g_*^*-I-Closed Sets and Their Properties in in Ideal Topological Space

      The subject of ideals in topological spaces has been studied by Kuratowski [1] and Vaidyanatha swamy [2]. In 1990, Jankovic and Hamlett [3] once again investigated applications of topological ideals. The concept of generalized closed sets plays a significant role in topology. There are many research papers that deal with different types of generalized closed sets. Levine [4] introduced generalized closed (briefly, g-closed) sets and studied their basic properties and Veera Kumar [5] introduced g^*-closed sets in topological spaces. The purpose of this present paper is to define a new class of generalized idea closed sets called g_*^*-I-closed sets by using g^*-open set (which is a complement of g^*-closed set) and also we obtain the basic properties of called g_*^*-I-closed set in ideal topological spaces.

 

 

Preliminaries

    An ideal I on a non-empty set X is a collection of subsets of X which satisfies the following properties [1], [2].

(i) A∈I,B∈I⇒A∪B∈I

(ii) A∈I,B⊂A⇒B∈I

     A topological space (X,τ) with an ideal I on X is called an ideal topological space and is denoted by (X,τ,I).

Let Y be a subset of X.I_Y={I∩Y/I∈I} is an ideal on Y and by (Y,τ/Y,I_Y ) we denote the ideal topological subspace. Let P(X) be the power set of X, then a set operator ( )^*:P(X)→P(X) called the local function [1] of A with respect to τ and I is defined as follows:

For A⊂X,A^* (I,τ)={x∈X/U∩A∉I for every open set U containing x}.

We write A^* instead of A^* (I,τ) in case there is no confusion. A Kuratowski closure operator cl^* ( ) for a topology τ^* (I,τ), called the τ^* - topology is defined by cl^* (A)=A∪A^* [6]

      A subset A of a space (X,τ) is said to be semi-open [7] if A⊂cl(int⁡(A)).

A set operator () ^(*S):P(X)→P(X) called a semi-local function and cl^(*s) () [7] of A with respect to τ and I are defined as follows:

For A⊂X,□( ) A^(*S) (I,τ)={x∈X/U∩A∉I□( ) for □( ) every □( ) semi □( ) open □( ) set □( ) U□( ) containing x}.□( ) and Cl^(*S) (A)=A∪A^(*S).

Note: A^(*S) defined in [7] and A_* defined in [8] are the same. For a subset A of X,cl⁡(A)(resp⁡scl⁡(A)) denotes the closure (resp semi closure) of A in (X,τ). Similarly cl^* (A) and int  ^* (A) denote the closure of A and interior of A in (X,τ^* ).

      A subset A of X is called * closed (resp.* S - closed) if A^*⊆A (resp A^(*S)⊆A) [3]. A is called * - dense  in itself (resp .*S-dense) [3]. If A⊂A^* (resp ⊂A^(*S) ) A is called * - perfect  (resp .*s - perfect). If A=A^* (resp A=A^(*S) ) [3]. A subsetA of a topological space (X,τ) is said to be generalized closed (briefly g-closed) if cl(A)⊂U whenever A⊂U and U is open in (X,τ)  [3]. The complement of g-closed set is said to be g-open.

Definition 2.1. A subset A of a topological space (X,τ) is said to be g^*-closed set if Cl(A)⊆U whenever A⊆U and U is g-open in (X,τ) [5].

Definition 2.2. A subset A of a space (X,τ,I) is said to be

(i) gI  – closed [9] if A^(*S)⊆U wherever A⊆U and U is open in X.

(ii) Ig- closed [10]  if A^*⊆U wherever A⊆U and U is open in X.

Definition 2.3. A space (X,τ,I) is said to be a T_I-space if every I-generalized closed subset of X is τ^*-closed [10] [14].

Definition 2.4. A subset A of an ideal topological space (X,τ,I) is said to be I- compact if for every τ-open cover {ω_α: α∈Δ} of A, there exists a finite subset Δ_0 of Δ such that (X-∪{ω_α: α∈Δ_0 })∈I [11], [12]. 

 

 Lemma 2.5. [13] Let (X,τ,I) be an ideal space and W⊆X. If W⊆W^*, then W^*= Cl(W^* )=Cl(W)=Cl^* (W). 

 

Theorem 2.6. Let (X,τ,I) be an ideal space. If W is an Ig-closed subset of X, then W is I-compact [14], Theorem 2.17]. 

 

Note: In general the intersection of g-closed sets need not be g-closed.

Definition 2.7. [7] A topological space (X,τ) is said to be a g-multiplicative space if the arbitrary intersection of g-closed sets in X is g-closed.

 

Remark 2.8. [7] 

In g-multiplicative spaces, gCl(W) which is the intersection of all g-closed sets in X containing W is also g-closed.

Any indiscrete topological space (X,τ) is g-multiplicative.

If X={x,y,z} and τ={X,∅,{x}} then {x,z} and {x,y} are g-closed but {x} is not g-closed and hence (X,τ) is not g-multiplicative.  

 

Theorem 2.9. [10] (Theorem 3.20). Let (X,τ,I) be an ideal space and W⊂Y⊂X where Y is α-open in X. Then W^* (I_Y,τ_Y )=W^* (I,τ)∩Y. 

 

Lemma 2.11. [3] Let (X,τ) be a space, I and J be ideals on X, and let A and B be subsets of X. Then

(1) A⊆B⇒A^*⊆B^*.

(2) If I⊆J, then A^* (I)⊇A^* (J).

(3) A^* (I)=Cl(A^* )⊆Cl(A) (i.e, A^* is a closed subset of (A) ).

(4) If A⊆A^*, then A^*=Cl(A^* )=Cl(A)=Cl^* (A).

(5) (A^* )^*⊆A^*.

(6) (A∪B)^*=A^*∪B^*.

(7) If U∈τ, then U∩A^*=U_x (U_x∩A)^*⊆(U∩A)^*.

(8) If A∈I, then A^*=∅.

 

Lemma 2.12. [3] For any two sets A and B  of an ideal topological space (X,τ,I),Cl^* (A∪B)=Cl^* (A)∪Cl^* (B).

Methodology

 

Definition 3.1: A subset W of an ideal space (X,τ,I) is said to be

g_*^*-I-closed, if Cl^* (W)⊂U whenever W⊂U and U is g^*-open in X.

g_*^*-I-open, if its complement is g_*^*-I-closed set.

The collection of all g_*^*-I-closed sets (resp g_*^*-I-open sets) is denoted by (g_*^* C(X)┤ ( ├ resp⁡g_*^* O(X)).

Remark 3.2: In any ideal topological space (X,τ,I),

Every g_*^*-I-closed set is Ig-closed set.

Every Ig-closed set is gI-closed set.

Every g_*^*-I-closed set is gI-closed set.

The converse of part (3) is not true in general, see the following example.

Example 3.3. Let X={a,b,c},τ={∅,X,{a},{b},{a,b}}, and I={∅,{c}}.

Put A={b} and the only open sets containing A are {a,b} and X, then A_*={b}⊆{a,b},

whenever {a,b} is open and {b}⊆{a,b}. So A is gI-closed set. But, since A^*={b}^*= {b,c}, so Cl^* ({b})={b,c}⊈{a,b}, whenever {b}⊆{a,b} and {a,b} is also g^*-open.

Theorem 3.4. Every *-closed set is g_*^*-I-closed but not conversely.

Proof. Let W be a *-closed, then W^*⊆W. Let W⊆U where U is g^*-open. Hence Cl^* (W)⊆U whenever W⊆U and U is g^*-open. Therefore W is g_*^*-I-closed.

Example 3.5. Let X={x,y,z} with a topology τ={∅,X,{x},{y,z}} and an ideal I= {∅,{z}}. Then g_*^*-I-closed sets are the power set of X and *-closed sets are ∅,X,{x},{z},{x,z},{y,z}. It is clear that {y} is g_*^*-I-closed set but it is not *-closed.

 

Theorem 3.6. If (X,τ,I) is an ideal topological space and W⊂X. Then the following are equivalent.

W is g_*^*-I-closed,

For all x∈Cl^* (W),g^* Cl({x})∩W≠∅,

Cl^* (W)-W contains no nonempty g^*-closed set,

W^*-W contains no nonempty g^*-closed set,

Proof. (1) ⇒ (2) Suppose x∈Cl^* (W). If g^* Cl({x})∩W=∅, then W⊆X-g^* Cl({x}). By Definition 3.1, Cl^* (W)⊆X-g^* Cl({x}), which is a contradiction, since x∈Cl^* (W).

(2) ⇒ (3) Suppose F⊆Cl^* (W)-W,F is g^*-closed and x∈F. Since F⊆X-W and F is g^*-closed, then W⊆X-F and F is g^*-closed, g^* Cl({x})∩W=∅. Which is a contradiction. Since x∈Cl^* (W) by (3), g^* Cl({x})∩W≠∅. Therefore Cl^* (W)-W contains no nonempty g^*-closed set.

(3) ⇒ (4) Since l^* (W)-W=(W∪W^* )-W=(W∪W^* )∩W^c=(W∩W^c )∪(W^*∩┤ ├ W^c )=W^*∩W^c=W^*-W. Therefore W^*-W contains no nonempty g^*-closed set.

(4) ⇒ (1) Let W⊆U where U is a g^*-open set. Therefore X-U⊆X-W and so Cl^* (W)∩(X-U)⊆Cl^* (W)∩(X-W)=W^*-W. Therefore Cl^* (W)∩(X-U)⊆W^*-W.

Since Cl^* (W) is always *-closed set, so Cl^* (W) is g^*-closed set and so Cl^* (W)∩(X-U) is a g^*-closed set contained in W^*-W. Therefore Cl^* (W)∩(X-U)=∅ and hence Cl^* (W)⊆U. Therefore W is g_*^*-I-closed.

Theorem 3.7. If (X,τ,I) is an ideal space, then W^* is always g_*^*-I-closed for every subset W of X.

Proof. Let W^*⊆U where U is g^*-open. Since (W^* )^*⊆W^* so by Lemma 2.11, we have Cl^* (W^* )⊆U whenever W^*⊂U and U is g^*-open. Hence W^* is g_*^*-I-closed.

Theorem 3.8. Let (X,τ,I) be an ideal space. For every W∈I, W is g_*^*-I-closed.

Proof. Let W⊆U where U is g^*-open set. Since W^*=∅ for every W∈I, then Cl^* (W)= W∪W^*=W⊆U. Therefore, W is g_*^*-I-closed.

Corollary 3.9. If (X,τ,I) is an ideal space and W is a g_*^*-I-closed set, Then the following are equivalent:

W is a *-closed set,

Cl^* (W)-W is a g^*-closed set,

W^*-W is a g^*-closed set.

Proof. (1) ⇒ (2) If W is *-closed, then W^*⊆W and so Cl^* (W)-W=(W∪W^* )-W=∅, so Cl(∅)=∅⊆U. Hence Cl^* (W)-W is g^*-closed set.

(2) ⇒ (3) Since Cl^* (W)-W=W^*-W and so W^*-W is g^*-closed set.

(3)⇒(1) If W^*-W is a g^*-closed set, since W is g_*^*-I-closed set, by Theorem 3.6, W^*-W=∅ and so W is *-closed.

Theorem 3.10. Let (X,τ,I) be an ideal space. Then every g_*^*-I-closed, g^*-open set is *-closed set.

Proof. Since W is g_*^*-I-closed and g^*-open. Then Cl^* (W)⊆W whenever W⊆W and W is g^*-open. Hence W is *-closed.

Corollary 3.11. If (X,τ,I) is a T_I ideal space and W is a  g_*^*-I-closed set, then W is *-closed set.

Proof. Since every g_*^*-I-closed set is an Ig-closed set in an ideal space (X,τ,I) and X is T_I space, so every Ig-closed set is *-closed. So W is *-closed.

Theorem 3.12. If (X,τ,I) is an ideal space, Then every g^*-closed set is an g_*^*-I-closed set but not conversely.

Proof. Let W be a g^*-closed set. If W⊆U, whenever U is g^*-open. Since every g^*-open is g-open and W is g^*-open, so Cl(W)⊆U. But, since Cl^* (W)⊆Cl(W)⊆U, whenever W⊆U and U is g^*-open, so W is g_i^*-I-closed.

Example 3.13. Let X={x,y,z} with a topology τ={∅,X,{x},{x,z}} and an ideal I={∅,{x}}. Then g_*^*-I-closed sets are ∅,X,{x},{y},{x,y},{y,z} and g^*-closed sets are ∅,X,{y},{y,z}. It is clear that {x} is a g_*^*-I-closed set but it is not g^*-closed in (X,τ).

Example 3.14. Let X={x,y,z} with a topology τ={∅,X,{x},{x,z}} and an ideal I={∅,{y},{z},{y,z}}. Clearly, the set {z} is a g_*^*-I-closed set but it is not g^*-closed in (X,τ,I).

Theorem 3.15. If (X,τ,I) is an ideal space, and W is a *-dense in itself, g_*^*-I-closed subset of X, then W is g^*-closed.

Proof. Suppose W is a *-dense in itself, g_*^*-I-closed subset of X. Let W⊆U where U is g-open. Then, Cl^* (W)⊆U whenever W⊆U and U is g-open. Since W is *-dense in itself, so every g^*-open is g-open and W is *-dense in itself, by Lemma 2.5, Cl(W)=Cl^* (W). Therefore Cl(W)⊆U whenever W⊆U and U is g-open. Hence W is g^*-closed.

Corollary 3.16. If (X,τ,I) is an ideal space where I={∅}, then W is g_*^*-I-closed if and only if W is g^*-closed

Proof. From the fact that for I={∅},W^*=Cl(W)⊇W. Therefore W is *-dense in itself. Since W is g_*^*-I-closed, by Theorem 3.15, W is g^*-closed.

Conversely, by Theorem 3.12, every g^*-closed set is a g_*^*-I-closed set.

Theorem 3.17. Let (X,τ,I) be an ideal space and W⊆X. Then W is g_*^*-I-closed if and only if W=F-N where F is *-closed and N contains no nonempty g^*-closed set.

Proof. If W is g_*^*-I-closed, then by Theorem 3.6 (4), N=W^*-W contains no nonempty g^*-closed set. If F=Cl^* (W), then F is *-closed such that F-N=(W∪W^* )-(W^*-┤ W)=(W∪W^* )∩(W^*∩W^c )^c=(W∪W^* )∩((W^* )^c∪W)=(W∪W^* )∩(W∪(W^* )^c )= W∪(W^*∩(W^* )^c )=W.

Conversely, suppose W=F-N where F is *-closed and N contains no nonempty g^*-closed set. Let U be a g^*-open set such that W⊆U. Then F-N⊆U which implies that F∩(X-U)⊆N. Now W⊆F and F^*⊆F then W^*⊆F^* and so (W^*∪W)∩(X-U)⊆ F^*∩(X-U)⊆F∩(X-U)⊆N. By hypothesis, since (W^*∪W)∩(X-U) is g^*-closed, (W^*∪W)∩(X-U)=∅ and so Cl^* (W)⊆U. Hence W is g_*^*-I-closed.

Theorem 3.18. Let (X,τ,I) be an ideal space and W⊆X. If W⊆B⊆W^*, then W^*=B^* and B is *-dense in itself.

Proof. Since W⊆B, then W^*⊆B^* and since B⊆W^*, then B^*⊆(W^* )^*⊆W^*. Therefore W^*=B^* and B⊆W^*⊆B^*. Hence proved.

Theorem 3.19. Let (X,τ,I) be an ideal space and W⊆X. If W and B are subsets of X such that W⊆B⊆Cl^* (W) and W is g_*^*-I-closed, then B is g_*^*-I-closed.

Proof. Let B⊆U and U is g^*-open. Since W⊆B and W is g_*^*-I-closed, so Cl^* (W)⊆U. But, since B⊆Cl^* (W), implies that Cl^* (B)⊆Cl^* (Cl^* (W))=Cl^* (W)⊆U. Therefor Cl^* (W)⊆U, whenever B⊆U. And U is g^*-open. Thus B is g_*^*-I-closed.

Corollary 3.20. Let (X,τ,I) be an ideal space. If W and B are subsets of X such that W⊆B⊆W^* and W is g_*^*-I-closed, then W and B are g^*-closed sets.

Proof. Let W and B be subsets of X such that W⊆B⊆W^* which implies that W⊆B⊆ W^*⊆Cl^* (W) and W is g_*^*-I-closed. By Theorem 3.19, B is g_*^*-I-closed. Since W⊆B⊆W^*, then W^*=B^* and so W and B are *-dense in itself. By Theorem 3.15, W and B are g^*-closed.

Theorem 3.21. Let (X,τ,I) be an ideal space and W⊆X. Then W is g_*^*-I-open if and only if F⊆ int  ^* (W) whenever F is g^*-closed and F⊆W.

Proof. Suppose W is g_*^*-I-open. If F is g^*-closed and F⊆W, then X-W⊆X-F and so Cl^* (X-W)⊆X-F. Therefore F⊆X-Cl^* (X-W)= int  ^* (W). Hence F⊆int^*⁡(W).

Conversely, suppose the condition holds. Let U be a g^*-open set such that X-W⊆U. Then by hypothesis X-U⊆W and so X-U⊆int^*⁡(W). Therefore Cl^* (X-W)⊆U. Thus, X-W is g_*^*-I-closed. Hence W is g_*^*-I-open.

Corollary 3.22. Let (X,τ,I) be an ideal space and W⊆X. If W is a g_*^*-I-open, then F⊆ int  ^* (W) whenever F is closed and F⊆W.

Proof. Since every closed set is g^*-closed set, so by Theorem 3.21 we get the result.

The following theorem gives a property of g_*^*-I-closed.

Theorem 3.23. Let (X,τ,I) be an ideal space and W⊆X. If W is g_*^*-I-open and int  ^* (W)⊆B⊆W, then B is g_*^*-I-open.

Proof. Since W is g_*^*-I-open, then X-W is g_*^*-I-closed. By Theorem 3.6 (4), Cl^* (X- W)-(X-W) contains no nonempty g^*-closed set. Since int (W)⊆int^* (B) which implies that Cl^* (X-B)⊆Cl^* (X-W) and so Cl^* (X-B)-(X-B)⊆Cl^* (X-W)-(X-W) by Theorem 3.6 we get, X-B is g_*^*-I-closed. Thus, B is g_*^*-I - open.

       The following theorem gives a characterization of g_*^*-I-closed sets in terms of g_*^*-I-open sets.

Theorem 3.24. If (X,τ,I) be an ideal topological space and W⊆X. Then the following are equivalent:

W is g_*^*-I-closed,

W∪(X-W^* ) is g_*^*-I-closed,

W^*-W is g_*^*-I-open.

Proof. (1) ⇒ (2) Suppose W is g_*^*-I-closed. If U is any g^*-open set such that ∪(X- ├ W^* )⊆U, then X-U⊆X-(W∪(X-W^* ))=X∩(W∪(W^* )^c )^c=W^*∩W^c= W^*-W. Since W is g_*^*-I-closed, by Theorem 3.6 (4), it follows that X-U=∅ and so X=U. Therefore W∪(X-W^* )⊆U which implies that W∪(X-W^* )⊆X and so Cl^* (W∪(X-W^* ))⊆X=U. Hence W∪(X-W^* ) is g_*^*-I-closed.

(2) ⇒ (1) Suppose W∪(X-W^* ) is g_*^*-I-closed. If F is any g^*-closed set such that F⊆W^*-W, then F⊆W^* and F⊆X∖W which implies that X-W^*⊆X-F and W⊆X-F. Therefore W∪(X-W^* )⊆W∪(X-F)=X-F and X-F is g^*-open. Since Cl^* (W∪(X-W^* ))⊆X-F and since (W∪(X-W^* ))^*⊆Cl^* (W∪(X-W^* ))⊆X-F which implies that W^*∪(X-W^* )^*⊆X-F and so W^*⊆X-F which implies that F⊆X-W^*. Since F⊆W^*, it follows that F=∅. Hence by Theorem 3.6 W is g_*^*-I closed.

(2) ⇒(3) Since -(W^*-W)=X∩(W^*∩W^c )^c=X∩((W^* )^c∪W)=(X∩ ├ (W^* )c)∪(X∩W)=W∪(X-W^* ). Therefore, X-(W^*-W) is g_*^*-I-closed. Hence, W^*-W is g_*^*-I- open. The equivalence is clear.

Theorem 3.25. If (X,τ,I) is an ideal topological space. Then every subset of X is g_*^*-I closed if and only if every g^*-open set is *-closed.

Proof. Suppose every subset of X is g_*^*-I-closed. If U⊆X is g^*-open, then U is g_*^*-I closed and so Cl^* (U)⊆U, then U^*⊆Cl^* (U)⊆U. Hence U is *-closed.

Conversely, suppose that every g^*-open set is *-closed. If U is g^*-open set such that ⊆ U⊆X, then W^*∪W=Cl^* (W)⊆U^*∪U=U and so W is g_*^*-I-closed.

Corollary 3.26. Let (X,τ,I) be an ideal space. If W is a g_*^*-I-closed subset of X, then W is I-compact.

Proof. The proof follows from the fact that every g_*^*-I-closed is Ig-closed.

Definition 3.27. Let N be a subset of (X,τ,I) and x∈X. The subset N of X is called a g_*^*-I-open neighbourhood of x if there exists g_*^*-I-open set U containing x such that U⊂N.

Theorem 3.28. For each (X,τ,I) either {x} is g^*-closed or {x}^c is g_*^*-I-closed in X.

Proof. {x} is not g^*-closed, then {x}^c is not g^*-open.Therefore the only g^*-open set containing {x}^c is X and Cl^* ({x}^c )⊆X which proves that {x}^c is g_*^*-I-closed.

Theorem 3.29. If W and B are g_*^*-I-closed sets in an ideal space (X,τ,I), then W∪B is also a g_*^*-I-closed set.

Proof. Let U be a g^*-open subset of (X,τ,I) containing W∪B. Then W⊂U and ⊂ U. Since W and B are g_*^*-I-closed, Cl^* (W)⊂U and Cl^* (B)⊂U. By Lemma 2.12, Cl^* (W∪B)=Cl^* (W)∪Cl^* (B)⊆U∪U=U. where W∪B⊂U and U is g^*-open which implies W∪B is g_*^*-I-closed.

Theorem 3.30. Let (X,τ,I) be a g-multiplicative ideal space and let W be g_*^*-I-closed. Then W is τ^*-closed ⟺W^*-W is closed.

Proof. Necessity: W is τ^*-closed ⟹W^*⊂W⟹W^*-W=∅ which is closed.

Sufficiency: Let W^*-W be closed. Then it is g-closed By (4) of theorem 3.6, W^*-W=∅ which implies W^*⊂W.

Theorem 3.31. Let (X,τ,I) be a g-multiplicative ideal space and W⊂X. If W is g_*^*-I closed then W∪(X-W^* ) is g_*^*-I-closed.

Proof. Let U be g^*-open and W∪(X-W^* )⊂U

Then X-U⊂X-[W∪(X-W^* )]=W^*-W. Since W is g_*^*-I-closed, W^*-W contains no non-empty g^*-closed set. Therefore X-U=∅ which implies X=U. Thus X is the only g^*-open set containing W∪(X-W^* ), then Cl^* (W∪(X-W^* ))⊆X, which proves W∪(X-W^* ) is g_*^*-I-closed.

Theorem 3.32. Let W be a subset of a g-multiplicative ideal space (X,τ,I). If W is g_*^*-I closed then W^*-W is g_*^*-I-open

Proof. Since X-(W^*-W)=W∪(X-W^* ), the proof follows from Theorem 3.30.

Theorem 3.33. Let (X,τ,I) be an ideal space and W⊂Y⊂X. If W is g_*^*-I-closed in (Y,τ_Y,I_Y ),Y is α-open and τ^*-closed in X. Then W is g_*^*-I-closed in X.

Proof. Let W⊂U and U be g^*-open in X. Then W^* (I_Y,τ/Y)=W^* (I,τ)∩Y⊂U∩Y. Then Y⊂U∪(X-W^* (I,τ)). Since Y is τ^*-closed, Y^*⊂Y. Therefore W^*⊂Y^*⊂Y⊂

U∪(X-W^* (τ,I)). This implies W^*⊂U and hence Cl^* (W)⊂U. So W is g_*^*-I-closed in X.

Theorem 3.34. Let (X,τ,I) be an ideal space. If every g^*-open set is τ^*-closed, then every subset of X is g_*^*-I-closed.

Proof. Let W⊂U and U be a g^*-open set in X. Then Cl^* (W)⊂Cl^* (U)=U which proves W is g_*^*-I-closed.