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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 g-local function, the set operator Ψg and g-compatibility 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:*g-additive space, *g- finitely additive space,g-local function,( ) *g operator, Ψg operator,g-compatibility 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 g-local function, the set operator Ψg , and g-compatibility 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 g-closed)[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. g-Local function
In this section we introduce new class of the set operator using g-neighbourhood and discuss various properties.
Definition 3.1: Given an ideal space , a set operator , called the
g-local 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 g-multiplicative.
8. If then
9. If then
Proof:
(1) and (2) are obvious by definition of g-local 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 g-open sets and hence it is g-open. So is g-closed. Therefore (since (X,t) is g-multiplicative, gcl(A) is g-closed). But . Hence is a g-closed 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 g-open 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 g-local 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 g-closed. 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. g-compatability of with
Definition 5.1 Given a space is said to be g-compatible 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 *g-finitely additive ideal space then the following are equivalent.
(1)
(2) If has a cover of g-open 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 non-empty subset then
Proof: Let and where each is g-open 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 *g-finitely additive ,
is *g-closed. 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 *b-finitely 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), 89-96, 1970.
-
R.Vaidyanathasamy, The localization theory in set-topology, Proc. Indian Acad. Sci., 20, 51-61, 1945.
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