 Open Access
 Total Downloads : 10
 Authors : M. Jeyaraman, S. Chinthamani Vasthirani
 Paper ID : IJERTCONV3IS33012
 Volume & Issue : RACMS – 2015 (Volume 3 – Issue 33)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
on _{}Generalized Closed Sets in Generalized Topological Spaces
1M. Jeyaraman,
1 Assistant Professor, Department of Mathematics, RDM Government Arts College,
Sivagangai.
2S. Chinthamani Vasthirani,
2Assosiate Professor, Department of Mathematics, Sree Sevugan Annamalai College,
Devakottai.
Abstract We introduce new classes of sets called Âµ gclosed sets and Âµ gopen sets in GTS. We also investigate several properties of such sets. It turns out that Âµ gclosed sets and Âµ gopen sets are weaker forms of closed sets and open sets, and stronger forms of gÂµclosed sets and gÂµopen sets, respectively.
2010 Mathematics Subject Classification: 57C10, 57C08, 57C05
Keywords and phrases: Âµgclosed set, locally closed sets, gÂµclosed sets.

INTRODUCTION
In 2010 E. Ekici and B. Roy introduced the notion of Âµsets in GTS. A Âµset is a set A which is equal to its kernel. In 2011 Bishwabhar and Ekici introduced and investigate (, Âµ)closed sets by involving Âµsets and
Âµclosed sets. The objective of this paper is to introduce new classes of sets called Âµ gclosed sets and Âµ gopen sets in GTS. It turns out that Âµ gclosed sets and Âµ gopen sets are weaker forms of closed sets and open sets, and stronger forms of gÂµclosed sets and gÂµopen sets, respectively.
We recall some notion defined in [ 1, 2]. Let X be a nonempty set and Âµ be a collection of subsets of
X. Then Âµ is called a generalized topology(briefly GT) on X if Âµ and Gi Âµ for i I implies G = iI Gi Âµ. We say Âµ is strong if X Âµ, and we call the pair (X, Âµ) a generalized topological space (briefly GTS) on X. The elements of Âµ are called Âµopen sets and their complements are called Âµclosed sets. For A X, we denote by cÂµ(A) the intersection of all Âµclosed sets containing A, i.e., the smallest Âµclosed set containing A; and by iÂµ(A) the union of all Âµopen sets contained in A, i.e., the largest Âµopen set contained in A.
Definition: 1.1[3]
Let (X, Âµ) be a GTS and A X. Then the subset Âµ(A) is defined by
( )
( )
{: , }, , = { , .
Definition: 1.2[3]
In a GTS (X, Âµ) a subset A is called a Âµset if A = Âµ(A).
Definition: 1.3[3]
A subset A of a GTS (X, Âµ) is called (, Âµ)closed set if A = L D, where L is a Âµsets and D is a Âµclosed set.
The intersection of all (, Âµ)closed sets containing a subset A of X is called the (, Âµ)closure of A and is denoted by c(, Âµ)(A). The complement of (, Âµ) closed set is called (, Âµ)open set. We denote the collection of (, Âµ)open sets (resp.(, Âµ)closed sets) by ÂµO(X, Âµ) (resp. ÂµC(X, Âµ)).
Lemma:1.1 [4]
For subsets Ai (i I) of a GTS (X, Âµ), the following properties hold:

If Ai is (, Âµ)closed for each i I, then iIAi is (, Âµ)closed.

If Ai is (, Âµ)open for each i I, then iIAi is (, Âµ)open.


ÂµGENERALIZED CLOSED SETS Definition: 2.1
A subset A of a generalized topological space (X, Âµ) is called Âµgeneralized closed, briefly Âµgclosed, if cÂµ(A) U whenever A U and U is (, Âµ)open.
Definition: 2.2
A subset A of a generalized topological space (X, Âµ) is called Âµgclosed, if c(, Âµ)(A) U whenever A U and U is (, Âµ)open.
Definition: 2.3
A subset A of a generalized topological space (X, Âµ) is called gÂµclosed, if c(, Âµ)(A) U whenever A U and U is Âµopen.
Remark: 2.4
From above definitions, we have the following
diagram

Âµgclosed sets and (, Âµ)closed sets are independent concepts.

Âµgclosed sets and Âµgclosed sets are independent concepts.

(, Âµ)closed sets and Âµgclosed sets are also independent concepts
Proof.
Necessity: Suppose that A is Âµgclosed. Let S be
Âµclosed
closed
a (, Âµ)closed subset of cÂµ(A) \ A. Then A Sc. Since A
Âµg
Âµgclosed
is Âµgclosed, we have cÂµ(A) Sc consequently S (cÂµ(A))c.. Hence S cÂµ(A) (cÂµ(A))c = . Therefore S is empty.
(, Âµ)closed Example: 2.5;
Âµgclosed
gÂµclosed
Sufficiency: Suppose that cÂµ(A) \ A contains no nonempty (, Âµ)closed sets. Let A G and G be (, Âµ)open. If cÂµ(A) G, then cÂµ(A) GC is a nonempty (, Âµ)closed subset of cÂµ(A) \ A. Therefore, A is Âµgclosed.
Let X = {a, b, c,} and Âµ = {,{a},{a, b},{b, c},X}.
Thus ÂµO(X, Âµ) = {,{a},{c},{a, b},{b, c},{a, c},X}. Take A = {a, c}. Then we obtained that A is a Âµgclosed set but it is not Âµgclosed.
Let X = {a, b, c,} and Âµ = {,{a},{a, b},{b, c},X}. Then A = {b} is a (, Âµ)closed set but it is not Âµgclosed. Let X = {a, b, c,} and Âµ = {, {a}, X}. Then
A = {a, b} is Âµgclosed set but it is not (, Âµ)closed Remark 2.6
The union of two Âµgclosed sets need not be Âµgclosed as can be verified from the following example Example: 2.7
Let X = {a, b, c} and Âµ = {,{a},{a, b},{b, c},X}. Now put A = {a} and B = {c} are two Âµgclosed sets. But A B = {a, c} is not a Âµgclosed set.
Remark: 2.8
The intersection of two Âµgclosed sets need not be Âµgclosed as can be verified from the following lemma
Example: 2.9
Let X = {a, b, c} and Âµ = {, {a}}. Now put A = {a} and B = {b} are two Âµgclosed. But A = is not a Âµgclosed set.
Theorem: 2.10
A subset A of GTS (X, Âµ) is Âµgclosed, then cÂµ(A) \ A contains no non empty Âµclosed subset of (X, Âµ). Proof.
Let F be a Âµclosed subset contained in cÂµ(A) \ A. Clearly A Fc where A is Âµgclosed and Fc is an Âµopen set of X. Thus cÂµ(A) Fc or F (cÂµ(A))c. Then F (cÂµ(A))c (cÂµ(A) \ A) (cÂµ(A))c cÂµ(A) = . This shows that F = .
The converse of the above theorem is not true in general as it is shown in the following example.
Example: 2.11
Let X = {a, b, c,} and Âµ = {,{a},{a, b},{b, c},X}. If A = {a, c}, then cÂµ(A) \ A = {b} does not contain a nonempty Âµclosed set. But A is not Âµgclosed in (X, Âµ). Corollary: 2.12
In a T1 space, every Âµgclosed set is Âµclosed.
Proof.
Let A be a Âµgclosed set in a T1 space (X, Âµ). Let x cÂµ(A) \ A. Since (X, Âµ) is T1, {x} is a Âµ closed set in (X, Âµ). By theorem 2.10, there exists no non empty Âµ closed set in cÂµ(A) \ A = . Therefore cÂµ(A) = A. Hence A is Âµclosed.
Theorem: 2.13
A set A is Âµgclosed if and only if cÂµ(A) \ A contains no non empty (, Âµ)closed sets.
Theorem: 2.14
If A is a Âµgclosed set of (X, Âµ) and A B cÂµ(A), then B is a Âµgclosed set in (X, Âµ).
Proof.
Let A B and cÂµ(A) cÂµ(B). Hence cÂµ(B) \ B cÂµ(A) \ A. But by the theorem 2.13, cÂµ(A) \ A contains no non empty (, Âµ)closed subsets of X and hence neither cÂµ(B) \ B. Again by theorem 2.13, B is a Âµgclosed set.
Theorem: 2.15
If A is (, Âµ)open and Âµgclosed set in (X, Âµ), then A is Âµclosed in (X, Âµ).
Proof.
Since A is (, Âµ)open and Âµgclosed, cÂµ(A)
A and hence A is Âµclosed. Theorem: 2.16
For each {x} X, either {x} is (, Âµ)closed or
{x}c is Âµgclosed in (X, Âµ). Proof.
Suppose {x} is not (, Âµ)closed in(X, Âµ) then{x}c is not (, Âµ)open and the only (, Âµ)open set containing{x}c is the space X itself. Therefore cÂµ({x}c) ) X and so {x}c is Âµgclosed in (X, Âµ).
Definition: 2.17
A subset S of X is said to be locallyÂµ closed if S = U F, where U is Âµopen F is Âµclosed in (X, Âµ).
Theorem: 2.18.
Let A be locallyÂµ closed subset of(X, Âµ). For the set A the following properties are equivalent:
(i). A is Âµclosed; (ii). A is Âµgclosed; (iii). A is Âµgclosed. Proof.
By Remark 2.4, it suffices to prove that (iii) implies (i). By A (cÂµ(A)) c is Âµopen in (X, Âµ) since A is locallyÂµclosed. Now A (cÂµ(A)) c is an Âµopen set of (X, Âµ) such that A A (cÂµ(A)) c. Since A is Âµgclosed, then cÂµ(A) A (cÂµ(A)) c . But cÂµ(A) (cÂµ(A))c = . Thus we have cÂµ(A) A and hence A is Âµclosed.
Definition: 2.19
A subset A in (X, Âµ) is said to be Âµgopen in (X, Âµ) if and only if Ac is Âµgclosed.
Theorem: 2.20
A set A is said to be Âµgopen in (X, Âµ) if and only if F iÂµ(A) whenever F is (, Âµ)closed in (X, Âµ) and F A.
Proof.
Suppose that F iÂµ(A) whenever F is , Âµ) closed and F A. Let Ac G, where G is(, Âµ)open.
Proof.
Necessity: Suppose that A is Âµgclosed in (X,
Hence Gc A. By assumption Gc iÂµ(A) which implies that (iÂµ(A))c G, so cÂµ(Ac) G. Hence Ac is Âµgclosed. i.e., A is Âµgopen
Conversely, let A be Âµgopen. Then Ac is Âµgclosed. Also let F be a (, Âµ)closed set contained in

Then Fc is (, Âµ)open. Therefore whenever Ac Fc,
cÂµ(Ac) Fc. This implies that F (cÂµ(Ac))c = iÂµ(A). Thus F iÂµ(A)..
Theorem: 2.21
A set A is said to be Âµgopen in (X, Âµ) if and only if G = X whenever G is (, Âµ)open and iÂµ(A) Ac
G.
Proof.
Let A be Âµgopen, G be (, Âµ)open and iÂµ(A) Ac G. This gives Gc (iÂµ(A))c (Ac)c = (iÂµ(A))c \ Ac = cÂµ(Ac) \ Ac. Since Ac is Âµgclosed and Gc is (, Âµ)closed, by theorem 2.13, it follows that Gc = . Therefore X = G. Conversely, suppose that F is (, Âµ)closed and F A. Then iÂµ(A) Ac iÂµ(A) Fc. It follows that iÂµ(A) Fc = X and hence F iÂµ(A). Therefore A is Âµgopen.
Theorem: 2.22
If iÂµ(A) B A and A is Âµgopen in (X, Âµ), then B is Âµgopen in (X, Âµ).
Proof.
Suppose iÂµ(A) B A and A is Âµgopen in (X,
Âµ). Then Ac Bc cÂµ(Ac) and Ac is Âµg closed. Theorem: 2.23
A set A is Âµgclosed in (X, Âµ) if and only if cÂµ(A) \ A is Âµgopen in (X, Âµ).
Âµ). Let F cÂµ(A) \ A, where F is (, Âµ)closed. By theorem 2.10, F = . Therefore F iÂµ(cÂµ(A) \ A) and by theorem 2.20, cÂµ(A) \ A is Âµgopen in (X, Âµ).
Sufficiency: Let A G, where G is (, Âµ)open set. Then cÂµ(A) Gc cÂµ(A) Ac = cÂµ(A) \ A. Since cÂµ(A) Gc is (, Âµ)closed and cÂµ(A) \ A is Âµgopen in by theorem 2.20, we have cÂµ(A) Gc iÂµ(cÂµ(A) \ A) = . Hence A is Âµgclosed in (X, Âµ).
Theorem: 2.24
For the subset A X the following properties are equivalent:

A is Âµgclosed;

cÂµ(A) \ A contains no nonempty is (, Âµ)closed set;

(iii).cÂµ(A) \ A is Âµgopen Proof.
This follows from the theorem 2.10 and 2.23
REFERENES

A.CsÃ¡szÃ¡r, Generalized open sets, Acta Math. Hungar., 75(1 2)(1997), 6587.

A.CsÃ¡szÃ¡r, Generalized open sets in generalized topologies, Acta Math. Hungar., 106 (2005), 5366.

B. Roy and Erdal Ekici, On (, )closed sets in generalized topological spaces, Method of Functional Analysis and Topology, 17(2) (2011), 174179.

E. Ekici and B. Roy, New generalized topologies on generalized topological spaces due to CsÃ¡szÃ¡r, Acta Math. Hungar., 132(12) (2011), 117124.

B. Roy, On a type of generalized open sets, Applied General topology,12 (2) (2011), 163173.

B. Roy, On generalized of R0 and R1 spaces, Acta Math. Hungar., 127(2010), 291300.

M. Caldas, S. Jafari and T. Noiri, On – generalized closed sets in topological spaces, Acta. Math. Hungar., 118(4) (2008), 337 343.