Nano ∧β-sets and nano ∧β-continuity

The concept of nano near open sets was originally proposed by Thivagar and Richard (Int. J. Math. Stat. Inven 1:31-37). The main aspect of this paper is to introduce a new sort of nano near open sets namely, nano ∧β-sets. Fundamental properties of these sets are studied and compared to the previous one. It turns out that every nano β-open set is a nano ∧β-set. So, nano ∧β-sets are an extension of the previous nano near open sets, such as nano regular open, nano α-open, nano semi-open, nano pre-open, nano γ-open, and nano β-open sets. Meanwhile, it is shown that the concepts of nano ∧β-sets and nano δβ-open sets are different and independent. Based on these new sets, nano ∧β-continuous functions are defined and some results involving their characterizations are derived. In addition, the concepts of nano ∨β-closure and nano ∧β-interior are presented. Their properties are used to introduce and study the nano ∧β-continuous functions.


Introduction
Thivagar and Richard [1] established and constructed nano topological spaces, nano open sets, nano closed sets, nano interior, and nano closure. The nano α-open sets, nano semiopen sets, nano pre-open sets, and nano regular-open sets are also studied in [1]. After the work of Thivagar and Richard [1] on nano near open sets, various mathematicians turned their attention to the generalizations of these sets. In this direction, the nano βopen sets and some of their properties were discussed in [2] and shown that this notion was a generalization of the other types of nano near sets [1]. More importantly, Nasef et al. [3] provided new properties of some weak forms of nano open sets and studied the relationships between them. In 2018, Hosny [4] proposed the notion of nano δβ-open sets as a generalization of nano β-open sets. Consequently, it was an extension of all the previous weak forms of nano open sets in [1]. Nano continuous functions were defined in terms of nano open sets in [5]. On the other hand, nano α-continuous, nano semicontinuous, nano pre-continuous, and nano γ-continuous were investigated in [6,7]. The concept of the nano β-continuity was studied by Nasef et al. [3]. It was shown that this concept was an extension of the previous concepts of nano near continuous functions [5][6][7]. More recently, the notion of nano δβ-continuous functions was introduced in [4].
This notion was a generalization of nano β-continuous. Therefore, it was a generalization of the other types of nano near continuous functions [5][6][7].
The purpose of this paper is to continue the research along these directions. The intention of the present work is to generalize the notion of nano β-open sets by proposing a study of a new structure in nano topology which is the nano ∧ β -sets. It should be noted that the generalization of nano β-open sets by using ∧ β -sets is very different from the generalization of nano β-open sets by using nano δβ-open sets [4]. The main difference states that the family of all nano δβ-open sets does not form a topology, as the intersection of two nano δβ-open sets need not be a nano δβ-open set as shown in [4]. While, the family of all nano ∧ β -sets forms a topology as it is shown in this work. This new notion is not only an extension of nano β-open sets, but also can be regarded as a generalization of the other kinds of nano near open sets. This paper is organized as follows: The "Preliminaries" section contains the basic concepts of nano topological spaces. The nano ∧ β -sets and their properties are presented in the "Nano ∧ β -sets" section. The basic nano topological properties of this concept are also studied in this section. The relationships between the nano ∧ β -sets and nano β-open sets are revealed through Lemma 3.2. Moreover, it is shown that the concepts of nano ∧ β -sets and nano δβ-open sets are different and independent (see Examples 3.1 and 3.2). At the end of this section, we draw a diagram to describe the relationships among nano ∧ β -sets and the previous nano near sets. The objective of the "Nano ∧ β -sets and lower and upper approximations" section is to obtain various forms of nano ∧ β -sets corresponding to different cases of approximations. In the "Nano ∧ β -continuity" section, nano ∧ β -continuous functions are introduced as a generalization of nano β-continuous functions. Additionally, Remark 5.2 shows that the difference between nano ∧ β -continuous functions and nano δβ-continuous functions. The notion and the fundamental properties of nano ∨ β -closure and nano ∧ β -interior are given. The nano ∧ β -continuous functions are redefined by using the concepts of nano closed sets, nano closure, nano interior, nano ∨ β -sets (the complement of nano ∧ β -sets), nano ∨ βclosure and nano ∧ β -interior. The "Conclusions" section concludes this study.

Preliminaries
Before proceeding further, let me first recall some fundamental concepts and properties in nano topological spaces.
Definition 2.1 [8] Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U. The pair (U,R) is said to be the approximation space. Let X ⊆ U: [1] Let (U, τ R (X)) be a nano topological space with respect to X, where X ⊆ U. For a subset A ⊆ U: 1. The nano interior of A is defined as the union of all nano open subsets contained in A and is denoted by nint(A). 2. The nano closure of A is defined as the intersection of all nano closed subsets containing A, and is denoted by ncl(A).
It should be noted that the concepts of the nano interior and the nano closure have the same topological characterizations and properties of the concepts of interior and the closure in the general topology.
Definition 2.4 [1,2,4] Let (U, τ R (X)) be a nano topological space and A ⊆ U. The set A is said to be:  Definition 2.6 Let (U, τ R (X)) and ðV ; τ Ã R 0 ðY ÞÞ be nano topological spaces. A mapping f : ðU; τ R ðXÞÞ→ðV ; τ Ã R 0 ðY ÞÞ is said to be: In the paper [4] the relationships between the different types of near nano continuous functions are studied as shown in the following diagram in Fig. 2.
Throughout this paper (U, τ R (X)) is a nano topological space with respect to X where X ⊆ U, R is an equivalence relation on U, and U/R denotes the family of equivalence classes of U by R. Nano ∧ β -sets This section presents a new notion in nano topology which is called nano ∧ β -set. In addition, i indicates some nano topological properties of these sets. The results show that the proposed sets generalize the usual notions of nano near open sets [1,2], whereas it is independent and different from nano δβ-open sets [4].
In the following lemma, I summarize the fundamental properties of the set N − ∧ β . Lemma 3.1 For subsets A, B and A α (α ∈ Δ) of a nano topological space (U, τ R (X)), the following hold: Proof. I prove only (5) and (6) since the others are consequences of Definition 3.1.
for each α ∈ Δ and hence there exists G α ∈ N β (U, X) such that A α ⊆ G α and x ∉ G α for each α ∈ Δ. We have that ∪ α ∈ Δ A α ⊆ ∪ α ∈ Δ G α and ∪ α ∈ Δ G α is a nano β-open set which does not contain x.
In Example 3.1, N−τ ∧ β ¼ fU; ϕ; fcg; fa; cg; fb; cg; fc; dg; fa; b; cg; fa; c; dg; fb; c; dgg and N−Γ ∨ β ¼ fU; ϕ; fag; fbg; fdg; fa; bg; fa; dg; fb; dg; fa; b; dgg: In the following lemma, i summarize the fundamental properties of nano ∧ β -sets which show that nano ∧ β -sets are a generalization of nano β-open sets [2]. Lemma 3.2 For subsets A, B and A α (α ∈ Δ) of a nano topological space (U, τ R (X)), the following properties hold: Proof. This follows from Lemma 3.1. In (2) in Lemma 3.2, the converse is not necessarily true as shown in the following example. Then, Hence, B ⊆ ncl(nint(ncl(B))). Therefore B is nano β-open in U. Thus, by Lemma 3.2 (2) B is a nano ∧ β -set in U. Definition 3.3 Let (U, τ R (X)) be a nano topological space and A ⊆ U. The nano ∨ βclosure of a set A, denoted by ncl ∨ β ðAÞ, is the intersection of nano ∨ β -sets including A. The nano ∧ β -interior of a set A, denoted by nint ∧ β ðAÞ, is the union of nano ∧ β -sets included in A.
The following theorem presents the main properties of nano ∨ β -closure and nano ∧ βinterior which are required in the sequel to study the properties of nano ∧ β -continuous functions.
Theorem 3.1 Let (U, τ R (X)) be a nano topological space and A, B ⊆ U. Then, the following properties hold: Proof. Obvious. Remark 3.5 Example 3.1 shows that 1. The inclusion in Theorem 3.1 parts 2, 5, and 6 can not be replaced by equality relation: Nano ∧ β -sets and lower and upper approximations The goal of this section is to investigate various forms of nano ∧ β -sets corresponding to different cases of approximations.
Proposition 4.2 If U R (X) ≠ U in a nano topological space, then U, ϕ and any sets which intersects U R (X) are nano ∧ β -sets in U.

If
The concepts of nano near continuous functions are extended to nano ∧ β -continuous functions. It is shown that every nano β-continuous function is nano ∧ β -continuous function. Therefore, the nano ∧ β -continuous functions are generalization of the other types of nano near continuous functions [3,[5][6][7]. Definition 5.1 Let (U, τ R (X)) and ðV ; τ Ã R 0 ðY ÞÞ be nano topological spaces. A mapping f : ðU; τ R ðXÞÞ→ðV ; τ Ã R 0 ðY ÞÞ is said to be a nano ∧ β -continuous function if f −1 (B) is a nano ∧ β -set in U for every nano open set B in V.
The relationships between nano β-continuous and nano ∧ β -continuous functions are given in the following remark.
Remark 5.1 Every nano β-continuous is nano ∧ β -continuous. The converse of Remark 5.1 is not necessarily true as shown in the following example. It should be noted that from Remarks 5.1 and 5.2 and Fig. 2, I present Fig. 4 which shows that the current Definition 5.1 is a generalization of Definition 2.6 in [3,[5][6][7]. Additionally, it is different and independent of Definition 2.6 [4].