# Some New Sets and Topologies in Ideal Topological Spaces

Authors: R. Manoharan; P. Thangavelu
Journal: Chinese Journal of Mathematics
 (2013)

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## Abstract

An ideal topological space is a triplet (X, τ, ℑ), where X is a nonempty set, τ is a topology on X, and ℑ is an ideal of subsets of X. In this paper, we introduce L∗-perfect, R∗-perfect, and C∗-perfect sets in ideal spaces and study their properties. We obtained a characterization for compatible ideals via R∗-perfect sets. Also, we obtain a generalized topology via ideals which is finer than τ using R∗-perfect sets on a finite set.


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## Body

## 1. Introduction and Preliminaries

The contributions of Hamlett and Jankovic [1–4] in ideal topological spaces initiated the generalization of some important properties in general topology via topological ideals. The properties like decomposition of continuity, separation axioms, connectedness, compactness, and resolvability [5–9] have been generalized using the concept of ideals in topological spaces.By a space(X,τ), we mean a topological space X with a topology τ defined on X on which no separation axioms are assumed unless otherwise explicitly stated. For a given point x in a space (X,τ), the system of open neighborhoods of x is denoted by N(x)={U∈τ:x∈U}. For a given subset A of a space (X,τ),cl(A) and int(A) are used to denote the closure of A and interior of A, respectively, with respect to the topology.A nonempty collection of subsets of a setX is said to be an ideal on X, if it satisfies the following two conditions: (i) If A∈ℑ and B⊆A, then B∈ℑ; (ii) If A∈ℑ and B∈ℑ, then A∪B∈ℑ. An ideal topological space (or ideal space) (X,τ,ℑ) means a topological space (X,τ) with an ideal ℑ defined on X. Let (X,τ) be a topological space with an ideal ℑ defined on X. Then for any subset A of X,A*(ℑ,τ)={x∈X/A∩U∉ℑ for every U∈N(x)} is called the local function of A with respect to ℑ and τ. If there is no ambiguity, we will write A*(ℑ) or simply A* for A*(ℑ,τ). Also, cl*(A)=A∪A* defines a Kuratowski closure operator for the topology τ*(ℑ) (or simply τ*) which is finer than τ. An ideal ℑ on a space (X,τ) is said to be codense ideal if and only if τ∩ℑ={∅}. X* is always a proper subset of X. Also, X=X* if and only if the ideal is codense.Lemma 1 ([see 12]).
Let(X,τ) be a space with ℑ1 and ℑ2 being ideals on X, and let A and B be two subsets on X. Then(i)
A⊆B⇒A*⊆B*;(ii)
ℑ1⊆ℑ2⇒A*(ℑ2)⊆A*(ℑ1);(iii)
A*=cl(A*)⊆cl(A) (A* is a closed subset of cl(A));(iv)
(A*)*⊆A*;(v)
(A∪B)*=A*∪B*;(vi)
A*-B*=(A-B)*-B*⊆(A-B)*;(vii)
for everyI∈ℑ,(A∪I)*=A*=(A-I)*.Definition 2 (see [3]).
Let(X,τ) be a space with an ideal ℑ on X. One says that the topology τ is compatible with the ideal ℑ, denoted by τ~ℑ, if the following holds, for every A⊆X: if for every x∈A, there exists a U∈N(x) such that U∩A∈ℑ, then A∈ℑ.Definition 3.
A subsetA of an ideal space (X,τ,ℑ) is said to be(i)
τ*-closed [3] if A*⊆A,(ii)
*-dense-in-itself [10] if A⊆A*,(iii)
I-open [11] if A⊆int(A*),(iv)
almostI-open [12] if A⊆cl(int(A*)),(v)
I-dense [7] if A*=X,(vi)
almost strongβ-I-open [13] if A⊆cl*(int(A*)),(vii)
*-perfect [10] if A=A*,(viii)
regularI-closed [14] if A=(int(A))*,(ix)
anfI-set [15] if A⊆(int(A))*.Theorem 4 ([3]).
Let(X,τ) be a space with an ideal ℑ on X. Then the following are equivalent.(i)
τ~ℑ.(ii)
IfA has a cover of open sets each of whose intersection with A is ℑ, then A is in ℑ.(iii)
For everyA⊆X, A∩A*=∅⇒A∈ℑ.(iv)
For everyA⊆X, A-A*∈ℑ.(v)
For everyτ*-closed subset A,A-A*∈ℑ.(vi)
For everyA⊆X, if A contains no nonempty subset B with B⊆B*, then A∈ℑ.


## 2.L*-Perfect, R*-Perfect, and C*-Perfect Sets

In this section, we define three collections of subsets𝔏, ℜ and ℭ in an ideal space and study some of their properties.Definition 5.
Let(X,τ,ℑ) be an ideal topological space. A subset A of X is said to be(i)
L*-perfect if A-A*∈ℑ,(ii)
R*-perfect if A*-A∈ℑ,(iii)
C*-perfect if A is both L*-perfect and R*-perfect. The collection of L*-perfect sets, R*-perfect sets, and C*-perfect sets in (X,τ,ℑ) is denoted by 𝔏, ℜ, and ℭ, respectively.Remark 6.
(i) Ifℑ={∅}, then
(1)𝔏={A⊆X:A⊆A*=cl(A)}=thecollectionofall*-dense-in-itselfsets=℘(X),ℜ={A⊆X:A*⊆A}=thecollectionofallτ*-closedsets={A⊆X:A=A*=cl(A)}=thecollectionofallclosedsetsin(X,τ)=thecollectionofall*-perfectsets,ℭ={A⊆X:A*⊆A,A⊆A*}=AX:A=A*=cl(A)=thecollectionofallclosedsetsin(X,τ)=thecollectionofall*-perfectsets.
(ii) Ifℑ=𝔓(X), then 𝔏=𝔗=℘(X).
(iii) Ifτ~ℑ, then 𝔏=℘(X) (by Theorem 4(iv)).Remark 7.
Every *-perfect set is bothR*-perfect and L*-perfect (and hence C*-perfect). In Proposition 15, we proved that every members of an ideal is both R*-perfect and L*-perfect (and hence C*-perfect). But any nonempty member of an ideal is not a *-perfect set. Hence R*-perfect and L*-perfect sets (and hence C*-perfect sets) need not be *-perfect.Proposition 8.
If a subsetA of an ideal space (X,τ,ℑ) is C*-perfect, then AΔA*∈ℑ.Proof.
SinceA is both L*-perfect and R*-perfect, A-A*∈ℑ and A*-A∈ℑ. By the finite additive property of ideals, (A-A*)∪(A*-A)∈ℑ. Hence AΔA*∈ℑ.Proposition 9.
In an ideal space(X,τ,ℑ), every τ*-closed set is R*-perfect.Proof.
LetA be a τ*-closed set. Therefore, A*⊆A. Hence A*-A=ϕ=ℑ. Therefore, A is an R*-perfect set.Corollary 10.
In an ideal space(X,τ,ℑ),(i)
X and ϕ are R*-perfect sets,(ii)
everyτ-closed set is R*-perfect,(iii)
for any subsetA of an ideal topological space (X,τ,ℑ),cl(A),A*,cl*(A) are R*-perfect sets,(iv)
every regular-I-closed set is R*-perfect.Proof.
The proof follows from Proposition9.The following example shows that the converses of Proposition9 and Corollary 10(iv) are not true.Example 11.
Let(X,τ,ℑ) be an ideal space with X={a,b,c}, τ={ϕ,X,{a}}, and ℑ={ϕ,{b}}. The set {a,c} is R*-perfect set which is not a τ*-closed set and hence not a regular-I-closed set.Proposition 12.
If a subsetA of an ideal topological space (X,τ,ℑ) is such that A∈ℑ, then A is C*-perfect.Proof.
SinceA∈ℑ,A*=ϕ. Then A-A*=A∈ℑ and A*-A=ϕ∈ℑ. Hence A is both an L*-perfect and R*-perfect set.Corollary 13.
LetA be a subset of an ideal space (X,τ,ℑ). Consider the following.(i)
IfA∈ℑ, then every subset of A is a C*-perfect set.(ii)
IfA is R*-perfect, then A*-A is C*-perfect.(iii)
IfA is an L*-perfect set, then A-A* is a C*-perfect set.(iv)
IfA is C*-perfect, then AΔA* is a C*-perfect set.Proof.
The proof follows from Proposition12.Corollary 14.
Let(X,τ) be a space with an ideal ℑ on X such that τ~ℑ. Then for every A⊆X,(i)
ifA∩A*=ϕ, then A is C*-perfect;(ii)
A-A* is C*-perfect;(iii)
ifA contains nonempty subset B with B⊆B*, then A is C*-perfect.Proof.
Follows from Theorem4 and Proposition 12.Proposition 15.
In an ideal space(X,τ,ℑ), every *-dense-in-itself set is an L*-perfect set.Proof.
LetA be a *-dense-in-itself set of X. Then A⊆A*. Therefore, A-A*=ϕ∈ℑ. Hence A is an L*-perfect set.Corollary 16.
In an ideal space(X,τ,ℑ),(i)
everyI-dense set is L*-perfect,(ii)
everyI-open set is L*-perfect,(iii)
every almost strongβ-I-open set is L*-perfect,(iv)
every almostI-open set is L*-perfect,(v)
every regular-I-closed set is L*-perfect,(vi)
everyfI-set is L*-perfect.Proof.
Since all the above sets are *-dense-in-itself, by Proposition15, these sets are L*-perfect.Remark 17.
The members of the ideal of an ideal space areL*-perfect, but the nonempty members of the ideal are not *-dense-in-itself. Therefore, the converses of the above Corollary and Proposition 15 need not to be true.Proposition 18.
In an ideal space(X,τ,ℑ),(i)
empty set is an  L*-perfect set,(ii)
X  is an  L*-perfect set if the ideal is codense.Proof.
(i) Sinceϕ-ϕ*=ϕ∈ℑ, the empty set is an L*-perfect set. (ii) We know that X=X* if and only if the ideal ℑ is codense. Then X-X*=ϕ∈ℑ. Hence the result follows.


## 3. Main Results

In this section, we prove that finite union and intersection ofR*-perfect sets are again R*-perfect set. Using these results, we obtain a new topology for the finite topological spaces which is finer than τ*-topology.In Ideal spaces, usuallyA⊂B implies A*⊂B*. We observe that there are some sets A and B such that A⊂B but A*=B*.Example 19.
Let(X,τ,ℑ) be an ideal space with X={a,b,c,d}, τ={ϕ,X,{a,c},{d},{a,c,d}}, ℑ={ϕ,{c},{d},{c,d}}. Here the sets A={a} and B={a,b} are such that A⊆B, but A*=B*={a,b,c}.Proposition 20.
Let(X,τ,ℑ) be an ideal space. Let A and B be two subsets of X such that A⊆B and A*=B*; then(i)
B  isR*-perfect ifAisR*-perfect;(ii)
A  isL*-perfect ifBisL*-perfect.Proof.
(i) LetA be an R*-perfect set. Then A*-A∈ℑ. Now, B*-B=A*-B⊆A*-A. By heredity property of ideals, B*-B∈ℑ. Hence B is R*-perfect.
(ii) LetB be an L*-perfect set. Then B-B*∈ℑ. Now, A-A*=A-B*⊆B-B*. By heredity property of ideals, A-A*∈ℑ. Hence A is L*-perfect.Corollary 21.
Let(X,τ,ℑ) be an ideal space. Let A and B be two subsets of X such that A⊆B⊆cl*A; then(i)
B is R*-perfect if A is R*-perfect,(ii)
A is L*-perfect if B is L*-perfect.Proof.
SinceA⊆B⊆cl*A, A*⊆B*⊆(cl*A)*=A*. Hence A*=B*. Therefore, the result follows from Proposition 20.Proposition 22.
LetA be a subset of an ideal topological space (X,τ,ℑ) such that A is L*-perfect set and A∩A* is R*-perfect; then both A and A∩A* are C*-perfect.Proof.
SinceA is L*-perfect, A-A*∈ℑ. By Lemma 1(vii), for every I∈ℑ, (A∪I)*=A*=(A-I)*. Therefore, (A∪(A-A*))*=A*=(A-(A-A*))*. This implies A*=(A∩A*)*. Therefore, we have A∩A*⊆A with (A∩A*)*=A*. By Proposition 20, A is R*-perfect if A∩A* is R*-perfect and A∩A* is L*-perfect if A is L*-perfect set. Hence A is R*-perfect and A∩A* is L*-perfect.Proposition 23.
If a subsetA of an ideal topological space (X,τ,ℑ) is R*-perfect set and A* is L*-perfect, then A∩A* is L*-perfect.Proof.
SinceA is R*-perfect, A*-A∈ℑ. By Lemma 1(vii), for every I∈ℑ, (A∪I)*=A*=(A-I)*. Therefore, (A*∪(A*-A))*=A**=(A*-(A*-A))*. This implies A**=(A∩A*)*. Therefore, we have A∩A*⊆A* with (A∩A*)*=A**. By Proposition 20, A∩A* is L*-perfect if A* is L*-perfect set. Hence A∩A* is L*-perfect.Proposition 24.
IfA and B are R*-perfect sets, then A∪B is an R*-perfect set.Proof.
LetA and B be R*-perfect sets. Then A*-A∈ℑ and B*-B∈ℑ. By finite additive property of ideals, (A*-A)∪(B*-B)∈ℑ. Since (A*∪B*)-(A∪B)⊆(A*-A)∪(B*-B), by heredity property (A*∪B*)-(A∪B)∈ℑ. Hence (A∪B)*-(A∪B)∈ℑ. This proves the result.Corollary 25.
Finite union ofR*-perfect sets is an R*-perfect set.Proof.
The proof follows from Proposition24.Proposition 26.
IfA and B are L*-perfect sets, then A∪B is an L*-perfect set.Proof.
SinceA and B are L*-perfect sets, A-A*∈ℑ and B-B*∈ℑ. Hence by finite additive property of ideals, (A-A*)∪(B-B*)∈ℑ. Since (A∪B)-(A∪B)*=(A∪B)-(A*∪B*)⊆(A-A*)∪(B-B*), by heredity property (A∪B)-(A∪B)*∈ℑ. This proves that A∪B is an L*-perfect set.Corollary 27.
Finite union ofL*-perfect sets is an L*-perfect sets.Proof.
The proof follows from Proposition26.Proposition 28.
IfA and B are R*-perfect sets, then A∩B is an R*-perfect set.Proof.
Suppose thatA and B are R*-perfect sets. Then A*-A∈ℑ and B*-B∈ℑ. By finite additive property of ideals, (A*-A)∪(B*-B)∈ℑ. Since (A*∩B*)-(A∩B)⊆(A*-A)∪(B*-B), by heredity property (A*∩B*)-(A∩B)∈ℑ. Also (A∩B)*-(A∩B)⊆(A*∩B*)-(A∩B)∈ℑ. This proves the result.Corollary 29.
Finite intersection ofR*-perfect sets is an R*-perfect set.Proof.
The proof follows from Proposition28.Proposition 30.
Finite union ofC*-perfect sets is a C*-perfect set.Proof.
From Corollaries27 and 29, finite union of C*-perfect sets is a C*-perfect set.Proposition 31.
If(X,τ,ℑ) is an ideal topological space with X being finite, then the collection ℜ is a topology which is finer than the topology of τ*-closed sets.Proof.
By Corollary10, X and ϕ are R*-perfect sets. By Corollary 25, finite union of R*-perfect sets is an R*-perfect set, and by Corollary 29, finite intersection of R*-perfect sets is R*-perfect. Hence the collection ℜ is a topology if X is finite. Also, by Proposition 9 every τ*-closed set is an R*-perfect set. Hence the topology ℜ is finer than the topology of τ*-closed sets if X is finite.Proposition 32.
In an ideal space(X,τ,ℑ), {τ*-closedsets}∪ℑ⊆ℜ.Proof.
The proof follows from Propositions9 and 12.The following example shows that{τ*-closedsets}∪ℑ≠ℜ.Example 33.
Let(X,τ,ℑ) be an ideal space with X={a,b,c}, τ={∅,X,{a}}, and ℑ={∅,{b}}. The collection of τ*-closed sets is {∅,X,{b},{b,c}} and ℜ={∅,X,{b},{b,c},{a,c}}.Now,{τ*-closed sets}∪ℑ={∅,X,{b},{b,c}}≠ℜ.Proposition 34.
Let(X,τ,ℑ) be an ideal space and A⊆X. The set A is R*-perfect if and only if F⊆A*-A in X implies that F∈ℑ.Proof.
Assume thatA is an R*-perfect set. Then A*-A∈ℑ. By heredity property of ideals, every set F⊆A*-A in X is also in ℑ. Conversely assume that F⊆A*-A in X implies that F∈ℑ. Since A*-A is a subset of itself, by assumption A*-A∈ℑ. Hence A is R*-perfect.Proposition 35.
Let(X,τ,ℑ) be an ideal space and A⊆X. The set A is L*-perfect if and only if F⊆A-A* in X implies that F∈ℑ.Proof.
Assume thatA is an L*-perfect set. Then A-A*∈ℑ. By heredity property of ideals, every set F⊆A-A* in X is also in ℑ. Conversely, assume that F⊆A-A* in X implies that F∈ℑ. Since A-A* is a subset of itself, by assumption A-A*∈ℑ. Hence A is L*-perfect.Proposition 36.
Let(X,τ) be a topological space and A⊆X. Let ℑ1 and ℑ2 be two ideals on X with ℑ1⊆ℑ2. Then A is R*-perfect with respect to ℑ2 if it is R*-perfect with respect to ℑ1.Proof.
Sinceℑ1⊆ℑ2,A*(ℑ2)⊆A*(ℑ1) by Lemma 1(ii) Let A be R*-perfect with respect to ℑ1. Then A*(ℑ1)-A∈ℑ1. Also, A*(ℑ2)-A⊆A*(ℑ1)-A. Hence by heredity property of ideals, A*(ℑ2)-A∈ℑ1⊆ℑ2. Therefore A is R*-perfect with respect to ℑ2.Theorem 37.
Let(X,τ) be a space with an ideal ℑ on X. Then the following are equivalent.(i)
τ~ℑ.(ii)
IfA has a cover of open sets each of whose intersection with A is ℑ, then A is in ℑ.(iii)
IfA⊆X, then A∩A*=ϕ⇒A∈ℑ.(iv)
IfA⊆X, then A-A*∈ℑ.(v)
IfA⊆X and A is R*-perfect set, then AΔA*∈ℑ.(vi)
For everyτ*-closed subset A,A-A*∈ℑ.(vii)
For everyA⊆X, if A contains no nonempty subset B with B⊆B*, then A∈ℑ.Proof.
To prove this theorem, it is enough to prove (iv)⇒ (v) ⇒ (vi). Others follow from Theorem 4. (iv) ⇒ (v) follows from Proposition 8. Suppose that AΔA*∈ℑ. Since A-A*⊆AΔA*, by heredity property A-A*∈ℑ. Hence (v) ⇒ (vi).


## 4.R*-Topology

By Corollary10 and Proposition 28, we observe that the collection ℜ satisfies the conditions of being a basis for some topology and it will be called as Rc*(τ,ℑ). We define R*(τ,ℑ)={A⊆X/X-A∈Rc*(τ,ℑ)} on a nonempty set X. Clearly, R*(τ,ℑ) is a topology if the set X is finite. The members of the collection R*(τ,ℑ) will be called R*-open sets. If there is no confusion about the topology τ and the ideal ℑ, then we call R*(τ,ℑ) as R*-topology when X is finite.Definition 38.
A subsetA of an ideal topological space (X,τ,ℑ) is said to be R*-closed if it is a complement of an R*-open set.Definition 39.
LetA be a subset of an ideal topological space (X,τ,ℑ). One defines R*-interior of the set A as the largest R*-open set contained in A.One will denote R*-interior of a set A by R*-int(A).Definition 40.
LetA be a subset of an ideal topological space (X,τ,ℑ). A point x∈A is said to be an R*-interior point of the set A if there exists an R*-open set U of x such that x∈U⊆A.Definition 41.
Let(X,τ,ℑ) be an ideal space and x∈X. One defines R*-neighborhood of x as an R*-open set containing x. One denotes the set of all R*-neighborhoods of x by R*-N(x).Proposition 42.
In an ideal space(X,τ,ℑ), every τ*-open set is an R*-open set.Proof.
LetA be a τ*-open set. Therefore, X-A is a τ*-closed set. That implies that X-A is an R*-closed set. Hence A is an R*-open set.Corollary 43.
The topologyR*(τ,ℑ) on a finite set X is finer than the topology τ*(τ,ℑ).Proof.
The proof follows from Proposition42.Corollary 44.
For any subsetA of an ideal topological space (X,τ,ℑ), int(A) is an R*-open set.Proof.
The proof follows from Proposition42.Remark 45.
(i) Since every open set is anR*-open set, every neighborhood U of a point x∈X is an R*-neighborhood of x.
(ii) Ifx∈X is an interior point of a subset A of X, then x is an R*-interior point of A.
(iii) From (ii), we haveint(A)⊆int*(A)⊆R*-int(A), where int*(A) denotes interior of A with respect to the topology τ*.Theorem 46.
LetA and B be subsets of an ideal space (X,τ,ℑ) with X being finite. Then the following properties hold.(i)
R*-int(A)=∪{U:U⊆A and U is an R*-open set}.(ii)
R*-int(A) is the largest R*-open set of X contained in A.(iii)
A is R*-open if and only if A=R*-int(A).(iv)
R*-int(R*-int(A))=R*-int(A).(v)
IfA⊆B, then R*-int(A)⊆R*-int(B).Proof.
The proof follows from Definitions39, 40, and 41.Definition 47.
LetA be a subset of an ideal topological space (X,τ,ℑ). One defines R*-closure of the set A as the smallest R*-closed set containing A. One will denote R*-closure of a set A by R*-cl(A).Remark 48.
For any subsetA of an ideal topological space (X,τ,ℑ), R*-cl(A)⊆cl*(A)⊆cl(A).Theorem 49.
LetA and B be subsets of an ideal space (X,τ,ℑ) where X is finite. Then the following properties hold:(i)
R*-cl(A)=∩{F:A⊆F and F is R*-closed set}.(ii)
A is R*-closed if and only if A=R*-cl(A)(iii)
R*-cl(R*-cl(A))=R*-cl(A)(iv)
IfA⊆B, then R*-cl(A)⊆R*-cl(B).Proof.
The proof follows from Definition47.Theorem 50.
LetA be a subset of an ideal space (X,τ,ℑ). Then the following properties hold:(i)
R*-int(X-A)=X-R*-cl(A);(ii)
R*-cl(X-A)=X-R*-int(A).Proof.
The proof follows from Definitions38, 39, and 47.



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