worked on generalized closed sets, their
generalizations
and
related
concepts
in
general
topology.P.Sundaram and A.Pushpalatha introduced λ-closed
sets in topological space[19]. N.Murugavalli and
A.Pushpalatha introduced g*-closed sets in topological
space[15].
Now, we introduce the concept of g**- closed sets and
g  -open sets in Topological space and study some of their
properties.
* *
II.
PRELIMINARIES
Definition 1.1[19]: A subset A of a topological space X is
called a -closed set in X if A  cl(G) whenever A  G and
G is open in X.
Definition1.2[15]: A subset A of a topological space X is
called a g*-closed set in X if A  cl(G) whenever A  G
and G is g-open in X.
Definition 1.3: A subset S of X is called (a) semi-open [11]
if there exists an open set G such that G  S  cl(G) and semiclosed [6] if there exists an closed set F such that int(F)  S 
F.
Equivalently, a subset S of X is called semi-open if S 
cl(int(S)) and semi-closed if Sint(cl(S)) [4].
Definition 1.4: A subset S of X is called
(a) generalized closed(g-closed)set if cl(S) G whenever
S G and G is open in X and generalized open if Sc is gclosed set in X [12]. Generalized closure of S, denoted by
c*(S), is defined as the intersection of all g-closed sets
containing S in X [9].
(b) semi generalized closed (sg-closed) if scl(S)G
whenever SG and G is semi open in X and semi generalized
open if Sc is sg-closed set in X [5].
(c) generalized semi closed (gs-closed) if scl(S)G
whenever SG and G is open in X and generalized semi open
if Sc is gs-closed set in X[3].
(d) generalized semi-preclosed set (gsp-closed) if
spcl(S)  G whenever S  G and G is open in X [8].
III.
G* Λ* - CLOSED SETS IN TOPOLOGICAL SPACES
In this section, we introduce the concept of g**-closed
sets in topological spaces and study some of its properties.
Definition 2.1: A subset A of a topological space X is
called a g**-closed set in X if A cl(G) whenever A  G
and G is g**-open in X.
Theorem 2.2: If a subset A of a topological space X is
closed in X, then it is g**-closed but not conversely.
Proof: Assume that A be a closed set in X. Let G be an
g*-open set in X such that AG. Then cl(A)  cl(G). But
cl(A) = A. Therefore A  cl(G) and hence A is g**-closed in
X.
The following example shows that the converse of the
above theorem need not be true.
Example 2.3: Let X = {a, b, c} be a topological space with
the topology  = {, X, {a}, {a, c}}. Then the set {a,b} is
g**-closed but not closed.
Theorem 2.4: If A and B are g**-closed sets in a
topological space X, then A  B is g**-closed in X.
(b) -closed if cl(int(cl(S)))  S and an -open if
Sint(cl(int(S))) [16].
(c) pre closed if cl (int(S))  S and pre-open if
Sint(cl(S)) [14].
101 | 1 2 4
Proof: Let A  B  G, where G is g**-open.
 A  G, B  G
 A cl(G), B  cl(G), since A, B are g**-closed.
International Journal of Conceptions on Computing and Information Technology
Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808
 A  B cl(G)
 A  B is g**-closed.
Theorem 2.5: If A is open and g**-closed, then A is
closed.
Proof: Assume that A is open and g**-closed in X. Then
A  cl(A), since A is open and g**-closed. But A  cl(A)
always. Therefore A = cl(A). Thus A is closed set in X.
Remark 2.17: The concepts of -closed sets and g**closed sets are independent as seen from the following
examples.For, in Example 2.3 the set {c} are -closed, but
not g**-closed. In example 2.10, The sets {a,c} & {b,c} are
g**-closed but not -closed.
From the definitions and from the above results, we obtain
the following diagram:
p-closed
sg-closed
gsp-closed
Remark 2.6: If A and B are g**-closed in X, then their
union is also g**-closed in X as seen from the following
example.
Example 2.7: Let X = {a, b, c} be a topological space with
topology  = { , X, {a, b}}. The sets {a,b} and {b} are g**closed, their union {a, b} is also g**-closed in X.
g**-closed
s-closed
-closed
* *
Remark 2.8: The concepts of g  -closed set and -closed
set are independent as seen from the following examples.
Example 2.9: Let X = {a, b, c} be a topological space with
the topology  = {, X, {a}}. The sets {a}, {a,c} &{a,b}are
g**-closed but not -closed.
Example 2.10: Let X = {a, b, c} be a topological space
with the topology  = {, X, {a, b}}. The sets {a}& {b} are closed but not g**-closed.
Remark 2.11: The concepts of semi-closed sets and closed sets are independent as seen from the following
examples.
Example 2.12: Let X = {a, b, c} be a topological space
with the topology  = {, X, {a}, {a, c}}. The set {c} is semiclosed but not g**-closed. The set {a,b} is g**-closed but not
semi-closed.
Remark 2.13: The concepts of sg-closed set and g**closed are independent. For, in Example 2.9, the sets {a},{a,c}
and {a,b} are g**-closed, but not sg-closed. In Example 2.12,
the set {b} is sg-closed but not g**-closed set in X.
Remark 2.14: The concepts of semi pre-closed sets and
g**-closed sets are independent as seen from the following
examples. For, in Example 2.9 the set {a},{a,c} and {a,b} are
g**-closed, but not sp-closed.In example 2.10, The sets {a} &
{b} are sp-closed but not g**-closed.
sp-closed
Tg** - space : A topological space X is called a Tg** - space
if every g**-closed set in X is g*-closed set in X.
IV.
Remark 2.16: The concepts of b-closed sets and g**closed sets are independent as seen from the following
examples. For, in Example 2.9 the set {a},{a,c} and {a,b} are
g**-closed, but not b-closed.In example 2.10, The sets {a} &
{b} are b-closed but not g**-closed.
G* Λ* - CLOSED SETS IN TOPOLOGICAL SPACES
In this section, we introduce the concept of g**-open sets,
which is weaker than that of open set in topological spaces and
study some of its properties.
Definition 3.1: A subset A of a topological space X is
called g**-open if Ac is g**-closed in X.
Theorem 3.2: A subset A of a topological space X is g**open if and only if A  int(F) whenever A  F and F is gclosed in X.
Proof: Assume that A is g**-open and F is a g-closed set
in X such that A  F.
 Ac Fc
 Ac cl(Fc), since Ac is g**-closed .
 Ac (int(F))c
* *
Remark 2.15: The concepts of gsp-closed sets and g  closed sets are independent as seen from the following
examples. For, in Example 2.9 the set {a} is g**-closed, but
not gsp-closed.In example 2.10, The sets {a} & {b} are gspclosed but not g**-closed.
-closed
 A  int(F)
Conversely assume that A  int(F) whenever A  F and F
is g-closed in X. We have to prove that Ac is g**-closed in
X. Let Ac  G, where G is g-open in X. Then A Gc, which
implies A  int(Gc) by our hypothesis.
102 | 1 2 4
 A  (cl(G))c, since (cl(G))c = int(Gc)
 Ac  cl(G)
 Ac is g**-closed. Thus A is g**-open.
International Journal of Conceptions on Computing and Information Technology
Vol.2, Issue 1, Jan’ 2014; ISSN: 2345 - 9808
Theorem 3.3: If a subset A of a topological space X is
open in X, then it is g**-open but not conversely.
[4]
Proof: Let A be a open set in X. Let A F, where F is
closed. Since every open set is g-open set, then int(A) 
int(F). But int(A) = A. Therefore A  int(F). Thus A is g**open in X.
[5]
The following example shows that the converse of the
above theorem need not be true.
[7]
Example 3.4: Let X = {a, b, c} be a topological space with
the topology = {, X, {b},{b,c}}. The set {c} is g**-open in
X but not open in X.
Theorem 3.5: If A and B are g**-open sets in X, then A 
B is also a g**-open set in X.
[6]
[8]
[9]
[10]
[11]
* *
Proof: Assume that A and B are g  -open sets in X. Then
Ac and Bc are g**-closed sets in X and so Ac Bc = (A  B)c
is g**-closed. Thus A  B is g**-open set in X.
[12]
[13]
* *
Remark 3.6: If A and B are g  -open sets then their
intersection, A  B is also g**-open as seen from the
following example.
Example 3.7: In Example 2.10 the sets {a} and {b} are
g  -open, their intersection {a,b} is also g**-open in X.
[14]
[15]
* *
REFERENCES
[1]
[2]
[3]
[16]
I.Arockiarani, Studies on generalizations of generalized closed sets and
maps
in
topological
spaces,
Ph.D
Thesis,
BharathiarUniversity,Coimbatore, (1997).
S.P.Arya and R.Gupta, On strongly continuous mappings, Kyungpook
Math., J. 14(1974), 131-143.
Arya, S. P. and Nour, T., Characterizations of s-normal spaces, Indian J.
Pure Appl. Math., 21(1990), 717-719.
[17]
[18]
[19]
103 | 1 2 4
K. Balachandran, P.Sundaram and H.Maki, On generalized continuous
maps in topological spaces, Mem. I ac Sci. Kochi Univ. Math.,
12(1991), 5-13.
P.Bhattacharyya and B.K. Lahiri, Semi-generalized closed sets in
topology, Indian J. Math., 29(1987), 376-382.
Biswas, N., On characterization of semi-continuous functions, Atti.
Accad. Naz. Lincei Rend, Cl. Sci. Fis. Mat. Natur., (8) 48(1970),
399-402.
R.Devi, K.Balachandran and H.Maki, On generalized α-continuous
maps, Far. East J. Math. Sci. Special Volume, part 1 (1997), 1-15.
Dontchev, J., On generalizing semi-pre open sets, Mem. Fac. Sci. Kochi
Univ. Ser. A. Math., 16(1995), 35-48.
W.Dunham, A new closure operator for non-T1 topologies, Kyungpook
Math. J., 22(1982), 55-60.
Y. Gnanambal, On generalized pre-regular closed sets in topological
spaces, Indian J. Pure Appl. Math., 28(1997), 351-360.
Levine, N., Semi-open sets and semi-continuity in topological spaces,
Amer. Math. Monthly, 70(1963), 36-41.
N.Levine, Generalized closed sets in topology, Rend. Circ. Mat.
Palermo,19(1970), 89-96.
S.R. Malghan, Generalized closed maps, J. Karnatak Univ. Sci.,
27(1982),82-88.
Mashhour, A. S., Abd.El-Monsef, M. E., and Deeb, S. N., On pre
continuous mappings and weak pre-continuous mappings, Proc.
Math, Phys. Soc. Egypt., 53(1982), 47-53
Murugavalli,N and A.Pushpalatha, Generalized * –closed sets in
Topological Space, Proc. of National Conference on Frontiers Analysis
and Differencial Calculus (NCFADE-2012), Organised by
Bharathidasan University, Tiruchirapalli, India- (2012), 42-43
Njastad, O., On some classes of nearly open sets, Pacific J.Math.,
15(1965), 961-970.
N.Palaniappan and K.C.Rao, Regular generalized closed sets,
Kyungpook,Math. J., 3(1993), 211-219.
J.K. Park and J.H. Park, Mildly generalized closed sets, almost normal
and mildly normal spaces, Chaos, Solutions and Fractals 20(2004),
1103-1111.
A.Pushpalatha, Studies on generalizations of mappings in topological
space, Ph.D., Thesis, Bharathiar University, Coimbatore(2000).