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Pauline Mary Helen
Associate Professor, Nirmala College, Coimbatore, Tamil Nadu, INDIA.
Corresponding Address:
[email protected]
Abstract: In this paper we introduce and study three different notions via ideals namely glocal function, the set operator Ψ_{g} and gcompatibility of τ with I. We characterize these new sorts. Several properties of them have been studied and their relationships with other types of similar operators are also investigated.
Keywords:*gadditive space, *g finitely additive space,glocal function,( ) ^{*g} operator, Ψ_{g }operator,gcompatibility of τ with I
1. Introduction
Ideals topological spaces have been first introduced by K. Kuratowski[2] in 1930.Vaidyanathaswamy [4] introduced local function in 1945 and defined a topology τ. In this paper we introduce and study three different notions via ideals namely glocal function, the set operator Ψ_{g }, and gcompatibility of τ with I and investigate their relationships with other types of similar operators .
2. Preliminaries
Let be a topological space and . We denote closure of A and interior of A by
and respectively.
Definition 2.1: A set A in a topological space (X, t) is said to be a generalized closed set (briefly gclosed)[3] if cl(A) ÍU whenever A ÍU and U is open in (X, t).
The class of g  open set in X will be denoted by GO(X,t).
Definition.2.3[2] An ideal on a non empty set is a collection of subsets of which satisfies the following properties:(i) , (ii) , . A topological space with an ideal on is called an ideal topological space and is denoted by . Let be a subset of . is an ideal on and by we denote the ideal topological subspace. Let be the power set of , then a set operator ( )*: called the local function [7] of A with respect to and is defined as follows:For , for every open set containing We simply write instead of in case there is no confusion. A Kuratowski closure operator for a topology , called the  topology is defined by . A set operator is defined as follows: For any , such that there exists open set such that . is said to be compatible with , denoted by if the following holds: for , if for every there exists open set such that then .
3. gLocal function
In this section we introduce new class of the set operator using gneighbourhood and discuss various properties.
Definition 3.1: Given an ideal space , a set operator , called the
glocal function of with respect to is defined as follows.
For , for every when there is no ambiguity,
we will simply write or instead of .
is defined as
Theorem 3.2: Let be an ideal topological space and .
Then the following statements hold.
1. , and if
2. For another ideal
3.
4.
5.
6.
7. if (X,t) is gmultiplicative.
8. If then
9. If then
Proof:
(1) and (2) are obvious by definition of glocal function.
(3) implies for every implies for every
implies .
(4) for every
for every
.
(5) and (6) follow from (1)
(7) If then there exists such that and this implies . Therefore is the union of gopen sets and hence it is gopen. So is gclosed. Therefore (since (X,t) is gmultiplicative, gcl(A) is gclosed). But . Hence is a gclosed sub set of .
(8) implies (A)
Let . Suppose , then there exists such that .
Then . This implies that . So, which is a
contradiction to the fact that . So, (B)
From (A) and (B) we get when .
(9) Let , and be a gopen set containing . Then and
hence which proves .Therefore .
So (A)
On the otherhand, implies
Therefore (B)
From (A) and(B) it follows that
Remark 3.3: In general and as seen from examples (3.4) and (3.5).
Example 3.4: Consider Z with cofinite topology and I = {φ}. A = {0,1,2,3,……..} and B = {0,1, 2, 3,…………….}. Then A^{*g }= X = B^{*g} and . But and . Therefore
Example 3.5: Let and Let and . The since and
Remark 3.6: In the ideal space because in general, we
are not able to define a topology using the operator . To define a topology we need the following definitions.
Definition 3.7: An ideal space said to be
1. finitely additive if for every finite positive integer n.
2. additive if for every indexing set .
3. finitely multiplicative if for every finite positive integer n.
4. multiplicative if for every indexing set .
Remark 3.8:
1. Every additive (resp multiplicative) space is finitely additive (resp finitely multiplicative).
2.
3. If is finitely additive then
(a) and
(b) .
Therefore in a finitely additive space, satisfies Kuratowski closure axioms.
Definition 3.9: Let be a finitely additive space. If a closed set A is defined to
be one for which , then the class of all complements of such sets is a topology on X
denoted by , whose closure operation is given as .
Example 3.10: Let be an infinite cofinite topological space and In this space is finite} and is finite}and if is finite and if is infinite. Then is *g – multiplicative but not *g – countably additive and g – additive.
Example 3.11: Let be an indiscrete topological space and and GO(X) = {all subsets}. if and if Since for all {all subsets). Then is *g – multiplicative, *g – finitely multiplicative, *g – countably multiplicative, *g – additive, *g – countably additive, and *g – finitely additive.
Example 3.12: Let be an indiscrete topological space, and GO(X) = {all subsets}. if and if Since and hence {all subsets). Then is *g – multiplicative, *g – finitely multiplicative, *g – countably multiplicative, *g – additive, *g – countably additive, and *g – finitely additive.
Remark 3.13: In a finitely additive space, ,
(1)
(2) and hence
Thus a new topology is defined in a finitely additive ideal space with the help of glocal function and this topology is finer than  topology.
Theorem 3.14: Let be an ideal space. For , we have the following results.
1.If then and .
2.If then and .
Proof: Obvious from the definition of
Remark 3.16: In a finitely additive space with , is the discrete
topology since every subset is open and closed.
Theorem 3.17: If I and J are two ideals in a finitely additive space such that .
Then .
Proof: Let A be closed in topology
.
Therefore
Then which proves and so A is closed in .
Definition 3.18: A subset A in an ideal space is said to be
1. dense subset in X if
2. perfect if .
3. closed in X if
Theorem 3.19: In a finitely additive ideal space the following are equivalent
1.
2. is  closed
3.
4.
Proof: Obvious.
4. The set operator
Definition 4.1: Let be an ideal space. A set operator is
defined as follows.
For any , there exists such that .
Remark 4.2:
1.Obviously if and only if . Therefore
2.We denote simply by when no ambiguity is present.
3.
4.
Theorem 4.3:For a subset A in an ideal space the following results are true.
1.If then .
2.If then .
Proof:
1. 2.
The following theorem gives many basic and useful facts for the operator .
Theorem 4.4: Let A and B subsets in an ideal space .
1. If then .
2. .
Proof:
2. Follows from 1.
Theorem 4.5: Let be a finitely additive space. Then
1. If then .
2. For every , then .
3. For every , then .
4. For every and then .
5. If then .
6. If then .
7. If then .
Proof:
1. . Then .
2.By theorem (3.2), is g closed. Therefore
Therefore is closed and hence .
3. By (2) and hence by (1).
4. .(by thm (4.3))
.
5. If then is gclosed. Therefore and this
implies A is open. So by (1) .
6. Follows from (5) since .
7.Let and . Then implies E and H are in I.
By (4) implies . (since ) .
Definition 4.6:In an ideal space , we say two subsets A and B are congruent modulo I
(in notation ) if . Obviously “ ” is an equivalence relation.
Theorem 4.7 Let A and B are two subsets in finitely additive ideal space .If then .
Proof:
It follows from definition of and by (10) of theorem (4.5).
5. gcompatability of with
Definition 5.1 Given a space is said to be gcompatible with , denoted by if the following holds: for , if for every there exists such that then .
Remark 5.2
Since
The following example shows the existence of this compatibility.
Example 5.3 Let be an indiscrete space, and . In this space .
Theorem 5.4 If is a *gfinitely additive ideal space then the following are equivalent.
(1)
(2) If has a cover of gopen set each of whose intersections with is in then is in .
(3) For every
(4) For every
(5) For every closed subset
(6) For every , if contains no nonempty subset then
Proof: Let and where each is gopen and . Then by definition
for .Let .So if then
Therefore there exists such hat . Then is an open cover for and hence .
Let . Suppose then for every .This implies which implies which is a contradiction.
and hence by (3).
proof is obvious.
Let and for every there exists such that .Then .
Since is *gfinitely additive ,
is *gclosed. By (5),
But
.Let By (4) . Let .
Suppose then there exists such that . This implies that which is a contradiction. Therefore .
By (6) and this implies .(Since
Let .since ,we have by (6).
Theorem 5.5 Let be an ideal space. Then if and only if for all
Proof: Necessity: Assume that .Let , .Then and there exists such that .Therefore for each there exists such that This implies .
Sufficiency: Let and for each there exists .By definition of
Theorem 5.6 Let be *bfinitely additive ideal space with . Then .
Proof: From theorem (4.1) .By theorem (5.5) for some . Therefore .
So, , by theorem (4.5).
Theorem 5.7 let be a*g finitely additive ideal space with . If and then .
Proof: By theorem (4.5)
Therefore
References

M.E.Abd EI Mosef, E.F.Lashien and A.A.Nasef, some topological operators via ideals, kyungpook Mathematical Journal, vol.32, 1992.

K.Kuratowski,Topologie,I.Warszawa, 1933.

N.Levine, Generalized closed sets in topology, Rend. Circ. Math. Palermo,19(2), 8996, 1970.

R.Vaidyanathasamy, The localization theory in settopology, Proc. Indian Acad. Sci., 20, 5161, 1945.
