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].
Definition 3.1 Let (U, τR(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 ∈ Nβ(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 Aα(α ∈ Δ) of a nano topological space (U, τR(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 ∈ Nβ(U, X)), then A = N-∧β(A).
- 5.
N-∧β(∪{Aα : α ∈ Δ}) = ∪ {N-∧β(Aα) : α ∈ Δ}.
- 6.
N-∧β(∩{Aα : α ∈ Δ}) ⊆ ∩ {N-∧β(Aα) : 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(Aα) ⊆ N-∧β(∪α ∈ ΔAα). Hence, ∪α ∈ ΔN-∧β(Aα) ⊆ N-∧β(∪α ∈ ΔAα). Conversely, suppose that x ∉ ∪α ∈ ΔN-∧β(Aα). Then, x ∉ N-∧β(Aα) for each α ∈ Δ and hence there exists Gα ∈ Nβ(U, X) such that Aα ⊆ Gα and x ∉ Gα for each α ∈ Δ. We have that ∪α ∈ ΔAα ⊆ ∪α ∈ ΔGα and ∪α ∈ ΔGα is a nano β-open set which does not contain x. Therefore, x ∉ N-∧β(∪α ∈ ΔAα). Thus, N-∧β(∪α ∈ ΔAα) ⊆ ∪α ∈ ΔN-∧β(Aα).
Suppose that, x ∉ ∩ {N − ∧β(Aα) : α ∈ Δ}. There exists α0 ∈ Δ such that \( x\notin N-{\wedge}_{\beta}\left({A}_{\alpha_0}\right) \), and there exists a nano β-open 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(∩{Aα : 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, τR(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, τR(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 β-open sets [2].
Lemma 3.2 For subsets A, B and Aα(α ∈ Δ) of a nano topological space (U, τR(X)), the following properties hold:
- 1.
N − ∧β(A), U, ϕ are nano ∧β-sets.
- 2.
If A is a nano-β-open set, then A is a nano ∧β-set.
- 3.
If Aα is a nano ∧β-set ∀α ∈ Δ, then ∪α ∈ ΔAα is a nano ∧β-set.
- 4.
If Aα is a nano ∧β-set ∀α ∈ Δ, then ∩α ∈ ΔAα 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, τR(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 β-open set.
Remark 3.3 The nano δβ-open 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 δβ-open 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 δβ-open set.
Corollary 3.1 Let (U, τR(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 β-open in U. Thus, by Lemma 3.2 (2) B is a nano ∧β-set in U.
Definition 3.3Let (U, τR(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, τR(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.