( ζ , δ ( μ ) )-closed sets in strong generalized topological spaces

Abstract: This paper deals with the concepts of ζδðμÞ-sets and ðζ ; δðμÞÞ-closed sets in a strong generalized topological space and investigate properties of several low separation axioms of strong generalized topologies constructed by the families of these sets. Some properties of ðζ ; δðμÞÞ-R0 and ðζ ; δðμÞÞ-R1 strong generalized topological spaces will be given. Finally, several characterizations of weakly ðζ ; δðμÞÞ-continuous functions are discussed.


Introduction
General topology is important in many fields of applied sciences as well as branches of mathematics. The theory of generalized topology, which was founded by Császár (Császár, 1997), is one of the most important development of general topology in recent years. Especially, the author defined some basic operators on generalized topological spaces. Noiri and Roy (Noiri & Roy, 2011) introduced a new kind of sets called generalized μ-closed sets in a topological space by using the concept of generalized open sets introduced by Császár. In 2007, Noiri (Noiri, 2007) introduced a new set called mg-closed which is defined on a family of sets satisfying some minimal conditions and obtained several basic properties of mg-closed sets. Moreover, the present author (Noiri, 2008) introduced and studied the notion of mng-closed sets defined in a set with two minimal structures. Ekici (Ekici, 2012) introduced the notion of generalized hyperconnected spaces and investigated various characterizations of generalized hyperconnected spaces and preservation theorem. In (Ekici, 2011), the present author introduced and studied the concept of generalized submaximal spaces. Ekici and Roy  introduced new types of sets called^μ-sets and _ μ -sets and investigated some of their fundamental properties. Roy and Ekici (Roy & Ekici, 2011) introduced and studied ð^; μÞ-open sets and ð^; μÞ-closed sets via μ-open and μ-closed sets in generalized topological spaces. Shanin (Shanin, 1943) introduced the notion of R 0 topological spaces. Davis (Davis, 1961) introduced the notion of a separation axiom called R 1 . These ABOUT THE AUTHOR Chawalit Boonpok received his Ph.D. (Applied Mathematics) from Brno University of Technology, Czech Republic. Currently, he is working as an assistant professor at the Department of Mathematics, Faculty of Science, Mahasarakham University, Thailand. His research interest focuses on Topology.
Continuity is basic concept for the study in topological spaces. The concept of weak continuity due to Levine (Levine, 1961) is one of the most important weak forms of continuity in topological spaces. Rose (Rose, 1984) has introduced the notion of subweakly continuous functions and investigated the relationships between subweak continuity and weak continuity. Popa and Stan (Popa & Stan, 1973) introduced and studied the notion of weakly quasi-continuous functions. Weak quasi-continuity is implied by both quasi-continuity and weak continuity which are independent of each other. Janković (Janković, 1985) introduced the concept of almost weakly continuous functions. It is shown in (Popa & Noiri, 1992) that almost weak continuity is equivalent to quasi precontinuity due to Paul and Bhattacharyya (Paul & Bhattacharyya, 1992). Noiri (Noiri, 1987) introduced the notion of weakly α-continuous functions. Several characterizations of weakly α-continuous functions are studied in (Noiri, 1987), (Rose, 1990) and (Sen & Bhattacharyya, 1993). In (Popa & Noiri, 1994), the present authors introduced and studied weakly β-continuous functions. Ekici et al. (Ekici, Jafari, Caldas, & Noiri, 2008) established a new class of functions called weakly λ-continuous functions which is weaker than λ-continuous functions and investigated some fundamental properties of weakly λ-continuous functions. Popa and Noiri (Popa & Noiri, 2002a) introduced the notion of weakly ðτ; mÞ-continuous functions as functions from a topological space into a set satisfying some minimal conditions and investigated several characterizations of such functions. Moreover, the present authors (Popa & Noiri, 2002b) introduced the concept of weakly M-continuous functions as functions from a set satisfying some minimal conditions into a set satisfying some minimal conditions and investigated some characterizations of weakly M-continuous functions. Min (Min, 2009) introduced the notions of weakly ðμ; μ 0 Þ-continuous functions and weakly ðψ; ψ 0 Þ-continuous functions on generalized topological spaces and generalized neighbourhood systems, respectively, and investigated several characterizations for such functions and the relationships between weak ðμ; μ 0 Þ-continuity and weak ðψ; ψ 0 Þ-continuity.
In this paper, we define ζ δðμÞ -sets, ðζ; δðμÞÞ-closed sets in a strong generalized topological ðX; μÞ and introduce the concepts of the ðζ; δðμÞÞ-closure and ðζ; δðμÞÞ-open sets by utilizing δðμÞ-open sets and δðμÞ-closure operators. In Section 3, we obtain fundamental properties of ðζ; δðμÞÞ-closed sets. In Section 4, we investigate properties of several low separation axioms of strong generalized topologies constructed by the concepts of ðζ; δðμÞÞ-closure operators and ðζ; δðμÞÞ-open sets. In the last section, we present the notion of weakly ðζ; δðμÞÞ-continuous functions and investigate some characterizations of such functions.

Preliminaries
Let X be a non-empty set and PðXÞ the power set of X. We call a class μ PðXÞ a generalized topology (briefly, GT) on X if [ 2 μ and an arbitrary union of elements of μ belongs to μ (Császár, 2002). A set X with a generalized topology μ on it is said to be a generalized topological space (briefly, GTS) and is denoted by ðX; μÞ. For a generalized topological space ðX; μÞ, the elements of μ are called μ-open sets and the complements of μ-open sets are called μ-closed sets. Let μ be a generalized topology on X. Observe that X 2 μ must not hold; if all the same X 2 μ, then we say that the generalized topology μ is strong (Császár, 2004). In general, let M μ denote the union of all elements of μ ; of course, M μ 2 μ and M μ ¼ X if μ is a strong generalized topology. For A X, we denote by c μ ðAÞ the intersection of all μ-closed sets containing A and by i μ ðAÞ the union of all μ-open sets contained in A (Császár, 2005). Moreover, i μ ðX À AÞ ¼ X À c μ ðAÞ. According to (Császár, 2008), for A X and x 2 X, we have x 2 c μ ðAÞ if x 2 M 2 μ implies M \ AÞ[. Consider a generalized topology μ on X. Let us define δðμÞ ¼ δ PðXÞ by A 2 δðμÞ iff A X and, if x 2 A, then there is a μ-closed set Q such that x 2 i μ ðQÞ A (Császár, 2008). Proposition 2.1. (Császár, 2008) Let ðX; μÞ be a generalized topological space. Then δðμÞ is a generalized topology on X.
A subset A of a generalized topological space ðX; μÞ is said to be μr-open (Császár, 2008)  A subset A of a generalized topological space ðX; μÞ is called δðμÞ-open if the complement of A is δðμÞ-closed. The family of all δðμÞ-closed sets in a generalized topological space ðX; μÞ is denoted by δðμÞC.
Theorem 2.7. (Min, 2010) For a subset A of a generalized topological space ðX; μÞ, the following properties hold: (1) x 2 i δðμÞ ðAÞ iff there exists a μr-open set R containing x such that R A.
Proof. (1) This is obvious from the definition.
Definition 3.6. A subset A of a strong generalized topological space ðX; μÞ is called a ζ δðμÞ -set if A ¼ ζ δðμÞ ðAÞ. The family of all ζ δðμÞ -sets in a strong generalized topological space ðX; μÞ is denoted by ζ δðμÞ ðX; μÞ. Lemma 3.7. For subsets A, B and C α ðα 2 ÑÞ of a strong generalized topological space ðX; μÞ, the following properties hold: (1) ζ δðμÞ ðAÞ A.
(4) If B α is a ζ δðμÞ -set for each α 2 Ñ, then [ α2Ñ B α is a ζ δðμÞ -set. Definition 3.9. A subset A of a strong generalized topological space ðX; μÞ is called ðζ; δðμÞÞ-closed if A ¼ T \ F, where T is a ζ δðμÞ -set and F is a δðμÞ-closed set. The family of all ðζ; δðμÞÞ-closed sets in a strong generalized topological space ðX; μÞ is denoted by ðζ; δðμÞÞC. Theorem 3.10. For a subset A of a strong generalized topological space ðX; μÞ, the following properties are equivalent: (1) A is ðζ; δðμÞÞ-closed.
Proof. ð1Þ It is sufficient to observe that A ¼ X \ A, where the whole set X is a ζ δðμÞ -set.
ð2Þ Let A be ðζ; δðμÞÞ-closed, then there exists a ζ δðμÞ -set T and a δðμÞ-closed set C such that Since    (6) c ðζ;δðμÞÞ ðX À AÞ ¼ X À i ðζ;δðμÞÞ ðAÞ. Definition 3.20. A subset A of a strong generalized topological space ðX; μÞ is said to be generalized ðζ; δðμÞÞ-closed (briefly g-ðζ; δðμÞÞ-closed) set if c ðζ;δðμÞÞ ðAÞ U whenever A U and U 2 ðζ; δðμÞÞO.  Conversely, suppose that x f g V 2 ðζ; δðμÞÞO, but c ðζ;δðμÞÞ ð x f gÞ is not a subset of V. This means that c ðζ;δðμÞÞ ð x f gÞ and the complement of V are not disjoint. Let y belongs to their intersection. Now, we have x 2 c ðζ;δðμÞÞ ð y f gÞ which is a subset of the complement of V and x 6 2 V. This is a contradiction.
Theorem 3.23. A subset A of a strong generalized topological space ðX; μÞ is g-ðζ; δðμÞÞ-closed if and only if c ðζ;δðμÞÞ ðAÞ À A contains no non-empty ðζ; δðμÞÞ-closed set.

Proof.
Suppose that x f g is not ðζ; δðμÞÞ-closed. Then X À x f g is not ðζ; δðμÞÞ-open and the only ðζ; δðμÞÞ-open set containing X À x f g is X itself. Therefore, c ðζ;δðμÞÞ ðX À x f gÞ X and hence, X À x f g is g-ðζ; δðμÞÞ-closed.
Lemma 3.31. Let A be a subset of a strong generalized topological space ðX; μÞ and G 2 ðζ; δðμÞÞO.
For a subset A of a strong generalized topological space ðX; μÞ, the following properties are equivalent: (1) A is g-ðζ; δðμÞÞ-closed.

Conversely, suppose that F A and
Theorem 3.40. For a subset A of a strong generalized topological space ðX; μÞ, the following properties are equivalent: (1) A is locally ðζ; δðμÞÞ-closed.
Proposition 3.46. For a subset D of a strong generalized topological space ðX; μÞ, the following properties are equivalent: (1) D is ðζ; δðμÞÞ-dense.
(2) If F is any ðζ; δðμÞÞ-closed set and D F, then F ¼ X. (4) The complement of D has empty ðζ; δðμÞÞ-interior.
(3) Each ðζ; δðμÞÞ-dense set of X is the intersection of a ðζ; δðμÞÞ-closed set and a ðζ; δðμÞÞ-open set.
Proof. It is similar to that of Theorem 3.56.
Suppose that x f g is ðζ; δðμÞÞ-closed. Then by Theorem 3.10, For any δðμÞ-open set U containing x, c δðμÞ ð x f gÞ U and hence, c δðμÞ ð x f gÞ ζ δðμÞ ð x f gÞ. Therefore, we have x f g ¼ ζ δðμÞ ð x f gÞ \ c δðμÞ ð x f gÞ c δðμÞ ð x f gÞ. This shows that x f g is δðμÞ-closed.

Conversely, suppose that
This shows that x f g is ðζ; δðμÞÞ-closed. Proof. Suppose that ðX; μÞ is δðμÞ-T 0 . For each x 2 X, it is obvious that x f g ζ δðμÞ ð x f gÞ \ c δðμÞ ð x f gÞ. If yÞx, (i) there exists a δðμÞ-open set V x such that y 6 2 V x and x 2 V x or (ii) there exists a δðμÞ-open set V y such that x 6 2 V y and y 2 V y . In case of (i), y 6 2 ζ δðμÞ ð x f gÞ and y 6 2 ζ δðμÞ ð x f gÞ \ c δðμÞ ð x f gÞ. This shows that x f g ζ δðμÞ ð x f gÞ \ c δðμÞ ð x f gÞ. In case (ii), y 6 2 c δðμÞ ð x f gÞ and y 6 2 ζ δðμÞ ð x f gÞ \ c δðμÞ ð x f gÞ. This shows that x f g ζ δðμÞ ð x f gÞ \ c δðμÞ ð x f gÞ. Consequently, we obtain x f g ¼ ζ δðμÞ ð x f gÞ \ c δðμÞ ð x f gÞ.
Conversely, suppose that ðX; μÞ is not δðμÞ-T 0 . There exist two distinct points x, y such that (i) y 2 V x for every δðμÞ-open set V x containing x and (ii) x 2 V y for every δðμÞ-open set V y containing y. From (i) and (ii), we obtain y 2 ζ δðμÞ ð x f gÞ and y 2 c δðμÞ ð x f gÞ, respectively. Therefore, we have y 2 ζ δðμÞ ð x f gÞ \ c δðμÞ ð x f gÞ. By Theorem 3.10, This is contrary to xÞy.
Corollary 3.70. Let ðX; μÞ be a δðμÞ-R 0 strong generalized topological space. Then ðX; μÞ is δðμÞ-T 0 if for each x 2 X, the singleton x f g is δðμÞ-closed.
Proof. It is an immediate consequence of Theorem 3.68 and Theorem 3.69.
Theorem 3.71. A strong generalized topological space ðX; μÞ is δðμÞ-T 1 if and only if for each x 2 X, the singleton x f g is a ζ δðμÞ -set.
Proof. Suppose that y 2 ζ δðμÞ ð x f gÞ for some point y distinct from x. Then, we have y 2 \fVxjx 2 V x ; V x 2 δðμÞg and so y 2 V x for every δðμÞ-open set V x containing x. This contradicts that ðX; μÞ is δðμÞ-T 1 .
Conversely, suppose that x f g is a ζ δðμÞ -set for each x 2 X. Let x and y be any distinct points. Then y 6 2 ζ δðμÞ ð x f gÞ and there exists a δðμÞ-open set V x such that x 2 V x and y 6 2 V x . Similarly, x 6 2 ζ δðμÞ ð y f gÞ and there exists a δðμÞ-open set V y such that y 2 V y and x 6 2 V y . This shows that ðX; μÞ is δðμÞ-T 1 .
(2) For any non-empty set A and G 2 ðζ; δðμÞÞO such that A \ GÞ[, there exists F 2 ðζ; δðμÞÞC such that A \ FÞ[ and F G.
ð2Þ , ð3Þ : This is a consequence of Theorem 4.6.
(2) For each non-empty set A of X and each U 2 ðζ; δðμÞÞO such that A \ UÞ[, there exists a ðζ; δðμÞÞ-closed set F such that A \ FÞ[ and F U.
Proof. Suppose that x n ¼ y for each n 2 N. Then x n f g n2N is a net in c ðζ;δðμÞÞ ð y f gÞ. By the fact that x n f g n2N converges to y, x n f g n2N converges to x and this means that x 2 c ðζ;δðμÞÞ ð y f gÞ.
ð2Þ ) ð1Þ : Let x and y are any two points of X such that y 2 c ðζ;δðμÞÞ ð x f gÞ. Suppose that x n ¼ y for each n 2 N. Then x n f g n2N is a net in ðX; μÞ such that ðζ; δðμÞÞ-converges to y. Since y 2 c ðζ;δðμÞÞ ð x f gÞ and x n f g n2N ðζ; δðμÞÞ-converges to y, it follows from ð2Þ x n f g n2N ðζ; δðμÞÞ-converges to x. Thus, x 2 c ðζ;δðμÞÞ ð y f gÞ. By Theorem 4.16, ðX; μÞ is ðζ; δðμÞÞ-R 0 . (1) ðX; μÞ is a ðζ; δðμÞÞ-R 1 space.
(2) For each x; y 2 X one of the following hold: (3) For each x; y 2 X such that c ðζ;δðμÞÞ ð x f gÞÞc ðζ;δðμÞÞ ð y f gÞ, there exist ðζ; δðμÞÞ-closed sets F x and F y such that x 2 F x ; y 6 2 F x ; y 2 F y ; x 6 2 F y and X ¼ F x [ F y . and y 6 2 G. This shows that (b) holds. There exist disjoint ðζ; δðμÞÞ-open sets U and V such that x 2 U and y 2 V. Put F x ¼ X À V and F y ¼ X À U. Then F x and F y are ðζ; δðμÞÞ-closed sets such that x 2 F x ; y 6 2 F x ; y 2 F y ; x 6 2 F y and X ¼ F x [ F y .
(2) For every pair of ðζ; δðμÞÞ-open sets U and V whose union is X, there exist ðζ; δðμÞÞ-closed sets F and H such that F U, H V and F [ H ¼ X.
Then X À U and X À V are disjoint ðζ; δðμÞÞ-closed sets. Since ðX; μÞ is ðζ; δðμÞÞ-normal, there exist disjoint ðζ; δðμÞÞ-open sets G and W such that X À U G and X À V W. Put F ¼ X À G and H ¼ X À W. Then F and H are ðζ; δðμÞÞ-closed sets such that F U, H V and F [ H ¼ X.
ð2Þ ) ð3Þ : Let F be a ðζ; δðμÞÞ-closed set and G be a ðζ; δðμÞÞ-open set containing F. Then X À F and G are ðζ; δðμÞÞ-open sets whose union is X. Then by ð2Þ, there exist ðζ; δðμÞÞ-closed sets M and N such that M X À F, N G and M [ N ¼ X. Then F X À M, X À G X À N and ðX À MÞ \ðX À NÞ ¼ [.
Put U ¼ X À M and V ¼ X À N. Then U and V are disjoint ðζ; δðμÞÞ-open sets such that F U X À V G. As X À V is a ðζ; δðμÞÞ-closed set, we have c ðζ;δðμÞÞ ðUÞ X À V and F U c ðζ;δðμÞÞ ðUÞ G. Theorem 5.5. For a function f : ðX; μÞ ! ðY; μ 0 Þ, the following properties are equivalent: (1) f is weakly ðζ; δðμÞÞ -continuous.