# θ−ℐg−Closed Sets

Authors: M. Navaneethakrishnan; S. Alwarsamy
Journal: ISRN Geometry
 (2012)

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## Abstract

We defineθ−ℐg−Closed sets and discuss their properties. Using these sets, we characterize 𝒯1/2-spaces and 𝒯ℐ-Spaces.


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## Body

## 1. Introduction and Preliminaries

Anidealℐ on a topological space  (X,τ) is a nonempty collection of subsets of X which satisfies (i) A∈ℐ and B⊂A implies B∈ℐ and (ii) A,B∈ℐ implies A∪B∈ℐ. Given a topological space (X,τ) with an ideal ℐ on X and if ℘(X) is the set of all subsets of X, a set operator (·)⋆:℘(X)→℘(X) called a local function[1] of A  with respect to τ and ℐ is defined as follows: for A⊂X,  A⋆(X,τ)={x∈X∣U∩A∉ℐ,foreveryU∈τ(x)},  where τ(x)={U∈τ∣x∈U}.  A Kuratowski closure operatorcl⋆(·) for a topology τ⋆(ℐ,τ) called the ⋆-topology, finer than τ,  is defined by cl⋆(A)=A∪A⋆(ℐ,τ) [2]. When there is no confusion we will simply write A⋆ for A⋆(ℐ,τ) and τ⋆ for τ⋆(ℐ,τ). If ℐ is an ideal on X, then (X,τ,ℐ) is called an ideal space. A subset A of an ideal space (X,τ,ℐ)  is said to be ⋆-closed [3] if A⋆⊂A. A subset A of an ideal space (X,τ,ℐ) is said to be an ℐg-closed [4] if A⋆⊂U whenever A⊂U and U is open. A subset A of an ideal space (X,τ,ℐ) is said to be ℐg-open if X-A is ℐg-closed. An ideal space (X,τ,ℐ) is said to be a 𝒯ℐ-space [4] if every ℐg-closed set is ⋆-closed. A subset A of an ideal space (X,τ,ℐ) is said to be ℐ-locally⋆-closed [5] if there exist an open set U and a ⋆-closed set F such that A=U∩F.  If ℐ={∅},  then ℐ-locally⋆-closed sets coincide with locally closed sets.By a space, we always mean a topological space(X,τ) with no separation properties assumed. If A⊂X,cl(A) and int(A) will, respectively, denote the closure and interior of A in (X,τ) and int⋆(A) will denote the interior of A in (X,τ⋆).  A subset A of a topological space (X,τ) is said to be a g-closed set [6] if cl(A)⊂U whenever A⊂U and U is open. A subset A of a topological space (X,τ) is said to be a g-open set if X-A is a g-closed set. A space (X,τ) is said to be a 𝒯1/2-space  [6] if every g-closed set is a closed set.For a subsetA of a space (X,τ), the θ-interior [7] of A is the union of all open sets of X whose closures contained in A and is denoted by intθ(A). The subset A is called θ-open if A=intθ(A). The complement of a θ-open set is called a θ-closed set. Equivalently, A⊂X is called θ-closed [7] if A=clθ(A), where clθ(A)={x∈X∣cl(U)∩A≠∅forallU∈τ(x)}. The family of all θ-open sets of X forms a topology [7] on X, which is coarser than τ and is denoted by τθ.  A subset A of a topological space (X,τ) is said to be a θ-g-closedset [8] if clθ(A)⊂U whenever A⊂U and U is open. A subset A of a space (X,τ) is said to be a θ-g-openset [8] if X-A is a θ-g-closed set. A subset A of a space (X,τ) is said to be a Λ-set [9, 10] if A=AΛ, where AΛ=∩{U∈τ∣A⊂U}.A subsetA of an ideal space (X,τ,ℐ) is said to be θ-ℐ-closed [11] if clθ⋆(A)=A,  where clθ⋆(A)={x∈X∣A∩cl⋆(U)≠ϕforallU∈τ(x)}.   A is said to be θ-ℐ-open if X-A is θ-ℐ-closed.  If ℐ={∅},  clθ⋆(A)=clθ(A).  If ℐ=℘(X),  clθ⋆(A)=cl(A).  For a subset A of X,  intθI(A)=∪{U∈τ∣cl⋆(U)⊂A} [11].  We denote this  intθI(A) by intθ⋆(A).  The family of all θ-ℐ-open sets of (X,τ,ℐ) is a topology and it is denoted by τθ-ℐ (see [11, Theorem  1]).Lemma 1.1 (see [11, Corollary  4 if Theorem  2]).
τθ⊂τθ-ℐ⊂τ.Lemma 1.2 (see [11, Proposition  3]).
Let(X,τ,ℐ) be an ideal space. Then, we have(1)
ifℐ={ϕ} or ℐ=𝒩,  where 𝒩 is the ideal of nowhere dense sets of (X,τ),  then τθ-ℐ=τθ,(2)
ifℐ={ϕ}, then τθ-ℐ=τ.Lemma 1.3 (see [5, Theorem  2.13]).
Let(X,τ,ℐ) be an ideal space. Then every subset of X is ℐg-closed if and only if every open set is ⋆-closed.Lemma 1.4 (see [11, Proposition  1]).
Let(X,τ,ℐ) be an ideal space and A a subset of X. Then A is θ-ℐ-open if and only if  intθ⋆(A)=A.Lemma 1.5.
Let(X,τ,ℐ) be an ideal space and A a subset of X. Then clθ⋆(A)={x∈X∣U∩cl⋆(A)≠ϕ for all U∈τ(x)} is closed.Proof.
Ifx∈cl(clθ⋆(A)) and U∈τ(x),  then U∩clθ⋆(A)≠ϕ. Then,  y∈U∩clθ⋆(A) for some y∈X. Since U∈τ(y) and y∈clθ⋆(A), from the definition of clθ⋆(A) we have A∩cl⋆(U)≠ϕ. Therefore, x∈clθ⋆(A). So cl(clθ⋆(A))⊂clθ⋆(A) and hence clθ⋆(A) is closed.Lemma 1.6.
Let(X,τ,ℐ) be an ideal space and A a subset of X. Then,  X-clθ⋆(X-A)=intθ⋆(A).Proof.
x∈X-clθ⋆(X-A) if and only if  x∉clθ⋆(X-A) if and only if there exist  U∈τ(x) such that (X-A)∩cl⋆(U)=ϕ if and only if  x∈U and,  cl⋆(U)⊂(A) if and only if  x∈U⊂intθ⋆(A).


## 2.θ-ℐg-Closed Sets

A subsetA of an ideal space (X,τ,ℐ) is said to be a  θ-ℐg-closed set   if clθ⋆(A)⊂U whenever A⊂U and U is open. Every θ-ℐ-closed set is a θ-ℐg-closed set. If ℐ={∅}, then clθ⋆(A)=clθ(A) and hence θ-ℐg-closed sets coincide with θ-g-closed sets. If ℐ=℘(X), then clθ⋆(A)=cl(A) and hence θ-ℐg-closed sets coincide with g-closed sets.  Since  cl⋆(A)⊂cl(A)⊂clθ⋆(A)⊂clθ(A),  we have the following inclusion diagram:
(2.1)θ-g-closed⟶θ-Ig-closed⟶g-closed⟶Ig-closed.Example2.1.  shows that a g-closed set needs not to be θ-ℐg-closed,  and Example 2.2 shows that θ-ℐg-closed set needs not to be a θ-g-closed set.Example 2.1.
LetX={a,b,c,d},  τ={ϕ,{b},{a,b},{b,c},{a,b,c},{a,b,d},X},  and ℐ={ϕ,{a},{c},{a,c}}. Let A={c}. Then A is closed and hence g-closed. But A is not θ-ℐg-closed because A⊂{b,c} and clθ⋆(A)=X⊄{b,c}.Example 2.2.
LetX and τ be the same as in Example 2.1.  Let  ℐ={ϕ,{a},{b},{a,b}} and A={c}. Then A is a θ-ℐ-closed and hence θ-ℐg-closed.  Since A⊂{b,c} and clθ(A)=X⊄{b,c},A is not θ-g-closed.Theorem 2.3.
If A is a subset of an ideal space(X,τ,ℐ), then the following are equivalent.(a)
A isθ-ℐg-closed.(b)
For allx∈clθ⋆(A), cl({x})∩A≠ϕ.(c)
clθ⋆(A)-A contains no nonempty closed set.Proof.
(a)⇒(b).  Suppose x∈clθ⋆(A). If cl({x})∩A=ϕ, then A⊂X-cl({x}).  Since A is θ-ℐg-closed,   clθ⋆(A)⊂X-cl({x}).  It is a contradiction to the fact that x∈clθ⋆(A).  This proves (b).
(b)⇒(c).  Suppose F⊂clθ⋆(A)-A,  F is closed and x∈F. Since F⊂X-A and F closed, cl({x})∩A⊂cl(F)∩A=F∩A=ϕ. Since x∈clθ⋆(A),  by (b), cl({x})∩A≠ϕ, a contradiction which proves (c).
(c)⇒(a).  Let U be an open set containing A.  Since clθ⋆(A) is closed, clθ⋆(A)∩(X-U) is closed and clθ⋆(A)∩(X-U)⊂clθ⋆(A)-A.  By hypothesis, clθ⋆(A)∩(X-U)=ϕ and hence  clθ⋆(A)⊂U.  Thus,  A is  θ-ℐg-closed.If we putℐ={ϕ} in Theorem 2.3, we get Corollary 2.4 which gives characterizations for θ-g-closed sets. If we put ℐ=℘(X) in Theorem 2.3,  we get Corollary 2.5 which gives characterizations for g-closed sets.Corollary 2.4.
IfA is a subset of a topological space  (X,τ), then the following are equivalent.(a)
A isθ-g-closed.(b)
For allx∈clθ(A),  cl({x})∩A≠ϕ.(c)
clθ(A)-A  contains no nonempty closed set.Corollary 2.5 (see [12, Theorem  2.2]).
IfA is a subset of a topological space  (X,τ), then the following are equivalent. (a)
A is g-closed.(b)
For allx∈cl(A), cl({x})∩A≠ϕ.(c)
cl(A)-A   contains no nonempty closed set.The following Corollary2.6 shows that in 𝒯1-space, θ-ℐg-closed sets are θ-ℐ-closed, the proof of which follows from Theorem 2.3(c). Corollary 2.7 gives the relation between θ-ℐg-closed and θ-ℐ-closed sets.Corollary 2.6.
If(X,τ,ℐ) is a  𝒯1-space and A is  θ-ℐg-closed then A is a  θ-ℐ-closed set.Corollary 2.7.
If(X,τ,ℐ) is  an ideal space and A is a   θ-ℐg-closed set,  then the following are equivalent.(a)
A is aθ-ℐ-closed set.(b)
clθ⋆(A)-A is a closed set.Proof.
(a)⇒(b).   If A is θ-ℐ-closed, then clθ⋆(A)-A=ϕ and so clθ⋆(A)-(A) is closed.
(b)⇒(a).   If clθ⋆(A)-(A) is closed, since A is θ-ℐg-closed, by Theorem 2.3(c), clθ⋆(A)-(A)=ϕ and so A is θ-ℐ-closed.If we putℐ={ϕ} in Corollary 2.7,  we get Corollary 2.8. If we put  ℐ=℘(X) in Corollary 2.7, we get Corollary 2.9.Corollary 2.8.
If(X,τ,) is  a topological space and  A is  a   θ-g-closed set,  then the following are equivalent.(a)
A is aθ-closed set.(b)
clθ(A)-A is a closed set.Corollary 2.9 (see [6, Corollary  2.3]).
If(X,τ) is  an topological space and  A is  a  g-closed set,  then the following are equivalent.(a)
A is a closed set.(b)
cl(A)-A is a closed set.Theorem 2.10.
If every open set of an ideal space(X,τ,ℐ) is  ⋆-closed,  then every  g-closed set is  θ-ℐg-closed.Proof.
Since every open set is⋆-closed,  cl⋆(U)=U for every U∈τ. Therefore, for every subset A of X,  intθ⋆(A)=∪{U∈τ∣cl⋆(U)⊂A}=∪{U∈τ∣U⊂A}=int(A).  So  clθ⋆(A)=cl(A) for every subset A of X. This implies that every g-closed set is θ-ℐg-closed.Corollary 2.11.
If every subset of an ideal space(X,τ,ℐ) is  ℐg-closed,  then every  g-closed set is  θ-ℐg-closed.The proof follows from Lemma1.3 and Theorem 2.10.Theorem 2.12.
Let(X,τ,ℐ) be an ideal space. Then every subset of X is θ-ℐg-closed if and only if every open set is  θ-ℐ-closed.Proof.
Suppose every subset ofX is θ-ℐg-closed. If U is open, then U is θ-ℐg-closed and so clθ⋆(U)⊂U. Hence U is θ-ℐ-closed. Conversely, suppose A⊂U and U is open. Since every open set is θ-ℐ-closed,   clθ⋆(A)⊂U and so A is θ-ℐg-closed.If we putℐ={ϕ} in Theorem 2.12, we get Corollary 2.13. If we put ℐ=℘(X) in Theorem 2.12,  we get Corollary 2.14.Corollary 2.13.
Let(X,τ) be a topological space. Then every subset of X is  θ-g-closed if and only if every open set is  θ-closed.Corollary 2.14 (see [6, Theorem  2.10]).
Let(X,τ) be a topological space. Then every subset of X is  g-closed if and only if every open set is closed.Theorem 2.15.
If everyθ-ℐg-closed set of an ideal space  (X,τ,ℐ) is  θ-closed,  then  (X,τ) is a  𝒯1space.Proof.
Suppose{x} is not closed for some x∈X. Then,  B=X-{x} is not open. So B is θ-ℐg-closed.  By hypothesis, B is θ-closed.  Therefore, {x} is θ-open.  So {x} is both open and closed, a contradiction. Hence,  (X,τ) is a 𝒯1-space.If we putℐ={ϕ}  in Theorem 2.15, we get Corollary 2.16.Corollary 2.16.
If everyθ-g-closed set of a space  (X,τ) is  θ-closed, then  (X,τ) is a  𝒯1space.Theorem 2.17.
Intersection of aθ-ℐg-closed set and a  θ-ℐ-closed set is always  θ-ℐg-closed.Proof.
LetA be a θ-ℐg-closed set and F  a θ-ℐ-closed set of an ideal space (X,τ,ℐ).  Suppose A∩F⊂U and U is open in X. Then,  A⊂U∪(X-F).   Now X-F is θ-ℐ-open and hence open. So U∪(X-F) is an open set containing A.   Since A is θ-ℐg-closed,  clθ⋆(A)⊂U∪(X-F). Therefore,  clθ⋆(A)∩F⊂U which implies that clθ⋆(A∩F)⊂U.  So A∩F is θ-ℐg-closed.If we putℐ={ϕ} in Theorem 2.17,  we get Corollary 2.18. If we put ℐ=℘(X) in Theorem 2.17,  we get Corollary 2.19.Corollary 2.18 (see [8, Proposition 3.11]).
Intersection of aθ-g-closed set and a  θ-closed set is always  θ-g-closed.Corollary 2.19 (see [6, Corollary  2.7]).
Intersection of ag-closed set and a closed set is always a  g-closed set.Theorem 2.20.
A subsetA of an ideal space  (X,τ,ℐ) is  θ-ℐg-closed if and only if  clθ⋆(A)⊂AΛ.Proof.
SupposeA is θ-ℐg-closed and x∈clθ⋆(A).  If x∉AΛ, then there exists an open set U such that A⊂U,  but  x∉U. Since A is θ-ℐg-closed,  clθ⋆(A)⊂U and so x∉clθ⋆(A),  a contradiction. Therefore, clθ⋆(A)⊂AΛ.   Conversely, suppose that clθ⋆(A)⊂AΛ.    If A⊂U and U is open, then AΛ⊂U and so  clθ⋆(A)⊂U. Therefore, A is θ-ℐg-closed.If we putℐ={ϕ} in Theorem 2.20, we get Corollary 2.21. If we put ℐ=℘(X) in Theorem 2.20,  we get Corollary 2.22.Corollary 2.21.
A subset A of a space(X,τ) is  θ-g-closed if and only if  clθ(A)⊂AΛ.Corollary 2.22.
A subset A of a space(X,τ) is  g-closed if and only if  cl(A)⊂AΛ.Theorem 2.23.
Let A be aΛ-set of an ideal space  (X,τ,ℐ). Then A is  θ-ℐg-closed if and only if   A  is  θ-ℐ-closed.Proof.
SupposeA is θ-ℐg-closed.  By Theorem 2.20,  clθ⋆(A)⊂AΛ=A,  since A is a  Λ-set. Therefore, A is θ-ℐ-closed. Converse follows from the fact that every θ-ℐ-closed is θ-ℐg-closed.If we putℐ={ϕ} in Theorem 2.23,  we get Corollary 2.24. If we put ℐ=℘(X) in Theorem 2.23,  we get Corollary 2.25.Corollary 2.24.
Let A be aΛ-set of a space  (X,τ). Then A is  θ-g-closed if and only if A is  θ-closed.Corollary 2.25.
Let A be aΛ-set of a space  (X,τ). Then A is  g-closed if and only if A is closed.Theorem 2.26.
Let(X,τ,ℐ) be an ideal space and  A⊂X.  If  AΛ is  θ-ℐg-closed,  then A is also  θ-ℐg-closed.Proof.
Suppose thatAΛ is a  θ-ℐg-closed set. If A⊂U and U is open,  then AΛ⊂U.  Since AΛ is θ-ℐg-closed,  clθ⋆(AΛ)⊂U.  But,  clθ⋆(A)⊂clθ⋆(AΛ).  Therefore, A is θ-ℐg-closed.If we putℐ={ϕ} in Theorem 2.26,  we get Corollary 2.27.  If we put  ℐ=℘(X) in Theorem 2.26,  we get Corollary 2.28.Corollary 2.27.
Let(X,τ) be a topological space and  A⊂X. If  AΛ is  θ-g-closed,  then A is also  θ-g-closed.Corollary 2.28.
Let(X,τ) be a space and  A⊂X. If  AΛ is  g-closed set,  then A is also  g-closed.Theorem 2.29.
For an ideal space(X,τ,ℐ), the following are equivalent.(a)
Everyθ-ℐg-closed set is  θ-ℐ-closed.(b)
Every singleton ofX is closed or  θ-ℐ-open.Proof.
(a)⇒(b).  Let x∈X.  If {x} is not closed, then A=X-{x}∉τ and then A is trivially θ-ℐg-closed.  By (a), A is θ-ℐ-closed. Hence {x} is θ-ℐ-open.
(b)⇒(a).  Let A be a θ-ℐg-closed set and let  x∈clθ⋆(A).  We have the following cases.
Case 1.
{x} is closed. By Theorem 2.3,  clθ⋆(A)-A does not contain a nonempty closed subset. This shows {x}∈A.Case 2.
{x} is θ-ℐ-open. Then,  {x}∩A≠ϕ. Hence,  x∈A.
Thus in both casesx∈A and so A=clθ⋆(A),  that is, A is θ-ℐ-closed,  which proves (a).If we putℐ={ϕ} in Theorem 2.29,  we get Corollary 2.30. If we put  ℐ=℘(X) in Theorem 2.29,  we get Corollary 2.31.Corollary 2.30.
For an ideal space(X,τ), the following are equivalent.(a)
Everyθ-g-closed set is  θ-closed.(b)
Every singleton ofX is closed or  θ-open.Corollary 2.31 (see [13, Theorem  2.5]).
For an ideal space(X,τ), the following are equivalent.(a)
Everyg-closed set is closed.(b)
Every singleton ofX is closed or open.Theorem 2.32.
Let(X,τ,ℐ) be an ideal space and  A⊂X. Then A is  θ-ℐg-closed if and only if  A=F-N,  where  F is  θ-ℐ-closed and  N contains no nonempty closed set.Proof.
IfA is  θ-ℐg-closed,  then by Theorem 2.3,  N=clθ⋆(A)-A contains no nonempty closed set. If F=clθ⋆(A),  then F is θ-ℐ-closed such that F-N=clθ⋆(A)-(clθ⋆(A)-A)=clθ⋆(A)∩((X-clθ⋆(A))∪A)=A. Conversely, suppose A=F-N,  where F is θ-ℐ-closed and  N contains no nonempty closed set. Let U be an open set such that A⊂U. Then,  F-N⊂U which implies that F∩(X-U)⊂N. Now, A⊂F and F is θ-ℐ-closed implies that clθ⋆(A)∩(X-U)⊂clθ⋆(F)∩(X-U)⊂F∩(X-U)⊂N. Since θ-ℐ-closed sets are closed,  clθ⋆(A)∩(X-U) is closed. By hypothesis, clθ⋆(A)∩(X-U)=ϕ and so clθ⋆(A)⊂U, which implies that A is θ-ℐg-closed.If we putℐ={ϕ} in Theorem 2.32,  we get Corollary 2.33. If we put ℐ=℘(X) in Theorem 2.32,  we get Corollary 2.34.Corollary 2.33.
Let(X,τ) be a space and  A⊂X. Then A is  θ-g-closed subset of X if and only if  A=F-N,  where F is  θ-closed and N contains no nonempty closed set.Corollary 2.34 (see [12, Corollary  2.3]).
Let(X,τ) be a space and  A⊂X. Then A is  g-closed if and only if  A=F-N,  where F is closed and N contains no nonempty closed set.Theorem 2.35.
Let(X,τ,ℐ) be an ideal space.  If A is  a  θ-ℐg-closed subset of X and A⊂B⊂clθ⋆(A), then B is also  θ-ℐg-closed.Proof.
clθ⋆(B)-B⊂clθ⋆(A)-A, and since clθ⋆(A)-A has no nonempty closed subset, neither does clθ⋆(B)-B.  By Theorem 2.3, B is  θ-ℐg-closed.If we putℐ={ϕ} in Theorem 2.35,  we get Corollary 2.36. If we put ℐ=℘(X) in Theorem 2.35,  we get Corollary 2.37.Corollary 2.36.
Let(X,τ) be a space. If  A is  a  θ-g-closed subset of X and  A⊂B⊂clθ(A), then  B is also  θ-g-closed.Corollary 2.37 (see [6, Theorem  2.8]).
Let(X,τ) be a space. If A is  a  g-closed subset of X and  A⊂B⊂cl(A), then B is also  g-closed.A subsetA of an ideal space (X,τ,ℐ) is said to be θ-ℐg-open if X-A is θ-ℐg-closed.Theorem 2.38.
A subset A of an ideal space(X,τ,ℐ) is  θ-ℐg-open if and only if  F⊂intθ⋆(A) whenever F is closed and  F⊂A.Proof.
SupposeA is a θ-ℐg-open set and  F  is a closed set contained in A, then X-A⊂X-F and X-F is open. Since X-A is θ-ℐg-closed,  clθ⋆(X-A)⊂(X-F) and so F⊂X-clθ⋆(X-A)=intθ⋆(A). Conversely, suppose X-A⊂U and X-U is closed.  By hypothesis,  X-U⊂intθ⋆(A),  which implies that clθ⋆(X-A)=X-intθ⋆(A)⊂U.  Therefore, X-A  is θ-ℐg-closed and hence A is θ-ℐg-open.If we putℐ={ϕ} in Theorem 2.38,  we get Corollary 2.39. If we put ℐ=℘(X) in Theorem 2.38,  we get Corollary 2.40.Corollary 2.39.
A subset A of a space(X,τ) is  θ-g-open if and only if F⊂intθ(A) whenever F is closed and  F⊂A.Corollary 2.40 (see [6, Theorem  4.2]).
A subset A of a space(X,τ) is  g-open if and only if  F⊂int(A) whenever F is closed and  F⊂A.Theorem 2.41.
Let(X,τ,ℐ) be an ideal space and  A⊂U. Then the following are equivalent.(a)
A isθ-ℐg-closed.(b)
A∪(X-clθ⋆(A)) is  θ-ℐg-closed.(c)
clθ⋆(A)-A is  θ-ℐg-open.Proof.
(a)⇒(b). Suppose A is θ-ℐg-closed. If U is any open set containing A∪(X-clθ⋆(A)),  then X-U⊂X-(A∪(X-clθ⋆(A))=clθ⋆(A)-A.  Since A is θ-ℐg-closed,  by Theorem 2.3(c), it follows that X-U=ϕ and so X=U.  Since X is the only open set containing A∪(X-clθ⋆(A)),  A∪(X-clθ⋆(A)) is θ-ℐg-closed.
(b)⇒(a). Suppose A∪(X-clθ⋆(A)) is θ-ℐg-closed.  If F is any closed set contained in clθ⋆(A)-A, then A∪(X-clθ⋆(A))⊂X-F and X-F is open. Therefore,  clθ⋆(A∪(X-clθ⋆(A))⊂X-F,  which implies that clθ⋆(A)∪clθ⋆(X-clθ⋆(A))⊂X-F and so X⊂X-F;  it follows that F=ϕ.  Hence A is θ-ℐg-closed.
The equivalence of (b) and (c) follows from the fact thatX-(clθ⋆(A)-A)=A∪(X-clθ⋆(A)).If we putℐ={ϕ} in Theorem 2.41, we get Corollary 2.42. If we put ℐ=℘(X) in Theorem 2.41,  we get Corollary 2.43.Corollary 2.42.
Let(X,τ) be a space and  A⊂U. Then the following are equivalent.(a)
A isθ-g-closed.(b)
A∪(X-clθ(A)) is  θ-g-closed.(c)
clθ(A)-A is  θ-g-open.Corollary 2.43.
Let(X,τ) be an ideal space and  A⊂U. Then the following are equivalent.(a)
A isg-closed.(b)
A∪(X-cl(A)) is  g-closed.(c)
cl(A)-A is  g-open.


## 3. Characterization of𝒯1/2 and 𝒯ℐ-Space

Theorem 3.1.
In an ideal space(X,τ,ℐ),  the following are equivalent.(a)
Everyθ-g-closed set is closed.(b)
(X,τ) is a  𝒯1/2-space.(c)
Everyθ-ℐg-closed set is closed.Proof.
(a)⇔(b). Equivalence of (a) and (b) follows from Theorem  4.1 of [8].
(b)⇒(c). Let A be a θ-ℐg-closed set. Since every θ-ℐg-closed set is g-closed,   A is g-closed. By hypothesis, A is closed.
(c)⇒(b). Let x∈X. If {x} is not closed, then B=X-{x} is not open. So B is θ-ℐg-closed. By hypothesis, B is closed and so {x} is open. By Corollary 2.31, (X,τ) is a 𝒯1/2-space.Theorem 3.2.
In an ideal space(X,τ,ℐ) the following,  are equivalent.(a)
Everyθ-g-closed set is  ⋆-closed.(b)
(X,τ,ℐ) is a  𝒯ℐ-Space.(c)
Everyθ-ℐg-closed set is  ⋆-closed.Proof.
(a)⇒(b). Let x∈X.  If {x}  is not closed,  then X is the only open set containing X-{x} and so X-{x} is θ-g-closed. By hypothesis,  X-{x} is ⋆-closed.   Equivalently {x} is ⋆-open.  Thus,  every singleton set in X is either closed or ⋆-open. By Theorem  3.3 of [4], (X,τ,ℐ) is a 𝒯ℐ-Space.
(b)⇒(a). The proof follows from the fact that every θ-g-closed set is   ℐg-closed.
(b)⇒(c). The proof follows from the fact that every set is θ-ℐg-closedℐg-closed.
(c)⇒(b). Let x∈X.  If {x} is not closed, then X is the only open set containing x-{x} and so x-{x}  is  θ-ℐg-closed.  By hypothesis, X-{x} is ⋆-closed.  Thus,  {x}  is ⋆-open. Therefore, every singleton set in X is either ⋆-open or closed. By Theorem of  3.3 [4], (X,τ,ℐ) is a 𝒯ℐ-Space.The proof of the Corollary3.3 follows from Theorem  3.2 and Theorem  3.10 of [5].If we putℐ={ϕ} in Corollary 3.3,  we get Corollary 3.4.Corollary 3.3.
In an ideal space(X,τ,ℐ),  the following are equivalent.(a)
Everyθ-g-closed set is  ⋆-closed.(b)
Everyθ-ℐg-closed set is  ⋆-closed.(c)
Everyℐg-closed set is an  ℐ-locally⋆-closed set.Corollary 3.4.
In a topological space(X,τ),   the following are equivalent.(a)
Everyθ-g-closed set is closed.(b)
Everyg-closed set is a locally closed set.



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*Source: 682829-2012-04-24.xml*