On generalized closed sets in generalized topological spaces - SciELO

CUBO A Mathematical Journal Vol.18, N¯o 01, (27–45). December 2016

On generalized closed sets in generalized topological spaces B. K. Tyagi1 , Harsh V. S. Chauhan2 1

Department of Mathematics, Atmaram Sanatan Dharma College, University of Delhi, New Delhi-110021, India. 2

Department of Mathematics, University of Delhi, New Delhi-110007, India

[email protected], [email protected] ABSTRACT In this paper, we introduce several types of generalized closed sets in generalized topological spaces (GTSs). Their interrelationships are investigated and several characterizations of µ-T0 , µ-T1 , µ-T1/2 , µ-regular, µ-normal GTSs and extremally µ-disconnected GTSs are obtained.

RESUMEN En este art´ıculo introducimos varios tipos de conjuntos cerrados generalizados en espacios topol´ ogicos generalizados (GTSs). Sus interrelaciones son investigadas y varias caracterizaciones de GTSs µ-T0 , µ-T1 , µ-T1/2 , µ-regulares, µ-normales y extremalmente µ-disconexos son obtenidas. Keywords and Phrases: Generalized topological spaces, generalized closed sets, extremally µ-disconnectedness, Separation axioms. 2010 AMS Mathematics Subject Classification: 54A05, 54D15.

28

1

B. K. Tyagi, Harsh V. S. Chauhan

CUBO 18, 1 (2016)

Introduction

Several types of generalized closed sets are investigated in the literature of topological spaces [3, 5, 6, 7, 16, 18, 19, 20, 23, 24, 26, 27, 28, 29, 30, 35, 38, 37, 39, 43, 44, 48, 49]. Their relationship with one another is shown by a diagram in Benchalli et al. [4] and Dontchev [17]. Using the concept of generalized closed sets, several separation axioms [17, 21] are introduced which are found to be useful in the study of digital topology (digital line) [25]. Cao et al. [9] obtained several characterizations of extremally disconnectedness in terms of generalized closed sets. The purpose of this paper is to show that these diagrams can be obtained in the setting of generalized topological spaces (GTSs) introduced by Cs´ asz´ ar [11]. Let X be a set and P(X) be the power set of X. A subset µ of P(X) is called generalized topology (GT) on X if µ is closed under arbitrary unions and in that case (X, µ) is called a generalized topological space (GTS). The elements of µ are called µ-open sets and their complements are called µ-closed sets. The closure of A, denoted by cµ A, is the intersection of µ-closed sets containing A. The interior of A, denoted by iµ A, is the union of µ-open sets contained in A. In a GTS (X, µ), we define Mµ = ∪{U : U ∈ µ}. A GTS (X, µ) is called strong if Mµ = X. The notions of various generalized closed sets depend on several types of stronger or weaker forms of open sets, for example, regular open set [44], semi open set [26], preopen set [31], semi preopen set [2], α-open set [36], θ-open set [50], δ-open set [50], π-open set [20] etc. All these notions are extended to the setting of generalized topological spaces. The concept of µ-T1/2 GTS depends in turn on the concept of a generalized closed set. We explore the relationship of generalized closed sets with several separation axioms, µ-T0 , µ-T1 , µ-T1/2 , µ-regularity, and µ-normality [32, 33]. A concept of extremally µ-disconnectedness was introduced in [46]; A GTS (X, µ) is extremallay µ-disconnected if cµ U ∩ Mµ ∈ µ for every U ∈ µ. It may be remarked that in strong GTS, this notion concide with the notion of extremally disconnectedness in Cs´ asz´ ar [12]. Several characterizations of extremally µ-disconnectedness in terms of generalized closed sets are obtained. Section 2 contains preliminaries. In section 3, we introduce various notions of generalized closed sets and obtain several implications among them. Section 4 contains characterizations of µ-T0 , µ-T1 and µ-T1/2 GTSs. In section 5, we study the characterization of µ-regularity and µ-normality. Section 6 obtains some characterizations of extremally µ-disconnected GTSs.

2

Preliminaries

Let (X, µ) be a GTS and A ⊆ X. Ac denotes the complement of A in X. The collection of all µ-closed sets in X is denoted by Ω.

CUBO 18, 1 (2016)

On generalized closed sets in generalized topological spaces.

29

Theorem 2.1. Let (X, µ) be a GTS and A, B ⊆ X. Then the following statements hold. (i) x ∈ cµ A if and only if x ∈ U ∈ µ implies U ∩ A ̸= ∅. (ii) cµ A = cµ (A ∩ Mµ ). (iii) cµ A = X − iµ (X − A). (iv) If U, V ∈ µ and U ∩ V = ∅ then cµ U ∩ V = ∅ and U ∩ cµ V = ∅. (v) Mµ − cµ A = X − cµ A. (vi) iµ A = iµ (A ∩ Mµ ). (vii) iµ (cµ A − A) = ∅. (viii) cµ and iµ are monotone: A ⊆ B implies cµ A ⊆ cµ B (respectively iµ A ⊆ iµ B), idempotent cµ cµ A = cµ A (respectively iµ iµ A = iµ A), cµ is enhancing (A ⊆ cµ A), iµ is contracting (iµ A ⊆ A). Proof. (vii). If x ∈ iµ (cµ A − A) then there exists a U ∈ µ such that x ∈ U ⊆ cµ A − A. Then x ∈ U ⊆ cµ A and U ∩ A = ∅. Now x ∈ U ⊆ cµ A implies U ∩ A ̸= ∅, a contradiction.

Let (X, µ) be a GTS and Y ⊆ X. Then the collection µY = {U ∩ Y : U ∈ µ} is a GT on Y and (Y, µY ) is called a generalized subspace of (X, µ). It may be noted that cµY A = cµ A ∩ Y for any A ⊆ Y. Thus, a set A ⊆ Y is µY -closed if and only if it is the intersection with Y of a µ-closed set. Definition 2.2. A subset A of a GTS (X, µ) is called (i) µ-regular open (or roµ -open) if iµ cµ A = A. (ii) µ-semi open (or sµ -open) if A ⊆ cµ iµ A ∩ Mµ . (iii) µ-preopen (or pµ -open ) if A ⊆ iµ cµ A. (iv) µ-α-open (or αµ -open) if A ⊆ iµ cµ iµ A. (v) µ-semi preopen (or spµ -open) if A ⊆ cµ iµ cµ A ∩ Mµ . (vi) µ-θ-closed (or θµ -closed) [34] if A = γθ A, where γθ (A) = {x ∈ X : cµ G ∩ Mµ ∩ A ̸= ∅ for all G ∈ µ, x ∈ G}. The complement of a θµ -closed set is called µ-θ-open (θµ -open). (vii) µ-δ-closed (or δµ -closed) [12] if A = cδ A, where cδ A = {x ∈ X : iµ cµ U ∩ A ̸= ∅ for U ∈ µ and x ∈ U}. The complement of a δµ -closed set is called µ-δ-open (δµ -open).

30


CUBO 18, 1 (2016)

(viii) µ-π-open (or πµ -open) if A is the union of finitely many µ-regular open sets. (ix) µ-regular semi open (or rsµ -open) if there exists a µ-regular open set U such that U ⊆ A ⊆ cµ U ∩ Mµ . The collections of all µ-( ) sets in (i) to (ix) of the above definitions are denoted by roµ , sµ , pµ , αµ , spµ , θµ , δµ , πµ , rsµ respectively. The complements of the sets in the above definitions are named similarly by replacing the word “open” by “closed”, for example µ-semi closed (or sµ closed) for the complement of a sµ -open set and vice-versa. It follows using Theorem 2.1, a subset A of GTS (X, µ) is a regular µ-closed (or roµ -closed) if and only if cµ iµ A = A; A set A is µ-semi open if and only if cµ A = cµ iµ A and A ⊆ Mµ . A is sµ -closed if and only if iµ cµ A ⊆ A and X − Mµ ⊆ A; A is pµ -closed if and only if cµ iµ A ⊆ A; A is αµ -closed if and only if cµ iµ cµ A ⊆ A; A is spµ -closed if iµ cµ iµ A ⊆ A and X − Mµ ⊆ A. For any set A, cµ iµ cµ A is αµ -closed. Also if A ∈ rsµ then A ∈ sµ but not conversely. Theorem 2.3. [46] For a GTS (X, µ), θµ , αµ , sµ , pµ and spµ are GTSs and (i) θµ ⊆ µ ⊆ αµ ⊆ sµ ⊆ spµ , (ii) αµ ⊆ pµ ⊆ spµ . Theorem 2.4. A is αµ -open if and only if A ∈ sµ ∩ pµ . Proof. If A ⊆ iµ cµ iµ A then A ⊆ cµ iµ A, A ⊆ Mµ and A ⊆ iµ cµ A. So A ∈ sµ ∩ pµ . Conversely, let A ∈ sµ ∩ pµ . Then A ⊆ cµ iµ A ∩ Mµ . Therefore, cµ A ⊆ cµ iµ A. Also A ⊆ iµ cµ A. Therefore, A ⊆ iµ cµ iµ A A subset A of a GTS (X, µ) is µ-nowhere dense if iµ cµ A = ∅. Lemma 2.5. Let x be a point in a GTS (X, µ). Then {x} is µ-nowhere dense or pµ -open. Proof. Suppose {x} is not µ-nowhere dense. Then iµ cµ {x} ̸= ∅. Then x ∈ iµ cµ {x}. So {x} ⊆ iµ cµ {x}. Lemma 2.6. If {x} is µ-nowhere dense in a GTS (X, µ) then {x} ∪ (X − Mµ ) is αµ -closed. Proof. cµ iµ cµ ({x} ∪ (X − Mµ )) = cµ iµ cµ {x} = cµ ∅ = X − Mµ . So cµ iµ cµ ({x} ∪ (X − Mµ )) ⊆ {x} ∪ (X − Mµ ). Lemma 2.7. For a subset A containing X − Mµ , csµ A = A ∪ iµ cµ A. Proof. Since csµ A is sµ -closed, iµ cµ (csµ A) ⊆ csµ A. On the other hand iµ cµ (A ∪ iµ cµ A) ⊆ iµ cµ cµ A = iµ cµ A. Therefore, iµ cµ (A ∪ iµ cµ A) ⊆ A ∪ iµ cµ A. Since X − Mµ ⊆ A, A ∪ iµ cµ A is sµ - closed.

CUBO 18, 1 (2016)


31

Lemma 2.8. For a subset A, cαµ A = A ∪ cµ iµ cµ A. Proof. Since cαµ A is αµ -closed, cµ iµ cµ cαµ A ⊆ cαµ A. Therefore, A ∪ cµ iµ cµ A ⊆ cαµ A. On the other hand cµ iµ cµ (A ∪ cµ iµ cµ A) ⊆ cµ iµ cµ cµ A = cµ iµ cµ A ⊆ A ∪ cµ iµ cµ A. Thus, A ∪ cµ iµ cµ A is αµ -closed set containing A. Lemma 2.9. For a subset A, A ∪ iµ cµ iµ A ⊆ cspµ A. Proof. iµ cµ iµ A ⊆ iµ cµ iµ (cspµ )A ⊆ cspµ A, since cspµ A is spµ -closed.

3

Various type of generalized closed sets

Definition 3.1. Let (X, µ) be a GTS. A subset A of X containing X − Mµ is called (i) a µ-generalized closed (or gµ -closed) set if cµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ µ. The complement of a gµ -closed set is called µ-generalized open (or gµ -open). The set of all gµ -open sets is denoted by gµ . (ii) a µ-semi generalized closed (or sgµ -closed) set if csµ A∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ sµ . (iii) a µ-generalized semi closed (or gsµ -closed) set if csµ A∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ µ. (iv) a µ-generalized α-closed (or gαµ -closed) set if cαµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ αµ . (v) a µα-generalized closed (or αgµ -closed) set if cαµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ µ. (vi) a µ-generalized semi preclosed (or gspµ -closed) set if cspµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ µ. (vii) a µ-regular generalized closed (or rgµ -closed) set if cµ A∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ roµ . (viii) a µ-generalized preclosed (or gpµ -closed) set if cpµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ µ. (ix) a µ-generalized preregular closed (or gprµ -closed) set if cpµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ roµ . (x) a µ-θ-generalized closed (or θgµ -closed) set if γθµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ µ. (xi) a µ-δ-generalized closed (or δgµ -closed) set if cδµ A ∩ Mµ ⊆ U whenever A ⊆ U ∈ µ. (xii) a µ-weakly generalized closed (or wgµ -closed) set if cµ iµ A∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ µ. (xiii) a µ-strongly generalized closed (or gµ ∗ -closed) set if cµ A∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ gµ . (xiv) a µ-π-generalized closed (or πgµ -closed) set if cµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ πµ .

32


CUBO 18, 1 (2016)

(xv) a µ-weakly closed (or wµ -closed) set if cµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ sµ . (xvi) a µ-mildly generalized closed (or mgµ -closed) set if cµ iµ A∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ gµ . (xvii) a µ-semi-weakly generalized closed (or swgµ -closed) set if cµ iµ A ∩ Mµ ⊆ U whenever A ∩ M µ ⊆ U ∈ sµ . (xviii) a µ-regular weakly generalized closed (or rwgµ -closed) set if cµ iµ A ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ roµ . (xix) a µ-regular generalized w-closed (or rwµ -closed) set if cµ A∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ rsµ . Lemma 3.2.

(i) A ∈ gµ then A ⊆ Mµ .

(ii) µ⊆ gµ . Proof. (i) Since X − Mµ is contained in a generalized closed set A, the complement of A is contained in Mµ . (ii) Let A ∈ µ and (X − A) ∩ Mµ ⊆ U ∈ µ. Then cµ (X − A) ∩ Mµ = (X − A) ∩ Mµ ⊆ U.

Theorem 3.3. A subset A of GTS (X, µ) is gµ -closed if and only if for any µ-closed set F such that F ∩ Mµ ⊆ cµ A − A implies F ∩ Mµ = ∅. Proof. Let F be a µ-closed set such that F∩Mµ ⊆ cµ A−A . Then A∩Mµ ⊆ Fc ∈ µ. Since A is gµ closed, cµ A ∩ Mµ ⊆ Fc . That is, F ∩ Mµ ⊆ (cµ A)c . Therefore, F ∩ Mµ ⊆ (cµ A − A) ∩ (cµ A)c = ∅. Conversely, let A ∩ Mµ ⊆ U ∈ µ and if cµ A ∩ Mµ is not contained in U then cµ A ∩ Mµ ∩ Uc ̸= ∅. Since cµ A ∩ Uc is µ-closed and cµ A ∩ Uc ∩ Mµ ⊆ cµ A − A, a contradiction. Theorem 3.4. If a gµ -closed subset A of a GTS (X, µ) be such that cµ A − (A ∩ Mµ ) is µ-closed then A is µ-closed. Proof. Let A be a gµ -closed set such that cµ A − (A ∩ Mµ ) is µ-closed. Then cµ A − (A ∩ Mµ ) is µ-closed subset of itself. Since cµ A − (A ∩ Mµ ) is gµ -closed subset of itself, by Theorem 3.3 (cµ A−(A∩Mµ ))∩MµY , where Y = cµ A−(A∩Mµ ), is empty. Since MµY = (cµ A−(A∩Mµ ))∩Mµ , A is µ-closed. Theorem 3.5. If A is a gµ -closed set and A ⊆ B ⊆ cµ A then B is gµ -closed. Proof. Let B ∩ Mµ ⊆ U ∈ µ. Since A is gµ -closed and A ∩ Mµ ⊆ U, cµ A ∩ Mµ ⊆ U. Then cµ B ∩ Mµ ⊆ cµ A ∩ Mµ ⊆ U. Theorem 3.6. In a GTS (X, µ), µ= Ω if and only if (X, µ) is strong and every subset of X is gµ -closed.

CUBO 18, 1 (2016)


33

Proof. If µ= Ω then obviously (X, µ) is strong. Now if A ⊆ U ∈ µ then cµ A ⊆ cµ U = U since U ∈ µ. Conversely, let U ∈ µ. Since U is gµ -closed, cµ U ⊆ U. Thus, U is µ-closed. On the other hand if F ∈ Ω then Fc ∈ µ. Since µ ⊆ Ω, F ∈ µ.

Theorem 3.7. A subset A of Mµ of a GTS (X, µ) is gµ -open if and only if F ∩ Mµ ⊆ iµ A whenever F is µ-closed and F ∩ Mµ ⊆ A.

Proof. Let A be a gµ -open set and F be a µ-closed set such that F ∩ Mµ ⊆ A. Then X − A ⊆ X − (F ∩ Mµ ). Since (X − A) ∩ Mµ ⊆ (X − (F ∩ Mµ )) ∩ Mµ = X − F and X − A is gµ -closed, cµ (X − A)∩Mµ ⊆ X − F. Then (X−iµ A)∩Mµ ⊆ X − F. That is, F∩Mµ ⊆ (X − (X − iµ A) ∩ Mµ ) ∩ Mµ = iµ A. Conversely, let A ⊆ Mµ and (X − A) ∩ Mµ ⊆ U ∈ µ. Then X − U ⊆ X − ((X − A) ∩ Mµ ). So (X − U) ∩ Mµ ⊆ A. Then (X − U) ∩ Mµ ⊆ iµ A. So X − iµ A ⊆ X − ((X − U) ∩ Mµ ). Therefore, cµ (X − A) ∩ Mµ ⊆ U. Thus, A is gµ -open.

Theorem 3.8. A set A in GTS (X, µ) is gµ -open if and only if iµ A ∪ (Ac ∩ Mµ ) ⊆ U ∈ µ implies U = Mµ .

Proof. Let A be a gµ -open set and iµ A ∪ (Ac ∩ Mµ ) ⊆ U ∈ µ. Then Uc ⊆ (iµ A)c ∩ (A ∪ Mcµ ) = cµ (X − A) ∩ (A ∪ Mcµ ). Therefore, Uc ∩ Mµ ⊆ (cµ (X − A) ∩ Mµ ) ∩ A = cµ (X − A) − (X − A). Then by Theorem 3.3 Uc ∩ Mµ = ∅, That is, U = Mµ . Conversely, let F be a µ-closed set such that F ∩ Mµ ⊆ A. Then iµ A ∪ (Ac ∩ Mµ ) ⊆ iµ A ∪ Fc ∈ µ. By the assumption, iµ A ∪ Fc = Mµ , that is, F ∩ Mµ ⊆ iµ A. Now apply Theorem 3.7.

Theorem 3.9. A subset A of a GTS (X, µ) is gµ -closed if and only if cµ A − A is gµ -open.

Proof. Suppose that A is gµ -closed and F ∩ Mµ ⊆ cµ A − A, where F is a µ-closed set. By Theorem 3.3 F ∩ Mµ = ∅. So F ∩ Mµ ⊆ iµ (cµ A − A). Therefore, cµ A − A is gµ - open by Theorem 3.7. Conversely, assume that X − Mµ ⊆ A and A ∩ Mµ ⊆ U ∈ µ. Now cµ A ∩ Uc ∩ Mµ ⊆ cµ A ∩ (Mµ − A) = cµ A − A. By Theorem 3.7 cµ A ∩ Uc ∩ Mµ ⊆ iµ (cµ A − A) = ∅. Thus, cµ A ∩ Mµ ⊆ U and A is gµ -closed.

The following diagram extends to the setting of GTSs the corresponding diagram of Benchalli and Wali [4] and Dontchev [17].

34


CUBO 18, 1 (2016)

For examples showing independence A ! B in the above diagram see [4]. Theorem 3.10. Let (X, µ) be a GTS and A ⊆ X. Then the following statements hold. (i) µ-closed ⇒ αµ -closed ⇒ sµ -closed ⇒ spµ -closed. (ii) αµ -closed ⇒ pµ -closed ⇒ spµ -closed. (iii) µ-closed ⇒ gµ -closed ⇒ rgµ - closed. (iv) gµ -closed ⇒ αgµ -closed ⇒ gsµ -closed ⇒ gspµ -closed. (v) αµ -closed ⇒ gαµ -closed ⇒ αgµ -closed. (vi) sµ -closed ⇒ sgµ -closed ⇒ gspµ -closed. (vii) sgµ -closed ⇒ gsµ -closed. (viii) spµ -closed ⇒ gspµ -closed.

CUBO 18, 1 (2016)


35

(ix) pµ -closed ⇒ gspµ -closed. (x) αgµ -closed ⇒ gspµ -closed. (xi) gαµ -closed ⇒ gsµ -closed. Proof. (i) Let A be µ-closed set. Then cµ A = A. Therefore, iµ cµ A = iµ A ⊆ A. Thus, cµ iµ cµ A ⊆ cµ A = A. Now let A be a αµ -closed set. Then iµ cµ A ⊆ cµ iµ cµ A ⊆ A. Now let A be a sµ -closed set. Then iµ cµ A ⊆ A and X − Mµ ⊆ A. Therefore, iµ cµ iµ A ⊆ iµ cµ A ⊂ A. This proves (i). The proofs of other parts also follow easily. Theorem 3.11.

(i) Every sgµ -closed sets is spµ -closed.

(ii) Every gαµ -closed set is pµ -closed. Proof. (i) Let A be a sgµ -closed set and x ∈ cspµ A ∩ Mµ . Then {x} is pµ -open or µ-nowhere dense. If {x} is pµ -open then by Theorem 2.3, {x} is spµ -open. Since x ∈ spµ A ∩ Mµ , {x}∩A ̸= ∅. Therefore, x ∈ A. If {x} is µ-nowhere dense then {x} ∪ (X − Mµ ) is αµ -closed and hence sµ -closed. Therefore, the complement B = Mµ − {x} is sµ -open. Assume that x ∈ / A, then A ∩ Mµ ⊆ B. Since A is sgµ -closed, and cspµ A ⊆ csµ A. cspµ A ∩ Mµ ⊆ B. Hence x∈ / cspµ A ∩ Mµ . By contradiction x ∈ A. Thus, A is spµ -closed. (ii) Let A be a gαµ -closed set. Let x ∈ cpµ A ∩ Mµ . If {x} is pµ -open, then {x} ∩ A ̸= ∅. So that x ∈ A. If {x} is µ-nowhere dense and does not meet A then {x} ∪ (X − Mµ ) is αµ -closed. Then B = Mµ − {x} is αµ -open and A ∩ Mµ ⊆ B. Since A is gαµ -closed, cαµ A ∩ Mµ ⊆ B. Therefore, x ∈ / cαµ A ∩ Mµ , a contradiction. Thus, x ∈ A and A is pµ -closed.

The following theorem also covers some immediate implications. Theorem 3.12. For a set in a GTS (X, µ), the following statements hold. (i) πµ -closed ⇒ δµ -closed. (ii) θµ -closed ⇒ θgµ -closed. (iii) πµ -closed ⇒ πgµ -closed. (iv) δµ -closed ⇒ δgµ -closed. (v) µ-closed ⇒ gµ ∗ -closed. (vi) µ-closed ⇒ wµ -closed.

36


CUBO 18, 1 (2016)

(vii) gµ ∗ -closed ⇒ mgµ -closed. (viii) gµ ∗ -closed ⇒ gµ -closed. (ix) gµ -closed ⇒ wgµ -closed. (x) rgµ -closed ⇒ gprµ -closed. (xi) gpµ -closed ⇒ gprµ -closed. (xii) αgµ -closed ⇒ gpµ -closed. (xiii) wµ -closed ⇒ rwµ -closed. (xiv) rwµ -closed ⇒ rgµ -closed. (xv) rwµ -closed ⇒ rwgµ -closed. Proof. (i) Let A be a πµ -closed set. Then there are µ-regular closed sets Ri , R2 , .....Rn such !n c c c that A = i=1 Ri . Let x ∈ X − A = ∪n i=1 Ri . Then x ∈ Ri for some i and iµ cµ Ri ∩ A = c Ri ∩ A = ∅. So x ∈ / cδµ A. The proofs of other parts are also easy and left to the reader.

4

µ-T0, µ-T1 and µ-T1/2 generalized topological spaces

Definition 4.1. A GTS (X, µ) is said to be (i) µ-T0 if x, y ∈ Mµ , x ̸= y implies the existence of a µ-open set containing precisely one of x and y. (ii) [32] µ-T1 if x, y ∈ Mµ , x ̸= y implies the existence of µ-open sets U1 and U2 such that x ∈ U1 and y ∈ / U1 and y ∈ U2 and x ∈ / U2 . (iii) µ-T1/2 if every gµ -closed set is µ-closed. Easy examples of GT-spaces which are not strong and having the properties of the above separation axioms may be provided. For example, let R be the set of real numbers and x, y, x ̸= y be any two real numbers. Then µ = {∅, {x}, {x, y}} is a GT which is not strong and has the property of µ-T0 but not µ-T1 . It is obvious that µ-T1 implies µ-T0 . Also (X, µ) is µ-T0 if and only if for each x, y ∈ Mµ , cµ ({x}) = cµ ({y}) implies x = y. Theorem 4.2. If a GTS (X, µ) is µ-T1/2 then it is µ-T0 .

CUBO 18, 1 (2016)


37

Proof. Suppose that (X, µ) is not a µ-T0 space. Then there exist distinct points x and y in Mµ such that cµ ({x}) = cµ ({y}). Let A = cµ ({x}) ∩ {x}c . We show that A is gµ -closed but not µ-closed. X − Mµ ⊆ A. Let A ∩ Mµ ⊆ V ∈ µ. Since A ⊆ cµ ({x}), cµ A ∩ Mµ ⊆ cµ ({x}) ∩ Mµ . Thus, we show c that cµ ({x}) ∩ Mµ ⊆ V. Since cµ ({x}) ∩ {x} ∩ Mµ ⊆ V, it is enough to show that x ∈ V. If x is not in V then y ∈ V and y ∈ cµ ({y}) = cµ ({x}) ⊆ V c as V c is a µ-closed set containing the set {x}. Thus, y ∈ V ∩ V c , a contradiction. Now if x ∈ U ∈ µ then U ∩ A ⊇ {y} ̸= ∅, and hence x ∈ cµ A. But x is not in A and thus, A is not a µ-closed set. Theorem 4.3. If a GTS (X, µ) is µ-T1 then for each x ∈ X, A = {x} ∪ (X − Mµ ) is µ-closed. Proof. Let y ∈ cµ A ∩ Mµ and y ̸= x. Then y ∈ cµ (A ∩ Mµ ) ∩ Mµ = cµ ({x}) ∩ Mµ . Then y ∈ cµ ({x}). So y ∈ U ∈ µ implies x ∈ U which is against our hypothesis. So cµ A ∩ Mµ = {x}, that is, cµ A = A. Theorem 4.4. If a GTS (X, µ) is µ-T1 then it is µ-T1/2 . Proof. Let A be a subset of X which is not µ-closed. If X − Mµ is not contained in A, then A is not gµ -closed. So let X − Mµ ⊆ A. Since A is not µ-closed, cµ A − A is non empty. Let x ∈ cµ A − A. By Theorem 4.3 {x} ∪ (X − Mµ ) is µ-closed. As ({x} ∪ (X − Mµ )) ∩ Mµ = {x} ⊆ cµ A − A, by Theorem 3.3 A is not gµ -closed. Definition 4.5. A GTS (X, µ) is said to be µ-symmetric if for each x, y ∈ Mµ , x ∈ cµ ({y}) implies y ∈ cµ ({x}). Theorem 4.6. A GTS (X, µ) is µ-symmetric if and only if {x} ∪ (X − Mµ ) is gµ -closed for each x ∈ X. Proof. Let A = {x} ∪ (X − Mµ ) and A ∩ Mµ ⊆ U ∈ µ. If A ∩ Mµ = ∅ then cµ A = cµ (A ∩ Mµ ) = cµ ∅ = X − Mµ . So cµ A ∩ Mµ ⊆ U. Otherwise cµ A ∩ Mµ = cµ (A ∩ Mµ ) ∩ Mµ = cµ ({x}) ∩ Mµ . If cµ ({x}) ∩ Mµ " U then assume that y ∈ cµ ({x}) ∩ Uc ∩ Mµ . Since (X, µ) is µ-symmetric, x ∈ cµ ({y}). Since x ∈ U, y ∈ U, then y ∈ U ∩ Uc , a contradiction. Conversely, let for each x ∈ X, {x}∪(X − Mµ ) is gµ -closed. Let x, y ∈ Mµ , x ∈ cµ ({y}) and y ∈ / cµ ({x}). Then y ∈ (cµ ({x}))c . Let A = {y} ∪ (X − Mµ ). Then A is gµ -closed and A ∩ Mµ = {y} ⊆ (cµ ({x}))c . So cµ A ∩ Mµ = (cµ ({y})) ∩ Mµ ⊆ (cµ ({x}))c . Then x ∈ (cµ ({y})) ∩ Mµ ⊆ (cµ ({x}))c , a contradiction. Corollary 4.7. If A GTS (X, µ) is µ-T1 then it is µ-symmetric. Proof. The proof follows from Theorem 4.3, Theorem 4.6 and Lemma 3.2. Theorem 4.8. A GTS (X, µ) is µ-symmetric and µ-T0 if and if only (X, µ) is µ-T1 .

38


CUBO 18, 1 (2016)

Proof. If (X, µ) is µ-T1 then by Corollary 4.7 (X, µ) is µ-symmetric and obviously µ-T0 . Conversely, let (X, µ) be µ-symmetric and µ-T0 . Let x, y ∈ Mµ and x ̸= y. Then by µ-T0 property there exists a U ∈ µ such that x ∈ U ⊆ ({y})c . Then x is not in cµ ({y}). Since (X, µ) is µ-symmetric, y is not in cµ ({x}). Then there exists V = (cµ ({x}))c such that y ∈ V and x ∈ / V. Theorem 4.9. If (X, µ) is µ-symmetric then (X, µ) is µ-T0 if and only if (X, µ) is µ-T1/2 if and only if (X, µ) is µ-T1 . Proof. The proof follows from Theorems 4.8, 4.4 and 4.2. Theorem 4.10. A GTS (X, µ) is µ-T1/2 if and only if for each x ∈ X, either {x} is µ-open or {x} ∪ (X − Mµ ) is µ-closed. Proof. Suppose X is µ-T1/2 and for some x ∈ X, {x} ∪ (X − Mµ ) is not µ-closed. Then Mµ is the only µ-open set containing Mµ − {x}. Therefore, (Mµ − {x}) ∪ (X − Mµ ) is gµ -closed. So it is µ-closed. Thus, {x} is µ-open. Conversely, let A be a gµ -closed set with x ∈ cµ A ∩ Mµ and x ∈ / A. If {x} is µ-open then ∅ ̸= {x} ∩ A. Thus, x ∈ A. Otherwise {x} ∪ (X − Mµ ) is µ-closed. Then({x} ∪ (X − Mµ )) ∩ Mµ = {x} ⊆ cµ A − A. Then by Theorem 3.3 {x} = ∅, a contradiction. Thus, x ∈ A and so A is µ-closed. Theorem 4.11. For a GTS (X, µ), the following statements are equivalent. (i) X is µ-T1/2 . (ii) Every αgµ -closed set is αµ -closed. Proof. (i) ⇒ (ii). Let A be a αgµ -closed set and x ∈ cαµ A ∩ Mµ . If {x} is µ-open then {x} ∈ αµ so that {x} ∩ A ̸= ∅. Thus, x ∈ A. Otherwise {x} ∪ (X − Mµ ) is µ-closed. Let x ∈ / A. Then Mµ − {x} is µ-open and A ∩ Mµ ⊆ Mµ − {x}. Since A is αgµ -closed, cαµ A ∩ Mµ ⊆ Mµ − {x}. Therefore, x∈ / cαµ A ∩ Mµ , a contradiction. Thus, x ∈ A and A is αµ -closed. (ii) ⇒ (i). If some set {x} ∪ (X − Mµ ) is not µ-closed then x ∈ Mµ and Mµ − {x} is not µ- open. Then (Mµ − {x}) ∪ (X − Mµ ) is trivially αgµ -closed. By (ii), (Mµ − {x}) ∪ (X − Mµ ) is αµ -closed. So {x} is αµ -open. Since a non-empty αµ -open set contains a non-empty µ-open set, {x} is µ-open. This shows that (X, µ) is µ-T1/2 .

5

µ-regular and µ-normal generalized topological spaces

Definition 5.1. [33] A GTS (x, µ) is said to be µ-regular if for each µ-closed set F of X not containing x ∈ X there exist disjoint µ-open subsets U and V of X such that x ∈ U and F∩Mµ ⊆ V. Theorems 5.2, 5.4, and 5.5 generalize the corresponding results in Roy [40].

CUBO 18, 1 (2016)


39

Theorem 5.2. For a GTS (X, µ), the following statements are equivalent. (i) X is µ-regular. (ii) x ∈ U ∈ µ implies that there exists V ∈ µ such that x ∈ V ⊆ cµ V ∩ Mµ ⊆ U. (iii) For each µ-closed set F, F = ∩{cµ V : F ∩ Mµ ⊆ V ∈ µ}. (iv) For each subset A of X and each U ∈ µ with A ∩ U ̸= ∅ there exists a V ∈ µ such that A ∩ V ̸= ∅ and cµ V ∩ Mµ ⊆ U. (v) For each non-empty set A ⊆ X and each µ-closed set F with A ∩ F = ∅ there exist U, V ∈ µ such that A ∩ V ̸= ∅, F ∩ Mµ ⊆ U and U ∩ V = ∅. (vi) For each µ-closed set F and x ∈ / F there exist U ∈ µ and a gµ -open set V such that x ∈ U, F ∩ Mµ ⊆ V and U ∩ V = ∅. (vii) For each non-empty A ⊆ X and each µ-closed set F with A ∩ F = ∅ there exist a U ∈ µ and a gµ -open set V such that A ∩ U ̸= ∅, F ∩ Mµ ⊆ V and U ∩ V = ∅. (viii) For each µ-closed set F of X, F = ∩{cµ V : F ∩ Mµ ⊆ V and V is gµ -open}. Proof. (i)⇔ (ii) [32]. (ii)⇒ (iii). Suppose x ∈ / F. Then by (ii) there exists a V ∈ µ such that x ∈ V ⊆ cµ V ∩ Mµ ⊆ X − F. Then F ∩ Mµ ⊆ (X − (cµ V ∩ Mµ )) ∩ Mµ = X − cµ V = W ∈ µ. Since cµ W ∩ V = ∅, (iii) follows. (iii) ⇒ (iv). x ∈ A ∩ U implies that x ∈ / X − U. By (iii) there exists a W ∈ µ such that (X − U) ∩ Mµ ⊆ W and x ∈ / cµ W. Let V = X − cµ W then x ∈ V ∩ A and V ⊆ X − W. Thus, cµ V ⊆ X − W. Therefore, cµ V ∩ Mµ ⊆ (X − W) ∩ Mµ ⊆ (X − ((X − U) ∩ Mµ )) ∩ Mµ = U. (iv) ⇒ (v). A ∩ (X − F) ̸= ∅. By (iv) there exists a µ-open set V such that A ∩ V ̸= ∅ and cµ V ∩ Mµ ⊆ X − F. Let W = X − cµ V. Then F ∩ Mµ ⊆ ⊆ (X − (cµ V ∩ Mµ )) ∩ Mµ = X − cµ V = W and W ∩ V = ∅. (v) ⇒ (i). Let F be a µ-closed set not containing x. By (v) there exist disjoint µ-open sets U and V such that x ∈ U and F ∩ Mµ ⊆ V. (i) ⇒ (vi). Follows from Lemma 3.2. (vi) ⇒ (vii). Note that A ⊆ Mµ . Since A is non-empty and A ∩ F = ∅ there exists a point x ∈ A such that x ∈ / F. By (vi) there exist a U ∈ µ and a gµ -open set V such that x ∈ U, F ∩ Mµ ⊆ V and U ∩ V = ∅. Then U ∩ A ̸= ∅. (vii) ⇒ (i). Let x ∈ / F, where F is a µ-closed set. Then {x} ∩ F = ∅. By (vii) there exist a U ∈ µ and a gµ -open set V such that x ∈ U, F ∩ Mµ ⊆ V and U ∩ V = ∅. Now F ∩ Mµ ⊆ iµ V by Theorem 3.7. (iii) ⇒ (viii). We have F ⊆ ∩{cµ V : F ∩ Mµ ⊆ V and V is gµ -open} ⊆ ∩{cµ V : F ∩ Mµ ⊆ V ∈ µ} = F. (viii) ⇒ (i). Let F be a µ-closed set such that x ∈ / F. Then by (viii) there exists gµ -open set W such that F ∩ Mµ ⊆ W and x ∈ / cµ W. Since F is µ-closed, W is gµ -open and F ∩ Mµ ⊆ W, by Theorem 3.7, F ∩ Mµ ⊆ iµ W.

40


CUBO 18, 1 (2016)

Definition 5.3. [32] A GTS (X, µ) is µ-normal if for any pair of µ-closed sets A and B such that A ∩ B ∩ Mµ = ∅ there exist disjoint µ-open sets U and V such that A ∩ Mµ ⊆ U and B ∩ Mµ ⊆ V. Theorem 5.4. For a GTS (X, µ), the following statements are equivalent. (i) X is µ-normal. (ii) For any µ-closed set A and µ-open set U such that A ∩ Mµ ⊆ U there is a µ-open set V such that A ∩ Mµ ⊆ V ⊆ cµ V ∩ Mµ ⊆ U. Proof. Let A be a µ-closed set such that A ∩ Mµ ⊆ U ∈ µ. Then B = X − U is µ-closed and A ∩ B ∩ Mµ is empty. Then by (i) there exist disjoint µ-open sets V and W such that A ∩ Mµ ⊆ V and B∩Mµ ⊆ W. Then A∩Mµ ⊆ V ⊆ cµ V ∩ Mµ ⊆ (X − W) ∩ Mµ ⊆ (X − (B ∩ Mµ )) ∩ Mµ = U. Conversely, assume that A and B be µ-closed sets such that A ∩ B ∩ Mµ = ∅. Then U = X − B is µopen and A∩Mµ ⊆ U. By (ii) there exists a µ-open set V such that A∩Mµ ⊆ V ⊆ Cµ V ∩ Mµ ⊆ U. Let W = X − cµ V. Since cµ V ∩ B ∩ Mµ = ∅, B ∩ Mµ ⊆ X − cµ V = W. Theorem 5.5. In a GTS (X, µ), the following statements are equivalent. (i) X is µ-normal. (ii) For any pair of µ-closed sets A and B such that A ∩ B ∩ Mµ = ∅ then there exist disjoint gµ -open sets U and V such that A ∩ Mµ ⊆ U and B ∩ Mµ ⊆ V. (iii) For every µ-closed set A and µ-open set U such that A ∩ Mµ ⊆ U there exists a gµ -open set V such that A ∩ Mµ ⊆ V ⊆ cµ V ∩ Mµ ⊆ U. (iv) For every µ-closed set A and every gµ -open set U containing A ∩ Mµ there exists a µ-open set V such that A ∩ Mµ ⊆ V ⊆ cµ V ∩ Mµ ⊆ U. (v) For every gµ -closed set A and every µ-open set U containing A ∩ Mµ there exists a µ-open set V such that cµ A ∩ Mµ ⊆ V ⊆ cµ V ∩ Mµ ⊆ U. Proof. (i) ⇒ (ii). Follows from Lemma 3.2. (ii) ⇒ (iii). Assume that B = X − U. By (ii) there exist disjoint gµ -open sets V and W such that A ∩ Mµ ⊆ V and B ∩ Mµ ⊆ W. Since B ∩ Mµ ⊆ W, (X − U) ∩ Mµ ⊆ W. Therefore, (X − W) ∩ Mµ ⊆ (X − (X − U) ∩ Mµ ) ∩ Mµ = U. Since X − W is gµ -closed, cµ (X − W) ∩ Mµ ⊆ U. Since cµ V ∩ Mµ ⊆ cµ (X − W) ∩ Mµ , the implication is established. (iii) ⇒ (iv). By Theorem 3.7 A ∩ Mµ ⊆ iµ U. Then by (iii) there exists a gµ -open set V such that A ∩ Mµ ⊆ V ⊆ cµ V ∩ Mµ ⊆ iµ U. By Theorem 3.7 A ∩ Mµ ⊆ iµ V ⊆ cµ (iµ V) ∩ Mµ ⊆ cµ V ∩ Mµ ⊆ U. (iv) ⇒ (v). Let A be a gµ -closed set and A ∩ Mµ ⊆ U ∈ µ. Then cµ A ∩ Mµ ⊆ U. By (iv) there exists a µ-open set V such that cµ A ∩ Mµ ⊆ V ⊆ cµ V ∩ Mµ ⊆ U. (v) ⇒ (i). Let A and B be µ-closed sets such that A ∩ B ∩ Mµ = ∅. Then A ∩ Mµ ⊆ X − B ∈ µ. By

CUBO 18, 1 (2016)


41

(v) there exists a µ-open set V such that cµ A ∩ Mµ ⊆ V ⊆ cµ V ∩ Mµ ⊆ X − B. Thus, A ∩ Mµ ⊆ V and B ⊆ X − (cµ V ∩ Mµ ). Therefore, B∩Mµ ⊆ (X − (cµ V ∩ Mµ )) ∩ Mµ = X−cµ V = W ∈ µ.

6

Extremally µ-disconnectedness

Theorem 6.1. For a GTS (X, µ), the following statements are equivalent. (i) (X, µ) is extremally µ-disconnected. (ii) Every spµ -closed set is pµ -closed. (iii) Every sgµ -closed set is pµ -closed. (iv) Every sµ -closed set is pµ -closed. (v) Every sµ -closed set is αµ -closed. (vi) Every sµ -closed set is gαµ -closed. Proof. (i) ⇒ (ii). Let A be a spµ -closed set. Then by Lemma 2.9 iµ cµ iµ A ⊆ A. Since X is extremally µ-disconnected, cµ iµ A ∩ Mµ = iµ (cµ iµ A ∩ Mµ ) ⊆ iµ cµ iµ A. Therefore, cµ iµ A ∩ Mµ ⊆ A. Since X − Mµ ⊆ A, cµ iµ A ⊆ A. (ii) ⇒ (iii). is Theorem 3.11(i). (iii) ⇒ (iv). Since a sµ -closed set is sgµ - closed, the result follows. (iv) ⇒ (v). Follows from Theorem 2.4. (v) ⇒ (vi). follows from Theorem 3.10(v). (vi) ⇒ (i). Let U be a µ-open set. We need to show that iµ (cµ U ∩ Mµ ) = cµ U ∩ Mµ . Now iµ (cµ U ∩ Mµ ) = iµ cµ U. Since iµ cµ U ⊆ cµ U ∩ Mµ , we prove the inclusion cµ U ∩ Mµ ⊆ iµ cµ U. Let A = iµ cµ U ∪ X − Mµ . Now iµ cµ A = iµ cµ iµ cµ U = iµ cµ U ⊆ A. So A is sµ -closed. By our assumption A is gαµ -closed. Since iµ cµ iµ (iµ cµ U) = iµ cµ U, iµ cµ U is αµ -open. Since A ∩ Mµ = iµ cµ U ∈ αµ and A is gαµ -closed, cαµ A ∩ Mµ ⊆ iµ cµ U ⊆ A. Thus, cαµ A ⊆ A. On the other hand cαµ A = A ∪ cµ iµ cµ A implies cµ iµ cµ A ⊆ A. Therefore, cµ iµ cµ A ∩ Mµ ⊆ A ∩ Mµ = iµ cµ A, which implies that A is µ-closed. Now U ⊆ iµ cµ U. Then cµ U ⊆ cµ iµ cµ U = cµ A = A. Therefore, cµ U ∩ Mµ ⊆ iµ cµ U.

Future scope: This paper may be useful in the study of digital topology since generalized closed sets and T1/2 separation axiom have already proved their utility in that area.

42

7


CUBO 18, 1 (2016)

Acknowledgements

The second author acknowledges the fellowship grant of University Grant Commission, India. The authors are very grateful to anonymous referee for his observations which improved the paper.

References [1] M.E. Abd El-Monsef, S.N. El-Deep and R.A. Mahmoud, β-open sets and β- continuous mappings, Bull. Fac. Sci. Assiut Univ. 12 (1983), 77–90. [2] D. Andrijevic, Semi-preopen sets Mat. Vesnik 38 (1986), 24–32. [3] S.P. Arya and T.M. Nour, Characterizations of s-normal spaces, Indian J. Pure Appl. Math, 21 (1990), 717–719. [4] S.S. Benchalli and R.S. Wali, On RW- closed sets in topological spaces Bull. Malays. Math. sci. soc. 30(2) (2007), 99–110. [5] P. Bhattacharyya and B.K. Lahiri, Semi-generalized closed sets in topology, Indian J. Math. 29 (1987), 376–382. [6] N. Biswas, On characterization of semi-continuous functions, Atti Accad. Naz. Lincei Rend, Cl. Sci. Fis. Mat. Natur 48 (8)(1970), 399–402. [7] D.E. Cameron, Properties of S-closed spaces, Proc. Amer Math. soc. 72 (1978), 581–586. [8] J. Cao, M. Ganster, Submaximal, extremal disconnectedness and generalized closed sets, Houston Journal of Mathematics 24(4) (1998), 681–688. [9] J. Cao, M. Ganster, I. Reilly, On generalized closed sets, Topology and its application 123 (2002), 17–47. ´ Cs´ asz´ ar, Generalized open sets, Acta Math. Hungar. 75 (1997), 65–87. [10] A. ´ Cs´ [11] A. asz´ ar, Generalized topology, generalized continuity, Acta Math. Hungar. 96 (2002), 351–357. ´ Cs´ [12] A. asz´ ar, δ and θ modification topologies, Annales Univ. Sci. Budapest 47 (2004), 91–96. ´ Cs´ [13] A. asz´ ar, Separation axioms for generalized topologies, Acta Math. Hungar. 104 (2004), 63–69. ´ Cs´ [14] A. asz´ ar, Normal generalized topologies, Acta Math. Hungar. 115(4) (2007), 309–313. ´ Cs´ [15] A. asz´ ar, Extremally disconnected generalized topologies, Acta Math. Hungar. 120 (2008), 275–279.

CUBO 18, 1 (2016)


43

[16] J. Dontchev, On generalizing semi-preopen sets , Mem. Fac Sci. Kochi. Univ. Ser. A. Math. 16 (1995), 35–48. [17] J. Dontchev, On some separation axioms associated with α- topology , Mem. Fac Sci. Kochi. Univ. Ser. A. Math. 18 (1997), 31–35. [18] J. Dontchev and M. Ganster, On δ- generalized set T3/4 Spaces, Mem. Fac Sci. Kochi. Univ. Ser. A. Math. 17 (1996), 15–31. [19] J. Dontchev and H. Maki, On θ-generalized www.Unipissing.ca/topology/p/a/b/a/08.htm.

closed

sets,

Topology

Atlass,

[20] J. Dontchev and T. Noiri, Quasi-normal spaces and πg-closed sets, Acta Math. Hungar. 89(3) (2000), 211–219. [21] W. Dunham, T1/2 spaces, Kyungpook Math. J. 17(2) (1997), 161–169. [22] E. Ekici, On γ- normal space, Bull. Math. soc. Math. Roumanie. Tome 50(98)(3) (2007), 259–272. [23] Y. Gnanambal, On generalized preregular closed sets in topological spaces, Indian J. Pure Appl. Math. 28 (1997), 351–360. [24] Y. Gnanambal and K. Balachandran, On gpr-continuous functions in topological spaces, Indian J. Pure Appl. Math. 30(6) (1999), 581–593. [25] E.D. Khalimsky, Applications of connected ordered topological spaces in topology, Conference of Math. Department of Povolsia, 1970. [26] N. Levine, semi open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36–41. [27] N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo 19 (1970), 89–96. [28] H. Maki, R. Devi and K. Balachandran, Generalized α-closed set in topology, Bull. Fukuoka Univ. Ed. part-III 42 (1993), 13–21. [29] H. Maki, R. Devi and K. Balachandran, Associated topologies of generalized α- closed sets and α- generalized closed sets, Mem. Sci. Kochi. Univ. Ser. A. Math. 15 (1994), 51–63. [30] H. Maki, J. Umehara and T. Noiri, Every topological space is pri-T1/2 , Mem. Fac Sci. Kochi. Univ. Ser. A. Math. 17 (1996), 33–42. [31] A.S. Mashhour, M.E. Abd. El-Monsef and S. N. El-deep, On pre continuous mappings and weak pre-continuous mappings Proc Math. Phys. Soc. Egypt 53 (1982), 47–53. [32] W.K. Min, Remarks on separation axioms on generalized topological space, Chungcheong mathematical society 23(2) (2010), 293–298.

44


CUBO 18, 1 (2016)

′

[33] W.K. Min, (δ, δ )-continuity on generalized topological spaces, Acta Math. Hungar. 129(4) (2010), 350–356. [34] W.K. Min, Remark on θ- open sets in generalized topological spaces, Applied math. letter’s 24 (2011), 165–168. [35] N. Nagaveni, Studies on generalizations of homemorphisms in topological spaces, Ph.D. Thesis, Bharathiar University, Coimbatore, 1999. [36] O. Njastad, On some classes of nearly open sets, Pacific J. Math. 15 (1965), 961–970. [37] N. Palaniappan and K.C. Rao, Regular generalized closed sets, Kyungpook Math J. 33 (1993), 211–219. [38] J.K. Park and J.H. Park, Mildly generalized closed sets, almost normal and mildly normal spaces, Chaos, Solitions and Fractals 20 (2004), 1103–1111. [39] A. Pushpalatha, Studies on generalizations of mappings in topological spaces, Ph.D. Thesis, Bharathiar University, Coimbatore, 2000. [40] B. Roy, On a type of generalized open sets, Applied general topology 12(2) (2011), 163–173. [41] R.D. Sarma, On extremally disconnected generalized topologies Acta Math. Hungar. 134(4) (2012), 583–588. [42] M. S. Sarsak, Weak separation axioms in generalized topological spaces, Acta Math. Hungar. 131(1-2) (2011), 110–121. [43] P. Sundaram and M. Sheik John, On w-closed sets in topology, Acta Ciencia Indica 4 (2000), 389–392. [44] M. Stone, Application of the theory of Boolean rings to generalized topology Trans. Amer. Math. Soc. 41 (1937), 374–481. [45] B.K. Tyagi, H.V.S. Chauhan, A remark on semi open sets in generalized topological spaces, communicated. [46] B.K. Tyagi, H.V.S. Chauhan, A remark on exremally µ-disconnected generalized topological spaces, Mathematics for applications, 5 (2016), 83-90. [47] B.K. Tyagi, H.V.S. Chauhan, R. Choudhary On γθ -operator and θ-connected sets in generalized topological space, Journal of Advanced Studies in Topology 6(4) (2015), 135–143. [48] J. Tong, Weak almost continuous mapping and weak nearly compact spaces, Boll. Un. Mat. Ital. 6(1982), 385–391. [49] M.K.R.S. Veera Kumar, Between closed sets and g-closed sets, Mem. Fac Sci. Kochi. Univ.(Math) 21 (2000), 1–19.

CUBO 18, 1 (2016)


45

[50] N.V. Velicko, H-closed topological space, Trans. Amer. Math. Soc. 78 (1968), 103–118. [51] GE Xun, GE Ying, µ-separations in generalized topological spaces, Appl. Math. J. Chiness Univ. 25(2) (2010), 243–252.