CUBO A Mathematical Journal Vol.18, No¯ 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 brijkishore.tyagi@gmail.com, harsh.chauhan111@gmail.com ABSTRACT In this paper, we introduce several types of generalized closed sets in generalized topo- logical spaces (GTSs). Their interrelationships are investigated and several characteri- zations of µ-T0, µ-T1, µ-T1/2, µ-regular, µ-normal GTSs and extremally µ-disconnected GTSs are obtained. RESUMEN En este art́ıculo introducimos varios tipos de conjuntos cerrados generalizados en es- pacios topológicos 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 B. K. Tyagi, Harsh V. S. Chauhan CUBO 18, 1 (2016) 1 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 rela- tionship 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 Császár [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 cµA, is the intersection of µ-closed sets containing A. The interior of A, denoted by iµA, 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 general- ized 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 extremal- lay µ-disconnected if cµU ∩ Mµ ∈ µ for every U ∈ µ. It may be remarked that in strong GTS, this notion concide with the notion of extremally disconnectedness in Császár [12]. Several character- izations 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 ∈ cµA if and only if x ∈ U ∈ µ implies U ∩ A ̸= ∅. (ii) cµA = cµ(A ∩ Mµ). (iii) cµA = X − iµ(X − A). (iv) If U, V ∈ µ and U ∩ V = ∅ then cµU ∩ V = ∅ and U ∩ cµV = ∅. (v) Mµ − cµA = X − cµA. (vi) iµA = iµ(A ∩ Mµ). (vii) iµ(cµA − A) = ∅. (viii) cµ and iµ are monotone: A ⊆ B implies cµA ⊆ cµB (respectively iµA ⊆ iµB), idempotent cµcµA = cµA (respectively iµiµA = iµA), cµ is enhancing (A ⊆ cµA), iµ is contracting (iµA ⊆ A). Proof. (vii). If x ∈ iµ(cµA − A) then there exists a U ∈ µ such that x ∈ U ⊆ cµA − A. Then x ∈ U ⊆ cµA and U ∩ A = ∅. Now x ∈ U ⊆ cµA 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 = cµA ∩ 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 iµcµA = A. (ii) µ-semi open (or sµ-open) if A ⊆ cµiµA ∩ Mµ. (iii) µ-preopen (or pµ-open ) if A ⊆ iµcµA. (iv) µ-α-open (or αµ-open) if A ⊆ iµcµiµA. (v) µ-semi preopen (or spµ-open) if A ⊆ cµiµcµA ∩ Mµ. (vi) µ-θ-closed (or θµ-closed) [34] if A = γθA, where γθ(A) = {x ∈ X : cµG ∩ 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 = cδA, where cδA = {x ∈ X : iµcµU ∩ A ̸= ∅ for U ∈ µ and x ∈ U}. The complement of a δµ-closed set is called µ-δ-open (δµ-open). 30 B. K. Tyagi, Harsh V. S. Chauhan 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 ⊆ cµU ∩ 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 cµiµA = A; A set A is µ-semi open if and only if cµA = cµiµA and A ⊆ Mµ. A is sµ-closed if and only if iµcµA ⊆ A and X − Mµ ⊆ A; A is pµ-closed if and only if cµiµA ⊆ A; A is αµ-closed if and only if cµiµcµA ⊆ A; A is spµ-closed if iµcµiµA ⊆ A and X − Mµ ⊆ A. For any set A, cµiµcµA 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 ⊆ iµcµiµA then A ⊆ cµiµA, A ⊆ Mµ and A ⊆ iµcµA. So A ∈ sµ ∩ pµ. Conversely, let A ∈ sµ ∩ pµ. Then A ⊆ cµiµA ∩ Mµ. Therefore, cµA ⊆ cµiµA. Also A ⊆ iµcµA. Therefore, A ⊆ iµcµiµA A subset A of a GTS (X, µ) is µ-nowhere dense if iµcµA = ∅. 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 iµcµ{x} ̸= ∅. Then x ∈ iµcµ{x}. So {x} ⊆ iµcµ{x}. Lemma 2.6. If {x} is µ-nowhere dense in a GTS (X, µ) then {x} ∪ (X − Mµ) is αµ-closed. Proof. cµiµcµ({x} ∪ (X − Mµ)) = cµiµcµ{x} = cµ∅ = X − Mµ. So cµiµcµ({x} ∪ (X − Mµ)) ⊆ {x} ∪ (X − Mµ). Lemma 2.7. For a subset A containing X − Mµ, csµA = A ∪ iµcµA. Proof. Since csµA is sµ-closed, iµcµ(csµ A) ⊆ csµA. On the other hand iµcµ(A ∪ iµcµA) ⊆ iµcµcµA = iµcµA. Therefore, iµcµ(A ∪ iµcµA) ⊆ A ∪ iµcµA. Since X − Mµ ⊆ A, A ∪ iµcµA is sµ- closed. CUBO 18, 1 (2016) On generalized closed sets in generalized topological spaces. 31 Lemma 2.8. For a subset A, cαµA = A ∪ cµiµcµA. Proof. Since cαµA is αµ-closed, cµiµcµcαµA ⊆ cαµA. Therefore, A ∪ cµiµcµA ⊆ cαµA. On the other hand cµiµcµ(A ∪ cµiµcµA) ⊆ cµiµcµcµA = cµiµcµA ⊆ A ∪ cµiµcµA. Thus, A ∪ cµiµcµA is αµ-closed set containing A. Lemma 2.9. For a subset A, A ∪ iµcµiµA ⊆ cspµA. Proof. iµcµiµA ⊆ iµcµiµ(cspµ )A ⊆ cspµA, since cspµA 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 cµA ∩ 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 csµA∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ sµ. (iii) a µ-generalized semi closed (or gsµ-closed) set if csµA∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ µ. (iv) a µ-generalized α-closed (or gαµ-closed) set if cαµA∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ αµ. (v) a µα-generalized closed (or αgµ-closed) set if cαµA ∩ 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 cµA∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ roµ. (viii) a µ-generalized preclosed (or gpµ-closed) set if cpµA ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ µ. (ix) a µ-generalized preregular closed (or gprµ-closed) set if cpµA ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ roµ. (x) a µ-θ-generalized closed (or θgµ-closed) set if γθµA ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ µ. (xi) a µ-δ-generalized closed (or δgµ-closed) set if cδµA ∩ Mµ ⊆ U whenever A ⊆ U ∈ µ. (xii) a µ-weakly generalized closed (or wgµ-closed) set if cµiµA∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ µ. (xiii) a µ-strongly generalized closed (or gµ ∗-closed) set if cµA∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ gµ. (xiv) a µ-π-generalized closed (or πgµ-closed) set if cµA ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ πµ. 32 B. K. Tyagi, Harsh V. S. Chauhan CUBO 18, 1 (2016) (xv) a µ-weakly closed (or wµ-closed) set if cµA ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ sµ. (xvi) a µ-mildly generalized closed (or mgµ-closed) set if cµiµA∩Mµ ⊆ U whenever A∩Mµ ⊆ U ∈ gµ. (xvii) a µ-semi-weakly generalized closed (or swgµ-closed) set if cµiµA ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ sµ. (xviii) a µ-regular weakly generalized closed (or rwgµ-closed) set if cµiµA ∩ Mµ ⊆ U whenever A ∩ Mµ ⊆ U ∈ roµ. (xix) a µ-regular generalized w-closed (or rwµ-closed) set if cµA∩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 ⊆ cµA − A implies F ∩ Mµ = ∅. Proof. Let F be a µ-closed set such that F∩Mµ ⊆ cµA−A . Then A∩Mµ ⊆ F c ∈ µ. Since A is gµ- closed, cµA ∩ Mµ ⊆ F c. That is, F ∩ Mµ ⊆ (cµA) c. Therefore, F ∩ Mµ ⊆ (cµA − A) ∩ (cµA) c = ∅. Conversely, let A ∩ Mµ ⊆ U ∈ µ and if cµA ∩ Mµ is not contained in U then cµA ∩ Mµ ∩ U c ̸= ∅. Since cµA ∩ U c is µ-closed and cµA ∩ U c ∩ Mµ ⊆ cµA − A, a contradiction. Theorem 3.4. If a gµ-closed subset A of a GTS (X, µ) be such that cµA − (A ∩ Mµ) is µ-closed then A is µ-closed. Proof. Let A be a gµ-closed set such that cµA − (A ∩ Mµ) is µ-closed. Then cµA − (A ∩ Mµ) is µ-closed subset of itself. Since cµA − (A ∩ Mµ) is gµ-closed subset of itself, by Theorem 3.3 (cµA−(A∩Mµ))∩MµY , where Y = cµA−(A∩Mµ), is empty. Since MµY = (cµA−(A∩Mµ))∩Mµ, A is µ-closed. Theorem 3.5. If A is a gµ-closed set and A ⊆ B ⊆ cµA then B is gµ-closed. Proof. Let B ∩ Mµ ⊆ U ∈ µ. Since A is gµ-closed and A ∩ Mµ ⊆ U, cµA ∩ Mµ ⊆ U. Then cµB ∩ Mµ ⊆ cµA ∩ 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) On generalized closed sets in generalized topological spaces. 33 Proof. If µ= Ω then obviously (X, µ) is strong. Now if A ⊆ U ∈ µ then cµA ⊆ cµU = U since U ∈ µ. Conversely, let U ∈ µ. Since U is gµ-closed, cµU ⊆ 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 ⊆ iµA 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−iµA)∩Mµ ⊆ X − F. That is, F∩Mµ ⊆ (X − (X − iµA) ∩ Mµ) ∩ Mµ = iµA. 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 ⊆ iµA. So X − iµA ⊆ 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 iµA∪(A c ∩ Mµ) ⊆ U ∈ µ implies U = Mµ. Proof. Let A be a gµ-open set and iµA ∪ (A c ∩ Mµ) ⊆ U ∈ µ. Then U c ⊆ (iµA) c ∩ (A ∪ Mcµ) = cµ(X − A) ∩ (A ∪ M c µ). Therefore, U c ∩ 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 iµA ∪ (A c ∩ Mµ) ⊆ iµA ∪ F c ∈ µ. By the assumption, iµA ∪ F c = Mµ, that is, F ∩ Mµ ⊆ iµA. Now apply Theorem 3.7. Theorem 3.9. A subset A of a GTS (X, µ) is gµ-closed if and only if cµA − A is gµ-open. Proof. Suppose that A is gµ-closed and F∩Mµ ⊆ cµA − A, where F is a µ-closed set. By Theorem 3.3 F ∩ Mµ = ∅. So F ∩ Mµ ⊆ iµ(cµA − A). Therefore, cµA − A is gµ- open by Theorem 3.7. Conversely, assume that X − Mµ ⊆ A and A ∩ Mµ ⊆ U ∈ µ. Now cµA ∩ U c ∩ Mµ ⊆ cµA ∩ (Mµ − A) = cµA − A. By Theorem 3.7 cµA ∩ U c ∩ Mµ ⊆ iµ(cµA − A) = ∅. Thus, cµA ∩ 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 B. K. Tyagi, Harsh V. S. Chauhan 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) On generalized closed sets in generalized topological spaces. 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 cµA = A. Therefore, iµcµA = iµA ⊆ A. Thus, cµiµcµA ⊆ cµA = A. Now let A be a αµ-closed set. Then iµcµA ⊆ cµiµcµA ⊆ A. Now let A be a sµ-closed set. Then iµcµA ⊆ A and X − Mµ ⊆ A. Therefore, iµcµiµA ⊆ iµcµA ⊂ 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 ∈ cspµA ∩ 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 ∈ spµA ∩ 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 ⊆ csµA. cspµA ∩ Mµ ⊆ B. Hence x /∈ cspµA ∩ Mµ. By contradiction x ∈ A. Thus, A is spµ-closed. (ii) Let A be a gαµ-closed set. Let x ∈ cpµA ∩ 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, cαµA ∩ Mµ ⊆ B. Therefore, x /∈ cαµA ∩ 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 B. K. Tyagi, Harsh V. S. Chauhan 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 that A = !n i=1Ri. Let x ∈ X − A = ∪ n i=1Ri c. Then x ∈ Ri c for some i and iµcµRi c ∩ A = Ri c ∩ A = ∅. So x /∈ cδµA. 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) On generalized closed sets in generalized topological spaces. 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}), cµA ∩ Mµ ⊆ cµ({x}) ∩ Mµ. Thus, we show that cµ({x}) ∩ Mµ ⊆ V. Since cµ({x}) ∩ {x} c ∩ 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 Vc is a µ-closed set containing the set {x}. Thus, y ∈ V ∩ Vc, a contradiction. Now if x ∈ U ∈ µ then U ∩ A ⊇ {y} ̸= ∅, and hence x ∈ cµA. 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 ∈ cµA ∩ 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 cµA ∩ Mµ = {x}, that is, cµA = 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, cµA − A is non empty. Let x ∈ cµA − A. By Theorem 4.3 {x} ∪ (X − Mµ) is µ-closed. As ({x} ∪ (X − Mµ)) ∩ Mµ = {x} ⊆ cµA − 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 cµA = cµ(A ∩ Mµ) = cµ∅ = X − Mµ. So cµA ∩ Mµ ⊆ U. Otherwise cµA ∩ 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 ∩ U c, 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 cµA ∩ 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 B. K. Tyagi, Harsh V. S. Chauhan 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 ∈ cµA ∩ 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} ⊆ cµA − 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 ∈ cαµA∩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, cαµA ∩ Mµ ⊆ Mµ − {x}. Therefore, x /∈ cαµA ∩ 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) On generalized closed sets in generalized topological spaces. 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 ⊆ cµV ∩ Mµ ⊆ U. (iii) For each µ-closed set F, F = ∩{cµV : 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 cµV ∩ 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 = ∩{cµV : 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 ⊆ cµV ∩Mµ ⊆ X − F. Then F ∩ Mµ ⊆ (X − (cµV ∩ Mµ)) ∩ Mµ = X − cµV = W ∈ µ. Since cµW ∩ 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 /∈ cµW. Let V = X − cµW then x ∈ V ∩ A and V ⊆ X − W. Thus, cµV ⊆ X − W. Therefore, cµV ∩ 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 cµV ∩Mµ ⊆ X − F. Let W = X−cµV. Then F∩Mµ ⊆ ⊆ (X − (cµV ∩ Mµ)) ∩ Mµ = X−cµV = 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 ⊆ iµV by Theorem 3.7. (iii) ⇒ (viii). We have F ⊆ ∩{cµV : F ∩ Mµ ⊆ V and V is gµ-open} ⊆ ∩{cµV : 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 /∈ cµW. Since F is µ-closed, W is gµ-open and F ∩ Mµ ⊆ W, by Theorem 3.7, F ∩ Mµ ⊆ iµW. 40 B. K. Tyagi, Harsh V. S. Chauhan 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 ⊆ cµV ∩ 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 ⊆ cµV ∩ 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 µ- open and A∩Mµ ⊆ U. By (ii) there exists a µ-open set V such that A∩Mµ ⊆ V ⊆ CµV ∩ Mµ ⊆ U. Let W = X − cµV. Since cµV ∩ B ∩ Mµ = ∅, B ∩ Mµ ⊆ X − cµV = 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 ⊆ cµV ∩ 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 ⊆ cµV ∩ 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 cµA ∩ Mµ ⊆ V ⊆ cµV ∩ 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 cµV ∩ Mµ ⊆ cµ(X − W) ∩ Mµ, the implication is established. (iii) ⇒ (iv). By Theorem 3.7 A ∩ Mµ ⊆ iµU. Then by (iii) there exists a gµ-open set V such that A ∩ Mµ ⊆ V ⊆ cµV ∩ Mµ ⊆ iµU. By Theorem 3.7 A ∩ Mµ ⊆ iµV ⊆ cµ(iµV) ∩ Mµ ⊆ cµV ∩ Mµ ⊆ U. (iv) ⇒ (v). Let A be a gµ-closed set and A ∩ Mµ ⊆ U ∈ µ. Then cµA ∩ Mµ ⊆ U. By (iv) there exists a µ-open set V such that cµA ∩ Mµ ⊆ V ⊆ cµV ∩ 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) On generalized closed sets in generalized topological spaces. 41 (v) there exists a µ-open set V such that cµA∩Mµ ⊆ V ⊆ cµV ∩ Mµ ⊆ X−B. Thus, A∩Mµ ⊆ V and B ⊆ X − (cµV ∩ Mµ). Therefore, B∩Mµ ⊆ (X − (cµV ∩ Mµ)) ∩ Mµ = X−cµV = 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 iµcµiµA ⊆ A. Since X is extremally µ-disconnected, cµiµA∩Mµ = iµ(cµiµA∩Mµ) ⊆ iµcµiµA. Therefore, cµiµA∩Mµ ⊆ A. Since X − Mµ ⊆ A, cµiµA ⊆ 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(cµU ∩ Mµ) = cµU ∩ Mµ. Now iµ(cµU ∩ Mµ) = iµcµU. Since iµcµU ⊆ cµU ∩ Mµ, we prove the inclusion cµU ∩ Mµ ⊆ iµcµU. Let A = iµcµU ∪ X − Mµ. Now iµcµA = iµcµiµcµU = iµcµU ⊆ A. So A is sµ-closed. By our assumption A is gαµ-closed. Since iµcµiµ(iµcµU) = iµcµU, iµcµU is αµ-open. Since A ∩ Mµ = iµcµU ∈ αµ and A is gαµ-closed, cαµA ∩ Mµ ⊆ iµcµU ⊆ A. Thus, cαµA ⊆ A. 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