Between Closed Sets and g

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 2 Ver. II (Mar. - Apr. 2017), PP 09-15 www.iosrjournals.org

Between Closed Sets and gđ??Ž-Closed Sets 1

C. Loganathan 2R. Vijaya Chandra And 3O. Ravi

1

Department of Mathematics,Maharaja Arts and Science College,Arasur,Coimbatore District, Tamil Nadu, India.

2 Department of Mathematics, Navarasam Arts and ScienceCollegefor Women,Arachalur, Erode District, Tamil Nadu, India.

3 Department of Mathematics,P. M. Thevar College, Usilampatti,MaduraiDistrict, Tamil Nadu, India.

Abstract: Levine [7] introduced the notion of g-closed sets and further proper ties of g-closed sets are investigated. In 1982, the notions of đ?œ”-open and đ?œ”-closed sets were introduced and studied by Hdeib [5]. Khalid Y. Al-Zoubi [6] introduced the notion of gđ?œ”-closed sets and further properties of gđ?œ”-closed sets are investigated. In this paper, we introduce the notion of mgđ?œ”-closed sets and obtain the unified characterizations for certain families of subsets between closed sets and gđ?œ”-closed sets.

Key words and phrases:gđ?œ”-closed set, m-structure, m-space, mgđ?œ” -closed set.

I. Introduction In 1970, Levine [7] introduced the notion of generalized closed (g-closed) sets in topological spaces. In 1982, Hdeib [5] introduced the notion of đ?œ”-closed sets in topo-logical spaces. Recently, many variations of g-closed sets are introduced and inves-tigated. One among them is g đ?œ”-closed sets which were introduced by Khalid Y. Al-Zoubi [6]. In 2006, Noiri and Popa [11] introduced the notion of mg⋆closed sets and studied the basic properties, characterizations and preservation properties. Also, they de fined several subsets which lie between closed sets and g-closed sets.In this paper, we introduce the notion of mgđ?œ”-closed sets and obtain the basic properties, characterizations and preservation properties.

In the last section, we define several new subsets which lie between closed sets and gđ?œ”-closed sets.

II. Preliminaries Let (X,đ?œ? ) be a topological space and A a subset of X. The closure of A and the interior of A are denoted by cl(A) and int(A), respectively. A subset A is said to be regular open if int(cl(A))=A. The finite union of regular open sets is said to beđ?œ‹-open.

Definition 2.1:A subset A of a topological space (X,đ?œ?) is said to beđ?›ź-openifA ⊂int(cl(int(A))). The complement of an đ?›ź-open set is said to be đ?›ź-closed.

Note: The family of all đ?›ź-open (resp. regular open, đ?œ‹--open) sets in X is denoted byđ?œ? Îą(resp. RO(X), đ?œ‹O(X)).

Definition 2.2:A subset A of a topological space (X, đ?œ?) is said to be g-closed [7](resp. g⋆-closed [20] or strongly g-closed [18], đ?œ‹g-closed [4], rg-closed [14])if cl(A)⊂U whenever A ⊂U and U is open (resp. gopen, đ?œ‹-open, regular open) in (X, đ?œ? ). The complements of the above closed sets are called their respective open sets.

The family of all g-open sets in (X, đ?œ? ) is denoted by gO(X). The g-closure (resp.Îą-closure) of a subset A of X, denoted by gcl(A) (resp.Îącl(A)), is defined to be theintersection of all g-closed sets (resp. Îą-closed sets) containing A. Definition 2.3:A subset A of a topological space (X,đ?œ?) is said to beđ?›źg-closed [8](resp. g#đ?›ź-closed [13], đ?œ‹gđ?›ź-closed [2], rđ?›źg-closed [10]) if đ?›źcl(A) ⊂ U whenever A ⊂U and U is open (resp. g-open, đ?œ‹-open, regular open) in (X,đ?œ? ).

The complements of the above closed sets are called their respective open sets.

Definition 2.4[21]:LetHbe a subset of a space (X,đ?œ?), a point p inXis called acondensation point of H if for each open set U containing p, U∊ H is uncountable.

Definition 2.5[5]:A subsetHof a space (X, đ?œ? ) is called đ?œ” -closed if it contains allits condensation points.The complement of an đ?œ”-closed set is called đ?œ”-open.It is well known that a subset W of a space (X,đ?œ?) is đ?œ”-open if and only if for eachx ∈ W, there exists U ∈đ?œ?such that x ∈ U and U−W is countable.The familyofall đ?œ”-open sets, denoted by đ?œ”, is a topology on X, which is finer than đ?œ?. The interior and closure operator in (X, đ?œ”) are denoted by đ?‘–đ?‘›đ?‘Ąđ?œ” and đ?‘?đ?‘™đ?œ” respectively.

Lemma 2.1 [5]: LetHbe a subset of a space (X,đ?œ?). Then DOI: 10.9790/5728-1302020915

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Between Closed Sets and Gđ?œ”-Closed Sets (1) H is đ?œ”-closed in X if and only if H =đ?‘?đ?‘™đ?œ” (H).

(2) đ?‘?đ?‘™đ?œ” (X\H) = X\đ?‘–đ?‘›đ?‘Ąđ?œ” (H).

(3) đ?‘?đ?‘™đ?œ” (H) is đ?œ”-closed in X. (4) x ∈đ?‘?đ?‘™đ?œ” (H) if and only if H∊ G ≠ ∅ for each đ?œ”-open set G containing x.

(5) đ?‘?đ?‘™đ?œ” (H)⊂ cl(H).

(6) int(H)⊂ đ?‘–đ?‘›đ?‘Ąđ?œ” (H).

Definition 2.6:LetAbe a subset of a space (X,đ?œ?). ThenAis said to be (1) (2) (3) (4)

g⋆đ?œ”-closed [17] if đ?‘?đ?‘™đ?œ” (A)⊂U whenever A ⊂ U and U is g-open in (X,đ?œ?). gđ?œ”-closed [6] if đ?‘?đ?‘™đ?œ” (A)⊂ U whenever A ⊂ U and U is open in (X, đ?œ?). đ?œ‹gđ?œ”-closed [3] if đ?‘?đ?‘™đ?œ” (A)⊂ U whenever A ⊂U and U is đ?œ‹-open in (X,đ?œ?). rđ?œ”-closed [1] if đ?‘?đ?‘™đ?œ” (A)⊂ U whenever A ⊂ U and U is regular open in (X, đ?œ? ).

Remark 2.1 [17]:For a subset of a topological space, we obtain the following implications: closed

→

↓ đ?œ”-closed

g⋆-closed

→

↓ →

g⋆ đ?œ”-closed

g-closed ↓

→

g đ?œ”-closed

None of the above implications is reversible.

Lemma 2.2 [6]:The open image of anđ?œ”-open set isđ?œ”-open.

Throughout the present paper, (X, đ?œ? ) and (Y, đ?œŽ) always denote topological spaces andf : (X, đ?œ?) → (Y, đ?œŽ) presents a function.

III. m-Structures Definition 3.1:A subfamily mXof the power set℘(X) of a nonempty set X is calleda Minimal Structure (briefly m-Structure) [15] on X if ∅∈ mX and X ∈ mX .

By (X, mX ), we denote a nonempty set X with a minimal structure mX on X and call it an m-space. Each member of mX is said to be mX -open (or briefly m-open) and the complement of an mX -open set is said to be mX -closed (or briefly m-closed).

Remark 3.1: Let (X, đ?œ?) be a topological space. Then the families,đ?œ?đ?œ” , đ?œ?Îą,đ?œ?, RO(X),đ?œ‹O(X) and gO(X) are all m-structures on X. Definition 3.2:Let (X, mX) be an m-space. For a subset A of X, the mX-closureof A and the mX -interior of A are defined in [9] as follows: (1) mX -cl(A) = ∊ {F : A ⊂ F, X−F ∈ mX }, (2) mX -int(A) = âˆŞ{U : U ⊂ A, U ∈ mX }.

Remark 3.2: Let (X, đ?œ?) be a topological space and A a subset of X. If m X=đ?œ?(resp.đ?œ?đ?œ” ,đ?œ?Îą, gO(X)), then we have mX -cl(A)=cl(A) (resp. đ?‘?đ?‘™đ?œ” (A), Îącl(A), gcl(A)).

Lemma 3.1 [15]: Let (X, mX) be an m-space and A a subset of X. Then x∈mX-cl(A)if and only if U∊ A ≠∅ for every U ∈ mX containing x.

Definition 3.3[9]: An m-structure mXon a nonempty set X is said to have property(B) if the union of any family of subsets belonging to mX belongs to mX .

Remark 3.3: Let (X, đ?œ?) be a topological space. Then the familiesđ?œ?đ?œ”,đ?œ?Îą,đ?œ?andđ?œ‹O(X) are all m-structures with property (B).

Lemma 3.2 [16]:Let X be a nonempty set and mXan m-structure on X satisfyingproperty (B). For a subset A of X, the following properties hold: (1) A ∈ mX if and only if mX -int(A)=A, (2) A is m-closed if and only if mX -cl(A)=A, (3) mX -int(A) ∈ mX and mX -cl(A) is m-closed.

Definition 3.4 [11]: Let (X, đ?œ?) be a topological space and mXan m-structure on X.A subset A of X is said to be DOI: 10.9790/5728-1302020915

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Between Closed Sets and Gđ?œ”-Closed Sets (1) mg⋆-closed if cl(A) ⊂ U whenever A ⊂U and U is mX -open, (2) mg⋆-open if its complement is mg⋆-closed.

Proposition 3.1 [11]: Letđ?œ? ⊂mX. Then the following implications hold: closed → mg⋆-closed → g-closed Proposition 3.2:Letđ?œ? ⊂mX. Then the following implications hold: closed → mg⋆-closed → g-closed → gđ?œ”-closed Proof: It follows from Remark 2.1.

Theorem 3.1 [11]: Letđ?œ? ⊂mXand mXhave property (B). A subset A of X ismg ⋆-closed if and only if cl(A)−A does not contain any nonempty m-closed set.

Theorem 3.2 [11]:LetmXhave property (B). A subset A of X ismg⋆-closed ifand only if mX -cl({x}∊ A ≠đ?œ™ for each x ∈ cl(A).

IV. mgđ??Ž-Closed Sets In this section, let (X, đ?œ? ) be a topological space and mX an m-structure on X. We obtain several basic properties of mgω-closed sets.

Definition 4.1:Let (X, đ?œ?) be a topological space and mXan m-structure on X. Asubset A of X is said to be (1) mgω-closed if clđ??Ž(A) ⊂ U whenever A ⊂U and U is mX -open, (2) mgω-open if its complement is mgđ?œ”-closed.

Remark 4.1: Let (X, đ?œ?) be a topological space and A a subset of X. If m X=gO(X)(resp. đ?œ?, đ?œ‹O(X), RO(X)) and A is mgω-closed, then A is g⋆đ?œ”-closed(resp.gω-closed,đ?œ‹gđ?œ”-closed, rgđ?œ”-closed).

Proposition 4.1: Letđ?œ? ⊂mX. Then the following implications hold:closed→ đ?œ”-closed → mgđ?œ”-closed → gđ?œ”-closed Proof:It is obvious that every closed set isω-closed [1,5] and everyđ?œ”-closed set is mgđ?œ”-closed by Lemma 2.1(1). Suppose that A is an mgđ?œ”-closed set. Let A⊂Uand U∈đ?œ?. Since đ?œ? ⊂mX , đ?‘?đ?‘™đ?œ” (A) ⊂U and hence A is gđ?œ”-closed.

Proposition 4.2: LetmXbe anm-structure onXin the topological space (X, đ?œ?). IfA and B are mgđ?œ”-closed, then AâˆŞB is mgđ?œ”-closed.

Proof:Let A âˆŞ B ⊂ U and U∈mX.Then A ⊂ U and B ⊂ U. Since A and Bare mg đ?œ” -closed, we have đ?‘?đ?‘™đ?œ” (AâˆŞB)=đ?‘?đ?‘™đ?œ” (A) âˆŞ đ?‘?đ?‘™đ?œ” (B) ⊂U. Therefore, AâˆŞB is mgđ?œ”-closed.

Proposition 4.3:LetmXbe anm-structure onXin the topological space (X, đ?œ?). IfA is mgđ?œ”-closed and mopen, then A is đ?œ”-closed.

Proof:This is obvious.

Proposition 4.4:Let (X, mX ) be an m-space andA⊆X. If A ismgđ?œ”-closed andA ⊂B ⊂ đ?‘?đ?‘™đ?œ” (A), then B is mgđ?œ”-closed.

Proof:Let B⊂U and U∈mX. Then A⊂U and A is mgđ?œ”-closed. Hence đ?‘?đ?‘™đ?œ” (B) =đ?‘?đ?‘™đ?œ” (A) ⊂ U and B is mgđ?œ”-closed.

Definition 4.2: [12] Let (X, mX) be an m-space and A a subset of X. The mX-frontier of A, mX -Fr(A), is defined as follows: mX -Fr(A) = mX -cl(A) ∊ mX -cl(X−A).

Proposition 4.5:If A is ađ?‘šđ?‘”đ?œ”-closed subset of X and A⊂U∈mX, then mX-Fr(U)⊂ iđ?‘›đ?‘Ąđ?œ” (X−A).

Proof: Let A be mgđ?œ”-closed and A⊂U∈mX. Then đ?‘?đ?‘™đ?œ” (A)⊂U. Suppose that x∈ mX -Fr(U).

Since U ∈mX , mX -Fr(U)= mX -cl(U)∊ mX -cl(X−U) = mX -cl(U) ∊ (X−U) = mX -cl(U)-U. Therefore, x∉U and x ∉ đ?‘?đ?‘™đ?œ” (A).

This shows that x ∈đ?‘–đ?‘›đ?‘Ąđ?œ” (X−A) and hence mX -Fr(U) ⊂ đ?‘–đ?‘›đ?‘Ąđ?œ” (X−A).

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Between Closed Sets and Gđ?œ”-Closed Sets Proposition 4.6:In the m-space (X, mX ), a subset A of X ismgđ?œ”-open if and onlyif F ⊂ đ?‘–đ?‘›đ?‘Ąđ?œ” (A) whenever F ⊂ A and F is m-closed.

Proof: Suppose that A is mgđ?œ”-open. Let F⊂A and F be m-closed.

Then X−A⊂X−F ∈ mX and X−A is mgđ?œ”-closed.

Therefore, we have X−đ?‘–đ?‘›đ?‘Ąđ?œ” (A) = đ?‘?đ?‘™đ?œ” (X−A) ⊂ X−Fand hence F ⊂ đ?‘–đ?‘›đ?‘Ąđ?œ” (A).

Conversely, let X−A ⊂ G and G ∈ mX .

Then X−G ⊂ A and X−G is m-closed.

By the hypothesis, we have X−G ⊂ đ?‘–đ?‘›đ?‘Ąđ?œ” (A) and hence đ?‘?đ?‘™đ?œ” (X−A) = X−đ?‘–đ?‘›đ?‘Ąđ?œ” (A)⊂ G. Therefore, X−A is mgđ?œ”-closed and A is mgđ?œ”-open.

Corollary 4.1:Letđ?œ? ⊂mX. Then the following properties hold: (1) Every open set is mgđ?œ”-open and every mgđ?œ”-open set is gđ?œ”-open, (2) If A and B are mgđ?œ”-open, then A ∊ B is mgđ?œ”-open, (3) If A is mgđ?œ”-open and m-closed, then A is đ?œ”-open, (4) If A is mgđ?œ”-open and intđ?œ”(A) ⊂B ⊂ A, then B is mgđ?œ”-open.

Proof:This follows from Propositions 4.1, 4.2, 4.3 and 4.4.

Proposition 4.7:Everymg⋆-closed set ismgđ?œ”-closed.

Proof:It follows from Lemma 2.1(5).

Proposition 4.8:Letđ?œ? ⊂mX. Then everymg⋆-closed set isgđ?œ”-closed.

Proof:It follows from Propositions 4.1 and 4.7.

Proposition 4.9: Letđ?œ? ⊂mX. Then the following implications hold: closed→ mg⋆-closed → mgđ?œ”-closed → gđ?œ”-closed Proof:It follows from Propositions 3.2, 4.1 and 4.7.

V. Characterizations Of mgđ??Ž-Closed Sets In this section, let (X, đ?œ? ) be a topological space and mX an m-structure on X. We obtain some characterizations of mgđ?œ”-closed sets.

Theorem 5.1: A subset A of X ismgđ?œ”-closed if and only if đ?‘?đ?‘™đ?œ” (A)∊ F =∅wheneverA ∊ F = ∅ and F is mclosed.

Proof:Suppose that A is mgđ?œ”-closed. Let A∊F =∅and F be m-closed. Then A⊂X−F ∈ mXandđ?‘?đ?‘™đ?œ” (A)⊂ X−F. Therefore, we have đ?‘?đ?‘™đ?œ” (A) ∊ F = ∅.

Conversely, let A⊂ U and U ∈mX. Then A ∊ (X−U) = ∅ and X−U is m-closed.

By the hypothesis, đ?‘?đ?‘™đ?œ” (A) ∊ (X−U) = ∅ and hence đ?‘?đ?‘™đ?œ” (A )⊂U. Therefore, A is mgđ?œ”-closed.

Theorem 5.2:Let đ?œ?đ?œ” ⊂ mXand mXhave property (B). A subset A of X ismg đ?œ” -closed if and only if đ?‘?đ?‘™đ?œ” (A)−A does not contain any nonempty m-closed set.

Proof:Suppose that A is mgđ?œ”-closed. Let F⊂ đ?‘?đ?‘™đ?œ” (A)−A and F be m-closed. Then F ⊂ đ?‘?đ?‘™đ?œ” (A) and A ⊂X−F ∈ mX. Henceđ?‘?đ?‘™đ?œ” (A) ⊂ X−F. Therefore, we have F ⊂X−đ?‘?đ?‘™đ?œ” (A).

Hence F ⊂ đ?‘?đ?‘™đ?œ” (A) ∊ (X−đ?‘?đ?‘™đ?œ” (A)) = ∅.

Conversely, suppose that A is not mgđ?œ”-closed. Then ∅ ≠đ?‘?đ?‘™đ?œ” (A)−U for some U ∈ mX containing A. Since đ?œ?đ?œ” ⊂mX and mX has property (B), đ?‘?đ?‘™đ?œ” (A)−U is m-closed.

Moreover, đ?‘?đ?‘™đ?œ” (A)−U ⊂ đ?‘?đ?‘™đ?œ” (A)−A. Thus đ?‘?đ?‘™đ?œ” (A)−A contains a nonempty m-closed set which is a contradiction.

Hence A is mgđ?œ”-closed.

Theorem 5.3:Letđ?œ?đ?œ” ⊂mXand mXhave property (B). A subset A of X ismgđ?œ”-closed if and only if đ?‘?đ?‘™đ?œ” (A)−A is mgđ?œ”-open.

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Between Closed Sets and Gđ?œ”-Closed Sets Proof: Suppose that A is mgđ?œ”-closed. Let F⊂ đ?‘?đ?‘™đ?œ” (A)−A and F be m-closed.

ByTheorem 5.2, we have F = ∅and F ⊂int đ?œ”(đ?‘?đ?‘™đ?œ” (A)−A).

It follows from Proposition 4.6, đ?‘?đ?‘™đ?œ” (A)−A is mgđ?œ”-open.

Conversely, let A⊂U and U ∈mX .

Then đ?‘?đ?‘™đ?œ” (A) ∊ (X−U) ⊂ đ?‘?đ?‘™đ?œ” (A)−A and đ?‘?đ?‘™đ?œ” (A)−A is mgđ?œ”-open.

Since đ?œ?đ?œ” ⊂mX and mXhas property (B), đ?‘?đ?‘™đ?œ” (A) ∊ (X−U) is m-closed and by Proposition 4.6, đ?‘?đ?‘™đ?œ” (A) ∊ (X−U) ⊂đ?‘–đ?‘›đ?‘Ąđ?œ” (đ?‘?đ?‘™đ?œ” (A)−A).

Now,đ?‘–đ?‘›đ?‘Ąđ?œ” (đ?‘?đ?‘™đ?œ” (A)−A) = đ?‘–đ?‘›đ?‘Ąđ?œ” (đ?‘?đ?‘™đ?œ” (A)) ∊ intđ?œ”(X−A) ⊂ đ?‘?đ?‘™đ?œ” (A) ∊ đ?‘–đ?‘›đ?‘Ąđ?œ” (X−A) = đ?‘?đ?‘™đ?œ” (A) ∊ (X−đ?‘?đ?‘™đ?œ” (A)) = ∅.

Thus đ?‘?đ?‘™đ?œ” (A) ∊ (X−U) =∅ and hence đ?‘?đ?‘™đ?œ” (A) ⊂U. This shows that A is mgđ?œ”-closed.

Theorem 5.4:Let mXhave property (B). A subset A of X ismgđ?œ”-closed if andonly if mX -cl({x} ∊ A ≠∅ for each x ∈ đ?‘?đ?‘™đ?œ” (A).

Proof:Suppose mX-cl({x}) ∊ A =∅for some x∈clđ?œ”(A).

By Lemma 3.2, mX-cl({x})is m-closed and A ⊂X−(mX -cl({x})) ∈ mX .

Ifđ?‘?đ?‘™đ?œ” (A) ⊂X− (mX -cl({x})) then x∈ đ?‘?đ?‘™đ?œ” (A) ⊂X−(mX -cl({x}))⊂X−{x} is a contradiction.

Thusđ?‘?đ?‘™đ?œ” (A) ⊈ X−(mX- cl({x}) and hence A in not mgđ?œ”-closed.

Conversely, suppose that A is not mgđ?œ”-closed.

Then there exists U ∈ mX such that A ⊂ U, but đ?‘?đ?‘™đ?œ” (A) ⊈ U. So there exists x ∈ đ?‘?đ?‘™đ?œ” (A) but x ∉U. Then x ∈ Uc which is mX -closed.

Thus mX -cl({x}) ⊂mX -cl(Uc) = Uc. This implies mX -cl({x})∊ U = ∅.

Hence mX -cl({x})∊A ⊂ mX -cl({x}) ∊ U = ∅.

Thus there exists x ∈ đ?‘?đ?‘™đ?œ” (A) such that mX -cl({x})∊ A = ∅. This proves the converse.

Corollary 5.1: Letđ?œ?đ?œ” ⊂mXand mXhave property (B). For a subset A of X, thefollowing properties are equivalent: (1) A is mg đ?œ”-open, (2) A−đ?‘–đ?‘›đ?‘Ąđ?œ” (A) does not contain any nonempty m-closed set, (3) A−đ?‘–đ?‘›đ?‘Ąđ?œ” (A) is mg đ?œ”-open, (4) mX-cl({x}∊(X−A) =Ě¸âˆ…for each x∈X−đ?‘–đ?‘›đ?‘Ąđ?œ” (A).

Proof:This follows from Proposition 4.6 and Theorems 5.2, 5.3 and 5.4.

VI. Preservation Theorems Definition 6.1 [11]:A function f : (X, mX)→(Y, mY) is said to be (1) M-continuous if for each x∈ X and each V∈ mY containing f(x), there existsU∈ mXcontaining x such that f(U) ⊂ V. (2) M-closed if for each m-closed set F of (X, mX ), f(F) is m-closed in (Y, mY ).

Theorem 6.1[15]: Let mXbe an m-structure on X with property (B) and mYbea minimal structure on Y. Let f :(X, mX ) → (Y, mY ) be a function. Then the following are equivalent: (1) f is M-continuous, (2) f -1(V)∈ mX for every V ∈ mY .

Lemma 6.1[11]: A function f : (X, mX)→(Y, mY) is M-closed if and only if foreach subset B of Y and each U ∈ mX containing f -1(B), there exists V ∈ mY such that B ⊂ V and f -1(V) ⊂ U. Theorem 6.2: If f : (X,đ?œ?)→(Y,đ?œŽ) is closed and f : (X, mX)→(Y, mY) is M-continuous, where mX has property (B), then f(A) is mgđ?œ”-closed in (Y, mY ) for each mgđ?œ”-closed set A of (X, mX ).

Proof:Let A be any mgđ?œ”-closed set of (X, mX) and f(A)⊂V∈mY. Since mXhas property (B), A ⊂f -1(V) ∈ mX by Theorem 6.1.

Since A is mgđ?œ”-closed, đ?‘?đ?‘™đ?œ” (A)⊂f-1(V) and f(đ?‘?đ?‘™đ?œ” (A))⊂V. Since f is closed, by Lemma 2.2, đ?‘?đ?‘™đ?œ” (f(A))⊂ f(đ?‘?đ?‘™đ?œ” (A))⊂ V. Hence f(A) is mgđ?œ”-closed in (Y, mY ).

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Between Closed Sets and Gđ?œ”-Closed Sets Definition 6.2 [6]: A function f : (X, đ?œ?)→(Y,đ?œŽ) is calledđ?œ”-irresolute if f-1(B) isđ?œ”-open in (X, đ?œ?) for every đ?œ”-open set B of (Y,đ?œŽ).

Theorem 6.3:If f : (X, đ?œ?)→(Y,đ?œŽ) isđ?œ”-irresolute and f : (X, mX)→(Y, mY) isM-closed, then f -1(B) is mgđ?œ”closed in (X, mX ) for each mgđ?œ”-closed set B of (Y, mY ).

Proof:Let B be any mgđ?œ”-closed set of (Y, mY) and f-1(B)⊂U∈mX. Since f isM-closed, by Lemma 6.1, there exists V ∈ mYsuch that B ⊂V and f -1(V) ⊂ U. Since B is mgđ?œ”-closed in Y, đ?‘?đ?‘™đ?œ” (B) ⊂ V and since f is đ?œ”-irresolute, đ?‘?đ?‘™đ?œ” (f-1(B)) ⊂f -1(đ?‘?đ?‘™đ?œ” (B)) ⊂f -1(V) ⊂ U. Hence f -1(B) is mgđ?œ”-closed in (X, mX ).

VII.

New Forms Of mgđ??Ž-Closed Sets

In a topological space (X, đ?œ?), from the definitions, we obtain the following diagram.

Diagram – I g*-open

ď‚žď‚Žď€

ď °g-open

ď‚žď‚Žď€

rg-open

g# ď Ą-open

ď‚žď‚Žď€

ď Ąg-open

ď‚žď‚Žď€

ď °gď Ą-open

ď‚žď‚Žď€

rď Ąg-open

ď‚žď‚Žď€

g-open

ď‚žď‚Žď€

ď‚žď‚Žď€

In (X, đ?œ?) we denote the collection of all g-open (resp. g⋆-open, đ?œ‹g-open, rg-open, g Îą-open, Îąg-open, #

đ?œ‹ gÎą-open, rgÎą-open) sets by gO(X) (resp. g⋆O(X), đ?œ‹ gO(X), rgO(X), g#ÎąO(X),ÎągO(X), đ?œ‹ gÎąO(X), rgÎąO(X)). These collections of are all m-structure onX. Using these m-structures gO(X) g⋆O(X), đ?œ‹gO(X), rgO(X), g#ÎąO(X), ÎągO(X), đ?œ‹gÎąO(X), rgÎąO(X)) for a subset A, we define new types of gđ?œ”-closed sets as follows.

Dfinition 7.1:A subset A of a topological space (X, đ?œ?) is said to beg⋆gđ?œ”-closed(resp. g⋆đ?œ”-closed [18], đ?œ‹ggđ?œ”-closed, rggđ?œ”-closed, g#Îągđ?œ”-closed, Îąggđ?œ”-closed, đ?œ‹gÎągđ?œ”-closed, rÎąggđ?œ”-closed) if clđ?œ”(A) ⊂ U whenever A ⊂ U and U is g⋆-open (resp. g-open, đ?œ‹g-open, rg-open, g#Îą-open, Îąg-open, đ?œ‹gÎą-open, rÎąg-open) in (X, đ?œ? ). By Diagram I and Definition 7.1, we have the following diagram: Diagram - II

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