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 δβopen sets [4].
Definition 3.1 Let (U, τ_{R}(X)) be a nano topological space and A naon ⊆ U.
A nano subset N − ∧_{β}(A) is defined as follows: N − ∧_{β}(A) = ∩ {G : A ⊆ G, G ∈ N_{β}(U, X)}. The complement of N − ∧_{β}(A)set is called N − ∨_{β}(A)set.
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:
 1.
A ⊆ N∧_{β}(A).
 2.
If A ⊆ B, then N∧_{β}(A) ⊆ N − ∧_{β}(B).
 3.
N∧_{β}(N∧_{β}(A)) = N∧_{β}(A).
 4.
If A ∈ N_{β}(U, X)), then A = N∧_{β}(A).
 5.
N∧_{β}(∪{A_{α} : α ∈ Δ}) = ∪ {N∧_{β}(A_{α}) : α ∈ Δ}.
 6.
N∧_{β}(∩{A_{α} : α ∈ Δ}) ⊆ ∩ {N∧_{β}(A_{α}) : α ∈ Δ}.
Proof. I prove only (5) and (6) since the others are consequences of Definition 3.1.
First, for each α ∈ Δ, N∧_{β}(A_{α}) ⊆ N∧_{β}(∪_{α ∈ Δ}A_{α}). Hence, ∪_{α ∈ Δ}N∧_{β}(A_{α}) ⊆ N∧_{β}(∪_{α ∈ Δ}A_{α}). Conversely, suppose that x ∉ ∪_{α ∈ Δ}N∧_{β}(A_{α}). Then, x ∉ N∧_{β}(A_{α}) 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 βopen set which does not contain x. Therefore, x ∉ N∧_{β}(∪_{α ∈ Δ}A_{α}). Thus, N∧_{β}(∪_{α ∈ Δ}A_{α}) ⊆ ∪_{α ∈ Δ}N∧_{β}(A_{α}).
Suppose that, x ∉ ∩ {N − ∧_{β}(A_{α}) : α ∈ Δ}. There exists α_{0} ∈ Δ such that \( x\notin N{\wedge}_{\beta}\left({A}_{\alpha_0}\right) \), and there exists a nano βopen set G such that x ∉ G and \( {A}_{\alpha_0}\subseteq G. \) We have that \( {\cap}_{\alpha \in \varDelta }{A}_{\alpha}\subseteq {A}_{\alpha_0}\subseteq G \) and x ∉ G. Therefore, x ∉ N − ∧_{β}(∩{A_{α} : α ∈ Δ}).
Remark 3.1 The inclusion in Lemma 3.1 parts 1 and 6 cannot be replaced by equality relation. Moreover, the converse of part 2 is not necessarily true as shown in the following example.
Example 3.1 Let U = {a, b, c, d} with U/R = {{a, b}, {c}, {d}} and X = {c}. Then, τ_{R}(X) = {U, ϕ, {c}}.
 i.
For part 1, if A = {a}, then N − ∧_{β}(A) = {a, c}, and N − ∧_{β}(A) ⊈ A.
 ii.
For part 6, if A = {a}, and B = {b}, then A ∩ B = ϕ, and N − ∧_{β}(A) = {a, c}, N − ∧_{β}(B) = {b, c}, N − ∧_{β}(A ∩ B) = ϕ, and N − ∧_{β}(A) ∩ N − ∧_{β}(B) = {c} ⊈ N − ∧_{β}(A ∩ B) = ϕ.
 iii.
For part 2, if A = {c}, and B = {a}, then N − ∧_{β}(A) = A, and N − ∧_{β}(B) = {a, c}. Therefore, N − ∧_{β}(A) ⊆ N − ∧_{β}(B), but A ⊈ B.
Definition 3.2 Let (U, τ_{R}(X)) be a nano topological space and A ⊆ U. A subset A is called nano ∧_{β}set if A = N − ∧_{β}(A). The complement of nano ∧_{β}set is nano ∨_{β}set. The family of all nano ∧_{β}sets and nano ∨_{β}sets are denoted by \( N{\tau}^{\wedge_{\beta }} \) and \( N{\varGamma}^{\vee_{\beta }}, \) respectively.
In Example 3.1, \( N{\tau}^{\wedge_{\beta }}=\left\{U,\phi, \left\{c\right\},\left\{a,c\right\},\left\{b,c\right\},\left\{c,d\right\},\left\{a,b,c\right\},\left\{a,c,d\right\},\left\{b,c,d\right\}\right\} \)
and \( N{\varGamma}^{\vee_{\beta }}=\left\{U,\phi, \left\{a\right\},\left\{b\right\},\left\{d\right\},\left\{a,b\right\},\left\{a,d\right\},\left\{b,d\right\},\left\{a,b,d\right\}\right\}. \)
In the following lemma, i summarize the fundamental properties of nano ∧_{β}sets which show that nano ∧_{β}sets are a generalization of nano βopen 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:
 1.
N − ∧_{β}(A), U, ϕ are nano ∧_{β}sets.
 2.
If A is a nanoβopen set, then A is a nano ∧_{β}set.
 3.
If A_{α} is a nano ∧_{β}set ∀α ∈ Δ, then ∪_{α ∈ Δ}A_{α} is a nano ∧_{β}set.
 4.
If A_{α} is a nano ∧_{β}set ∀α ∈ Δ, then ∩_{α ∈ Δ}A_{α} is a nano ∧_{β}set.
Proof. This follows from Lemma 3.1.
Remark 3.2 It is clear from (1), (3), and (4) in Lemma 3.2 that the family of all nano ∧_{β}sets forms a topology.
In (2) in Lemma 3.2, the converse is not necessarily true as shown in the following example.
Example 3.2 Let U = {a, b, c, d}, U/R = {{a, b},{c},{d}}, and X={a, c}. Then, τ_{R}(X) = {U, ϕ,{c},{a, b},{a, b, c}}. If A = {d}, then A is a nano ∧_{β}set, but A = {d} is not a nano βopen set.
Remark 3.3 The nano δβopen sets of Definition 2.4 [4] and the current Definition 3.2 of nano ∧_{β}sets are different and independent as shown in Fig. 3. Example 3.1 shows that {a} is a nano δβopen set, but it is not a nano ∧_{β}set. Moreover, Example 3.2 shows that {d} is a nano ∧_{β}set, but it is not a nano δβopen set.
Corollary 3.1 Let (U, τ_{R}(X)) be a nano topological space. Then,
 1.
\( \mathrm{NSO}\left(U,X\right)\cup \mathrm{NPO}\left(U,X\right)\subseteq N{\tau}^{\wedge_{\beta }}. \)
 2.
\( \mathrm{NSO}\left(U,X\right)\cap \mathrm{NPO}\left(U,X\right)\subseteq N{\tau}^{\wedge_{\beta }}. \)
Remark 3.4The equality in Corollary 3.1 does not hold in general. In Example 3.2, the set A = {d} is nano ∧_{β}set but it is neither in NSO(U, X) ∪ NPO(U, X) nor in NSO(U, X) ∩ NPO(U, X).
Proposition 3.1The intersection of a nano open and a nano ∧_{β}set is a nano ∧_{β}set.
Proof. Let A be a nano open and B be a nano ∧_{β}open. Then, A = N∧_{β}(A) and B = N∧_{β}(B).
$$ {\displaystyle \begin{array}{ll}A\cap B& =N{\wedge}_{\beta }(A)\cap N{\wedge}_{\beta }(B),\\ {}& \supseteq N{\wedge}_{\beta}\left(A\cap B\right).\end{array}} $$
(1)
Therefore, N − ∧_{β}(A ∩ B) ⊆ (A ∩ B), but N − ∧_{β}(A ∩ B) ⊇ (A ∩ B) from Lemma 3.1 (1).
Proposition 3.2If A and B are nano subsets of U such that A ⊆ B ⊆ ncl(nint(A)), then B is a nano ∧_{β}set in U.
Proof. It clear from Definition 2.3 [1] that,
$$ {\displaystyle \begin{array}{ll}\mathrm{nint}(A)& \subseteq \mathrm{ncl}(A),\\ {}\Rightarrow \mathrm{nint}\left(\mathrm{nint}(A)\right)& \subseteq \mathrm{nint}\left(\mathrm{ncl}(A)\right),\\ {}\Rightarrow \mathrm{nint}(A)& \subseteq \mathrm{nint}\left(\mathrm{ncl}(A)\right),\\ {}\Rightarrow \mathrm{ncl}\left(\mathrm{nint}(A)\right)& \subseteq \mathrm{ncl}\left(\mathrm{nint}\left(\mathrm{ncl}(A)\right)\right).\end{array}} $$
(2)
Then,
$$ {\displaystyle \begin{array}{ll}B& \subseteq \mathrm{ncl}\left(\mathrm{nint}(A)\right),\\ {}& \subseteq \mathrm{ncl}\left(\mathrm{nint}\left(\mathrm{ncl}(A)\right)\right),\\ {}& \subseteq \mathrm{ncl}\left(\mathrm{nint}\left(\mathrm{ncl}(B)\right)\right).\end{array}} $$
(3)
Hence, B ⊆ ncl(nint(ncl(B))). Therefore B is nano βopen in U. Thus, by Lemma 3.2 (2) B is a nano ∧_{β}set in U.
Definition 3.3Let (U, τ_{R}(X)) be a nano topological space and A ⊆ U. The nano ∨_{β}closure of a set A, denoted by\( {\mathrm{ncl}}^{\vee_{\beta }}(A) \), is the intersection of nano ∨_{β}sets including A. The nano ∧_{β}interior of a set A, denoted by\( {\mathrm{nint}}^{\wedge_{\beta }}(A) \), is the union of nano ∧_{β}sets included in A.
The following theorem presents the main properties of nano ∨_{β}closure and nano ∧_{β}interior which are required in the sequel to study the properties of nano ∧_{β}continuous functions.
Theorem 3.1Let (U, τ_{R}(X)) be a nano topological space and A, B ⊆ U. Then, the following properties hold:
 1.
\( {\mathrm{ncl}}^{\vee_{\beta }}(A) \) is a nano ∨_{β}set and \( {\mathrm{nint}}^{\wedge_{\beta }}(A) \) is a nano ∧_{β}set.
 2.
\( {\mathrm{nint}}^{\wedge_{\beta }}(A)\subseteq A\subseteq {\mathrm{ncl}}^{\vee_{\beta }}(A). \)
 3.
\( A={\mathrm{ncl}}^{\vee_{\beta }}(A) \) iff A is a nano ∨_{β}set and \( {\mathrm{nint}}^{\wedge_{\beta }}(A)=A \) iff A is a nano ∧_{β}set.
 4.
If \( A\subseteq B,\mathrm{then}\ {\mathrm{ncl}}^{\vee_{\beta }}(A)\subseteq {\mathrm{ncl}}^{\vee_{\beta }}(B) \) and \( {\mathrm{nint}}^{\wedge_{\beta }}(A)\subseteq {\mathrm{nint}}^{\wedge_{\beta }}(B). \)
 5.
\( {\mathrm{nint}}^{\wedge_{\beta }}(A)\cup {\mathrm{nint}}^{\wedge_{\beta }}(B)\subseteq {\mathrm{nint}}^{\wedge_{\beta }}\left(A\cup B\right). \)
 6.
\( {\mathrm{ncl}}^{\vee_{\beta }}\left(A\cap B\right)\subseteq {\mathrm{ncl}}^{\vee_{\beta }}(A)\cap {\mathrm{ncl}}^{\vee_{\beta }}(B). \)
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:
 i.
For part 2, if A = {a, b, c}, then \( {\mathrm{ncl}}^{\vee_{\beta }}(A)=U \) and so, \( {\mathrm{ncl}}^{\vee_{\beta }}(A)\nsubseteq A. \) If A = {a}, then \( {\mathrm{nint}}^{\wedge_{\beta }}(A)=\phi \) and thus, \( {\mathrm{nint}}^{\wedge_{\beta }}(A)\nsubseteq A. \)
 ii.
For part 5, if A = {a} and B = {c}, then \( A\cup B=\left\{a,c\right\},{\mathrm{nint}}^{\wedge_{\beta }}(A)=\phi, {\mathrm{nint}}^{\wedge_{\beta }}(B)=\left\{c\right\},{\mathrm{nint}}^{\wedge_{\beta }}\left(A\cup B\right)=A\cup B \) and so, \( {\mathrm{nint}}^{\wedge_{\beta }}\left(A\cup B\right)=\left\{a,c\right\}\nsubseteq \left\{c\right\}={\mathrm{nint}}^{\wedge_{\beta }}(A)\cup {\mathrm{nint}}^{\wedge_{\beta }}(B). \)
 iii.
For part 6, if A = {d} and B = {a, b, c}, then \( A\cap B=\phi, {\mathrm{ncl}}^{\vee_{\beta }}(A)=A,{\mathrm{ncl}}^{\vee_{\beta }}(B)=U,{\mathrm{ncl}}^{\wedge_{\beta }}\left(A\cap B\right)=\phi \) and so, \( {\mathrm{ncl}}^{\vee_{\beta }}(A)\cap {\mathrm{ncl}}^{\vee_{\beta }}(B)=\left\{d\right\}\nsubseteq {\mathrm{ncl}}^{\vee_{\beta }}\left(A\cap B\right)=\phi . \)

2.
The converse of part 4 is not necessarily true:
 i.
If A = {d} and B = {a, b, c}, then \( {\mathrm{ncl}}^{\vee_{\beta }}(A)=A \) and \( {\mathrm{ncl}}^{\vee_{\beta }}(B)=U \). Therefore, \( {\mathrm{ncl}}^{\vee_{\beta }}(A)\subseteq {\mathrm{ncl}}^{\vee_{\beta }}(B) \), but A ⊈ B.
 ii.
If A = {a} and B = {c}, then \( {\mathrm{nint}}^{\wedge_{\beta }}(A)=\phi \) and \( {\mathrm{nint}}^{\wedge_{\beta }}(B)=B \). Therefore, \( {\mathrm{nint}}^{\wedge_{\beta }}(A)\subseteq {\mathrm{nint}}^{\wedge_{\beta }}(B) \), but A ⊈ B.