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Some nano topological structures via ideals and graphs
Journal of the Egyptian Mathematical Society volume 28, Article number: 41 (2020)
Abstract
In this paper, new forms of nano continuous functions in terms of the notion of nano Iαopen sets called nano Iαcontinuous functions, strongly nano Iαcontinuous functions and nano Iαirresolute functions will be introduced and studied. We establish new types of nano Iαopen functions, nano Iαclosed functions and nano Iαhomeomorphisms. A comparison between these types of functions and other forms of continuity will be discussed. We prove the isomorphism between simple graphs via the nano continuity between them. Finally, we apply these topological results on some models for medicine and physics which will be used to give a solution for some reallife problems.
Introduction and preliminaries
The theory of nano topology was introduced by Lellis Thivagar et al. [1]. They defined a nano topological space with respect to a subset X of a universe U which is defined based on lower and upper approximations of X.
Definition 1.1 [2]. Let U be a certain set called the universe set and let R be an equivalence relation on U. The pair (U, R) is called an approximation space. Elements belonging to the same equivalence class are said to be indiscernible with one another. Let X ⊆ U.

(i)
The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by L_{R}(X). That is \( {L}_R(X)=\bigcup \limits_{x\in U}\left\{{R}_x:{R}_x\subseteq X\right\} \), where R_{x} denotes to the equivalence class determined by x.

(ii)
The upper approximation of X with respect to R is the set of all objects, which can be possibly classified as X with respect to R and it is denoted by U_{R}(X) . That is \( {U}_R(X)=\bigcup \limits_{x\in U}\left\{{R}_x:{R}_x\cap X\ne \varnothing \right\} \), where R_{x} denotes to the equivalence class determined by x.

(iii)
The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not X with respect to R and it is denoted by B_{R}(X). That is B_{R}(X) = U_{R}(X) − L_{R}(X), where R_{x} denotes the equivalence class determined by x.
According to Pawlak’s definition, X is called a rough set if U_{R}(X) ≠ L_{R}(X).
Definition 1.2 [3, 4]. Let U be the universe and R be an equivalence relation on U and τ_{R}(X) = {U, ∅ , L_{R}(X), U_{R}(X), B_{R}(X)}, where X ⊆ U and τ_{R}(X ) satisfies the following axioms:

(i)
$$ U\ and\varnothing \in {\tau}_R(X); $$

(ii)
The union of elements of any subcollection of τ_{R}(X) is in τ_{R}(X);

(iii)
The intersection of the elements of any finite subcollection of τ_{R}(X) in τ_{R}(X).
That is τ_{R}(X) forms a topology on U. (U, τ_{R}(X)) is called a nano topological space. Nanoopen sets are the elements of (U, τ_{R}(X)). It originates from the Greek word ‘nanos’ which means ‘dwarf’ in its modern scientific sense, an order to magnitudeone billionth. The topology is named as nano topology so because of its size since it has at most five elements [4]. The dual nano topology is [τ_{R}(X)]^{c} = F_{R}(X) and its elements are called nano closed sets.
Lellis Thivagar et al. [5] defined the concept of nano topological space via a direct simple graph.
Definition 1.3 [5, 6]. A graph G is an ordered pair of disjoint sets (V, E), where V is nonempty and E is a subset of unordered pairs of V. The vertices and edges of a graph G are the elements of V = V(G) and E = E(G), respectively. We say that a graph G is finite (resp. infinite) if the set V(G) is finite (resp. finite).
Definition 1.4 [5]. Let G(V, E) be a directed graph and u, v ∈ V(G), then:

(i)
u is invertex of v if \( \overrightarrow{uv}\in E(G) \).

(ii)
u is outvertex of v if \( \overrightarrow{vu}\in E(G) \).

(iii)
The neighborhood of v is denoted by N(v), and given by \( N(v)=\left\{v\right\}\cup \left\{u\in V\ (G):\overrightarrow{vu}\in E(G)\right\} \)
Definition 1.5. Let G(V, E) be a graph and H be a subgraph of G. Then

(i)
[5] The lower approximation L : P(V(G)) ⟶ P(V(G)) is \( {L}_N\left(V(H)\right)=\bigcup \limits_{v\in V(G)}\left\{v:N(v)\subseteq V(H)\right\} \);

(ii)
[7] The upper approximation U : P(V(G)) ⟶ P(V(G)) is \( {U}_N\left(V(H)\right)=\bigcup \limits_{v\in V(G)}\left\{v:N(v)\cap V(H)\ne \varnothing \right\} \);

(iii)
[5] The boundary is B_{N}(V(H)) = U_{N}(V(H)) − L_{N}(V(H)).
Let G be a graph, N(v) be a neighbourhood of v in V and H be a subgraph of G. τ_{N}(V(H)) = {V(G), ∅, L_{N}(V(H)), U_{N}(V(H)), B_{N}(V(H))} forms a topology on V(G) called the nano topology on V(G) with respect to V(H). (V(G), τ_{N}(V(H))) is a nano topological space induced by a graph G.
Nano closure and nano interior of a set are also studied by Lellis Thivagar and Richard and put their definitions as:
Definition 1.6 [1]. If (U, τ_{R}(X)) is a nano topological space with respect to X where X ⊆ U. If A ⊆ U, then the nano interior of A is defined as the union of all nanoopen subsets of A and it is denoted by NInt(A). That is, NInt(A) is the largest nanoopen subset of A. The nano closure of A is defined as the intersection of all nano closed sets containing A and it is denoted by NCl(A). That is, NCl(A) is the smallest nano closed set containing A.
Continuity of functions is one of the core concepts of topology. The notion of nano continuous functions was introduced by Lellis Thivagar and Richard [4]. They derived their characterizations in terms of nano closed sets, nano closure and nano interior. They also established nanoopen maps, nano closed maps and nano homeomorphisms and their representations in terms of nano closure and nano interior.
Definition 1.7 [4]. Let (U, τ_{R}(X)) and \( \left(V,{\tau}_{\acute{R}}(Y)\right) \) be nano topological spaces. Then a mapping \( f:\left(U,{\tau}_R(X)\right)\to \left(V,{\tau}_R(Y)\right) \) is nano continuous on U if the inverse image of every nanoopen set in V is nanoopen in U.
Definition 1.8 [4]. A function\( f:\left(U,{\tau}_{\acute{R}}(X)\right)\to \left(V,{\tau}_{\acute{R}}(Y)\right) \) is a nanoopen map if the image of every nanoopen set in U is nano open in V. The mapping f is said to be a nano closed map if the image of every nano closed set in U is nano closed in V.
Definition 1.9 [4]. A function \( f:\left(U,{\tau}_{\acute{R}}(X)\right)\to \left(V,{\tau}_{\acute{R}}(Y)\right) \) is said to be a nano homeomorphism if

(i)
f is 11 and onto,

(ii)
f is nano continuous and

(iii)
f is nano open.
Graph isomorphism is a related task of deciding when two graphs with different specifications are structurally equivalent, that is whether they have the same pattern of connections. Nano homeomorphism between two nano topological spaces are said to be topologically equivalent. Here, we are formalizing the structural equivalence for the graphs and their corresponding nano topologies generated by them.
Definition 1.10 [8]. Two directed graphs G and H are isomorphic if there is an isomorphism f between their underlying graphs that preserves the direction of each edge. That is, e is directed from u to v if and only if f(e) is directed from f(u) to f(v).
Definition 1.11 [8]. Two directed graphs C and D are isomorphic if D can be obtained by relabeling the vertices of C, that is, if there is a bijection between the vertices of C and those of D, such that the arcs joining each pair of vertices in C agree in both number and direction with the arcs joining the corresponding pair of vertices in D.
The subject of ideals in topological spaces have been studied by Kuratowski [9] and Vaidyanathaswamy [10]. There have been many great attempts, so far, by topologies to use the concept of ideals for maneuvering investigations of different problems of topology. In this connection, one may refer to the works in [11,12,13].
Definition 1.12 [9]. An ideal I on a set X is a nonempty collection of subsets of X which satisfies the conditions:

(i)
A ∈ I and B ⊆ A implies B ∈ I,

(ii)
A ∈ I and B ∈ I implies A ∪ B ∈ I.

(iii)
The concept of a set operator ()^{α∗} : Ρ(X) → Ρ(X) was introduced by Nasef [14] in 1992, which is called an αlocal function of I with respect to τ. In 2013, the notion of Iαopen set was introduced by Abd ElMonsef et al. [15] and has been studied by Radwan et al. [16, 17].
Definition 1.13 [15].: A subset A of an ideal topological space (X, τ, I) is said to be Iαopen if it satisfies that A ⊆ int (cl^{α∗}[int(A)]). The family of all Iaopen sets in ideal topological space (X, τ, I) is denoted by IαO(X).
It was made clear that each open set is Iαopen, but the converse may not be true, in general [16]. Radwan et al. have shown that the family of all Iαopen sets is a supra topology. In [18], the method of generating nano Iαopen sets are introduced and studied by Kozae et al.
Definition 1.14 [18]. A subset X of a nano ideal topological space (U, τ(X), I) is said to be Iαopen if it satisfies that A ⊆ NInt(NCl^{α∗}[NInt(A)]). The family of all nano Iαopen sets in nano ideal topological space (U, τ(X), I) is denoted by NIαO(U). The elements of [NIαO(U)]^{c} are nano Iαclosed sets in nano ideal topological space (U, τ(X), I) and denoted by NIαC(U).
Also, discussions of various properties of nano Iαopen sets are given, such as nano Iαclosure and nano Iαinterior of a set.
Definition 1.15 [18]. Let (U, τ(X), I) be a nano ideal topological space and A ⊆ U. The nano Iαinterior of A is defined as the union of all nano Iαopen subsets of A and it is denoted by NIαInt(A). That is, NIαInt(A) is the largest nano Iαopen subset of A. The nano Iαclosure of A is defined as the intersection of all nano Iαclosed sets containing A and it is denoted by NIαCl(A). That is, NIαCl(A) is the smallest nano Iαclosed set containing A.
2 NIαcontinuous functions and NIαhomeomorphims
We define some new functions in this section, say, nano Iαcontinuous, nano Iαopen (closed), nano Iαhomeomorphism and other functions. Also, study the relationship between these functions, one to other and between them and nano continuous function, nanoopen, nano closed and nano homeomorphism functions.
2.1 New types of NIαcontinuous functions:
Definition 2.1.1. Let \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) be a function. f is said to be

(i)
Nano Iαcontinuous function if \( {f}^{1}(B)\in NI\alpha O(U), for\ all\ B\in {\tau}_{\acute{R}}(Y) \).

(ii)
Strongly nano Iαcontinuous function if f^{−1}(B) ∈ τ_{R}(X), for all B ∈ NIαO(V).

(iii)
Nano Iα irresolute continuous function if f^{−1}(B) ∈ NIαO(U)for all B ∈ NIαO(V).
Proposition 2.1.2. A function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) is nano Iαcontinuous function if and only if one of the following is satisfied;

(i)
$$ {f}^{1}(B)\in NI\alpha C(U), for\ all\ B\in {F}_{\acute{R}}(Y). $$

(ii)
The inverse image of every member of the basis \( \overset{\hbox{'}}{B} \)of \( {\tau}_{\acute{R}}(Y) \) is NIαopen set in U.

(iii)
NIαcl [f^{−1} (B)] ⊆ f^{−1} [NCl(B)] , for all B ⊆ V.

(iv)
f^{−1} [NInt (B)] ⊆ NIαint [f^{−1}(B)] , for all B ⊆ V.
Proof:

(i)
Necessity: let f be nano Iαcontinuous and \( B\in {F}_{\acute{R}}(Y) \). That is, \( \left(VB\right)\in {\tau}_{\acute{R}}(Y) \). Since f is nano Iαcontinuous, f^{−1}(V − B) ∈ NIαO(U). That is, (U − f^{−1}(B )) ∈ NIαO(U). Therefore, f^{−1}(B ) ∈NIαC(U). Thus, the inverse image of every nano closed set in V is NIαclosed in U, if f is nano Iαcontinuous on U. Sufficiency: let \( {f}^{1}(B)\in NI\alpha C(U), for\ all\ B\in {F}_{\acute{R}}(Y) \). Let \( B\in {\tau}_{\acute{R}}(Y) \), then \( \left(VB\right)\in {F}_{\acute{R}}(Y) \) and f^{−1}(V − B) ∈ NIαC(U). That is, (U − f^{−1}(B)) ∈ NIαC(U) and therefore f^{−1}(B) ∈ NIαO(U). Thus, the inverse image of every nanoopen set in V is NIαopen in U. That is, f is nano Iαcontinuous on U.

(ii)
Necessity: let f be nano Iαcontinuous on U. Let \( B\in \overset{\hbox{'}}{B} \). Then \( B\in {\tau}_{\acute{R}}(Y) \). Since f is nano Iαcontinuous, f^{−1}(B) ∈ NIαO(U). That is, the inverse image of every member of \( \overset{\hbox{'}}{B} \)is NIαopen set in U. Sufficiency: let the inverse image of every member of \( \overset{\hbox{'}}{B} \)be NIαopen set in U. Let G be a nanoopen set in V. Then G = ∪ {B : B ∈ B_{1}}, where \( {B}_1\in \overset{\hbox{'}}{B} \). Then f^{−1}(G) = f^{−1} (∪{B : B ∈ B_{1}}) = ∪ {f^{−1}(B) : B ∈ B_{1}} , where each f^{−1}(B) ∈ NIαO(U) and hence their union, which is f^{−1}(G) is NIαopen in U. Thus f is nano Iαcontinuous on U.

(iii)
Necessity: if f is nano Iαcontinuous and B ⊆ V, \( NCl(B)\in {F}_{\overset{\hbox{'}}{R}}(Y) \)and from (i) f^{−1}(NCl(B)) ∈ NIαC(U). Therefore, NIαcl(f^{−1}(NCl(B))) = f^{−1}(NCl (B)) . Since B ⊆ NCl(B), f^{−1}(B) ⊆ f^{−1}(NCl(B)). Therefore, NIαcl(f^{−1}(B)) ⊆ NIαcl(f^{−1}(NCl(B))) = f^{−1}(NCl(B)) . That is, NIαcl(f^{−1}(B)) ⊆ f^{−1}(NCl(B)). Sufficiency: let NIαcl(f^{−1}(B)) ⊆ f^{−1}(NCl(B)) for every B ⊆ V. Let \( B\in {F}_{\acute{R}}(Y) \), then NCl(B) = B. By assumption, NIαcl(f^{−1}(B)) ⊆ f^{−1}(NCl(B)) = f^{−1} (B). Thus, NIαcl(f^{−1}(B)) ⊆ f^{−1} (B). But f^{−1} (B) ⊆ NIαcl(f^{−1}(B)) . Therefore, NIαcl(f^{−1}(B)) = f^{−1}(B). That is, f^{−1}(B) is NIαclosed in U for every nano closed set B in V. Therefore, f is nano Iαcontinuous on U.

(iv)
Necessity: let f be nano Iαcontinuous and B ⊆ V. Then \( NInt(B)\in {\tau}_{\acute{R}}(Y) \). Therefore, f^{−1}(NInt(B)) ∈ NIαO(U). That is, f^{−1}(NInt(B)) = NIα int (f^{−1}(NInt(B))). Also, NInt(B) ⊆ B implies that NIαint(f^{−1}(NInt(B))) ⊆ NIαint(f^{−1}(B)). Therefore f^{−1}(NInt(B)) = NIαint(f^{−1}(NInt(B))) ⊆ NIαint(f^{−1}(B)). That is, f^{−1}(NInt(B)) ⊆ NIαint(f^{−1}(B)) . Sufficiency: let f^{−1}(NInt(B)) ⊆ NIαint(f^{−1}(B)) for every subset B of V. If \( B\in {\tau}_{\acute{R}}(Y) \), B = NInt(B). Also, f^{−1}(B) = f^{−1}(NInt(B)), but f^{−1}(NInt(B)) ⊆ NIαint(f^{−1}(B)). That is, f^{−1}(B) = f^{−1}(NInt(B)) ⊆ NIαint(f^{−1}(B)). Therefore, f^{−1}(B) = NIαint(f^{−1}(B)). Thus, f^{−1}(B) is NIαopen in U for every nanoopen set B in V. Therefore, f is nano Iαcontinuous.
Proposition 2.1.3. A function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) is strongly nano Iαcontinuous function if and only if one of the following is satisfied;

(i)
$$ {f}^{1}(B)\in {F}_R(X), for\ all\ B\in NI\alpha C(V). $$

(ii)
The inverse image of every member of the basis \( \overset{\hbox{'}}{B} \)of NIαopen set of V is nanoopen set in U.

(iii)
NCl [f^{−1} (B)] ⊆ f^{−1}[NIαcl(B)], for all B ⊆ V.

(iv)
f^{−1}[NIαint(B)] ⊆ NInt[f^{−1}(B)] , for all B ⊆ V.
Proof:

(i)
Necessity: let f be strongly nano Iαcontinuous and B ∈ NIαC(V). That is, (V − B) ∈ NIαO(V), since f is strongly nano Iαcontinuous, f^{−1}(V − B) ∈ τ_{R}(X), and (U − f^{−1}(B )) ∈ τ_{R}(X). Therefore, f^{−1}(B ) ∈ F_{R}(X). Thus, f^{−1}(B) ∈ F_{R}(X), for all B ∈ NIαC(V) , if f is strongly nano Iαcontinuous on U. Sufficiency: let f^{−1}(B) ∈ F_{R}(X), for all B ∈ NIαC(V). Let B ∈ NIαO(V). Then (V − B) ∈ NIαC(V). Then, f^{−1}(V − B) ∈ F_{R}(X) that is, (U − f^{−1}(B)) ∈ F_{R}(X). Therefore, f^{−1}(B) ∈ τ_{R}(X). Thus, the inverse image of every NIαopen set in V is nanoopen in U. That is, f is strongly nano Iαcontinuous on U.

(ii)
Necessity: let f be strongly nano Iαcontinuous on U. Let \( B\in \overset{\hbox{'}}{B} \). Then B ∈ NIαO(V). Since f is strongly nano Iαcontinuous, f^{−1}(B) ∈ NIαO(U). That is, the inverse image of every member of \( \overset{\hbox{'}}{B} \)is nanoopen set in U. Sufficiency: let the inverse image of every member of \( \overset{\hbox{'}}{B} \)be nanoopen set in U. Let G be NIαopen set in V. Then G = ∪ {B : B ∈ B_{1}}, where \( {B}_1\in \overset{\hbox{'}}{B} \). Then f^{−1}(G) = f^{−1} (∪{B : B ∈ B_{1}}) = ∪ {f^{−1}(B) : B ∈ B_{1}}, where each f^{−1}(B) ∈ τ_{R}(X) and hence their union, which is f^{−1}(G) is nanoopen in U. Thus f is strongly nano Iαcontinuous on U.

(iii)
Necessity: if f is strongly nano Iαcontinuous and B ⊆ V , NIαcl(B) ∈ NIαC(V) and from (i) f^{−1}(NIαcl(B)) ∈ F_{R}(X). Therefore, NCl(f^{−1}(NIα cl(B))) = f^{−1}(NIα cl(B)). Since B ⊆ NIαcl(B) , f^{−1}(B) ⊆ f^{−1}(NIα cl(B)). Therefore, NCl(f^{−1}(B)) ⊆ NCl(f^{−1}(NIα cl(B))) = (f^{−1}(NIαcl(B))). That is, NCl(f^{−1}(B)) ⊆ (f^{−1}(NIα cl(B))). Sufficiency: let NCl(f^{−1}(B)) ⊆ (f^{−1}(NIα cl(B))) for every B ⊆ V. Let B ∈ NIαC(V). Then NIαcl (B) = B . By assumption, NCl(f^{−1}(B)) ⊆ (f^{−1}(NIα cl(B))) = f^{−1} (B). Thus, NCl(f^{−1}(B)) ⊆ f^{−1}(B). But f^{−1}(B) ⊆ NCl(f^{−1}(B)). Therefore, NCl(f^{−1}(B)) = f^{−1}(B) . That is, f^{−1}(B) ∈ F_{R}(X) for every NIαclosed set B in V. Therefore, f is strongly nano Iαcontinuous on U.

(iv)
Necessity: let f be strongly nano Iαcontinuous and B ⊆ V. Then NIαint(B) ∈ NIαO(V). Therefore, (f^{−1}(NIαint(B))) ∈ τ_{R}(X). That is, f^{−1}(NIαint(B)) = NInt(f^{−1}(NIαint(B))) . Also, NIαint(B) ⊆ B implies that NInt(f^{−1}(NIαint(B))) ⊆ NInt(f^{−1}(B)). Therefore f^{−1}(NIαint(B)) = NInt(f^{−1}(NIα int (B))) ⊆ NInt(f^{−1}(B)) . That is, f^{−1}(NIα int(B))) ⊆ NInt(f^{−1}(B)). Sufficiency: let f^{−1}(NIαint(B))) ⊆ NInt(f^{−1}(B)) for every subset B of V. If B is NIαopen set in V, B = (NIαint(B)). Also,f^{−1}(B) = f^{−1}(NIαint(B)), but f^{−1}(NIαint(B)) ⊆ NInt(f^{−1}(B)). That is, f^{−1}(B) = f^{−1}(NIαint(B)) ⊆ NInt(f^{−1}(B)). Therefore, f^{−1}(B) = NInt(f^{−1}(B)). Thus, f^{−1}(B) is nanoopen in U for every NIαopen set B in V. Therefore, f is strongly nano Iαcontinuous.
Proposition 2.1.4. A function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) is nano Iαirresolute continuous function if and only if one of the following is satisfied;

(i)
$$ {f}^{1}(B)\in NI\alpha C(U), for\ all\ B\in NI\alpha C(V). $$

(ii)
The inverse image of every member of the basis \( \overset{\hbox{'}}{B} \)of NIαopen set of V is NIαopen set in U.

(iii)
NIαcl [f^{−1} (B)] ⊆ f^{−1}[NIαcl(B)], for all B ⊆ V.

(iv)
f^{−1}[NIαint(B)] ⊆ NIαint [f^{−1}(B)], for all B ⊆ V.
Proof:

(i)
Necessity: let f be nano Iαirresolute continuous and B ∈ NIαC(V). That is, (V − B) ∈ NIαO(V). Since f is nano Iαirresolute continuous, f^{−1}(V − B) ∈NIαO(U). That is, (U − f^{−1}(B)) ∈ NIαO(U), and therefore f^{−1}(B ) ∈ NIαC(U). Thus, f^{−1}(B) ∈ NIαC(U), for all B ∈ NIαC(V), if f is nano Iαirresolute continuous on U. Sufficiency: let f^{−1}(B) ∈ NIαC(U), for all B ∈ NIαC(V). Let B ∈ NIαO(V). Then (V − B) is IαC(V). Then, f^{−1}(V − B)∈ IαC(U), that is, (U − f^{−1}(B))∈ IαC(U). Therefore, f^{−1}(B)∈ NIαO(U). Thus, f^{−1}(B) ∈ NIαO(U), for all B ∈ NIαO(V). That is, f is nano Iαirresolute continuous on U.

(ii)
Necessity: let f be nano Iαirresolute continuous on U. Let \( B\in \overset{\hbox{'}}{B} \). Then B ∈ NIαO(V). Since f is nano Iαirresolute continuous, f^{−1}(B) ∈ NIαO(U). That is, the inverse image of every member of \( \overset{\hbox{'}}{B} \)is NIαO(U). Sufficiency: let the inverse image of every member of \( \overset{\hbox{'}}{B} \)be NIαopen set in U. Let G ∈ NIαO(V). Then G = ∪ {B : B ∈ B_{1}} , where \( {B}_1\in \overset{\hbox{'}}{B} \). Then f^{−1}(G) = f^{−1} (∪{B : B ∈ B_{1}}) = ∪ {f^{−1}(B) : B ∈ B_{1}} , where each f^{−1}(B) ∈ NIαO(U) and hence their union, which is f^{−1}(G). Thus f is nano Iαirresolute continuous on U.

(iii)
Necessity: if f is nano Iαirresolute continuous and B ⊆ V , NIαcl(B) ∈ NIαC(V) and from (i) f^{−1}(NIαcl(B))∈ NIαC(U). Therefore, NIαcl(f^{−1}(NIα cl(B))) = f^{−1}(NIαcl(B)). Since B ⊆ NIαcl(B) , f^{−1}(B) ⊆ f^{−1}(NIαcl(B)). Therefore, NIαcl(f^{−1}(B)) ⊆ NIαcl(f^{−1}(NIαcl(B))) = (f^{−1}(NIαcl(B))). That is, NIαcl(f^{−1}(B)) ⊆ (f^{−1}(NIαcl(B))). Sufficiency: let NIαcl(f^{−1}(B)) ⊆ (f^{−1}(NIαcl(B))) for every B ⊆ V. Let B ∈ NIαC(V). Then NIαcl (B) = B . By assumption, NIαcl(f^{−1}(B)) ⊆ (f^{−1}(NIαcl(B))) = f^{−1} (B). Thus, NIαcl(f^{−1}(B)) ⊆ f^{−1} (B). But f^{−1} (B) ⊆ NIαcl(f^{−1}(B)) . Therefore, NIαcl(f^{−1}(B)) = f^{−1}(B) . That is, f^{−1}(B) is NIαclosed in U for every NIαclosed set B in V. Therefore, f is nano Iαirresolute continuous on U.

(iv)
Necessity: let f be nano Iαirresolute continuous and B ⊆ V. Then NIαint(B) ∈ NIαO(V). Therefore, (f^{−1}(NIαint(B)))∈ NIαO(U). That is, NIαint( f^{−1}(NIαint(B))) = (f^{−1}(NIαint(B))) . Also, NIαint(B) ⊆ B implies that NIαint(f^{−1}(NIαint(B))) ⊆ NIαint(f^{−1}(B)). Therefore f^{−1}(NIαint(B)) = NIαint(f^{−1}(NIαint(B))) ⊆ NIαint(f^{−1}(B)) . That is, f^{−1}(NIαint(B))) ⊆ NIαint(f^{−1}(B)). Sufficiency: let f^{−1}(NIαint(B))) ⊆ NIαint(f^{−1}(B)) for every subset B of V. If B ∈ NIαO(V), B = (NIαint(B)). Also, f^{−1}(B) = f^{−1}(NIαint(B)) but, f^{−1}(Iαint(B)) ⊆ NIαint(f^{−1}(B)). That is, f^{−1}(B) = f^{−1}(NIαint(B)) ⊆ NIαint(f^{−1}(B)). Therefore, f^{−1}(B) = NIαint(f^{−1}(B)). Thus, f^{−1}(B) is NIαopen in U for every NIαopen set B in V . Therefore, f is nano NIαirresolute continuous.
Remark 2.1.5. The following implication shows the relationships between different types of nano continuous functions.
The converse of the above diagram is not reversible, in general, as shown in Example 2.1.6.
Example 2.1.6. Consider the nano ideal topological spaces ( U, τ_{R}(X), I) and \( \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that \( U=\left\{x,y,z\right\},V=\left\{a,b,c\right\},U/R=\left\{\left\{x\right\},\left\{y\right\},\left\{z\right\}\right\},V/\acute{R}=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\} \), if we take X = {x}, Y = {b}, then \( {\tau}_R(X)=\left\{U,\varnothing, \left\{x\right\}\right\},{\tau}_{\acute{R}}(Y)=\left\{V,\varnothing, \left\{b\right\}\right\} \) and by taking I = {∅, {y}}, J = {∅, {a}, {c}, {a, c}}, so NIαO(U) = {U, ∅ , {x}, {x, y}, {x, z}}, NIαO(V) = {V, ∅ , {b}, {a, b}, {b, c}}. Define the function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that

(i)
f(x) = f(y) = a, f(z) = c. This function is nano Iαcontinuous and nano continuous, but it is not nano Iαirresolute continuous for {b, c} ∈ NIαO(V), but f^{−1}({b, c}) = {z} ∉ NIαO(U) . It is not strongly nano Iαcontinuous since {a, b} ∈ NIαO(V), but f^{−1}({a, b}) = {x, y} ∉ τ_{R}(X).

(ii)
f(x) = f(z) = b and f(y) = a. This function is nano Iαirresolute continuous and nano Iαcontinuous but neither strongly nano Iαcontinuous nor nano continuous function for \( \left\{b\right\}\in {\tau}_{\acute{R}}(Y)\subseteq NI\alpha O(V) \), but f^{−1}({b}) = {x, z} ∉ τ_{R}(X).
Remark 2.1.7. Consider the function \( :\left(U,{\tau}_R(X),I\right)\to \left(V,{\tau}_{\acute{R}}(Y),J\right) \) . The following statements are held.

(i)
If f is nano Iαcontinuous function, it is not necessary that the f(A) ∈ NIαC(V), for all \( A\in {F}_{\acute{R}}(Y) \).

(ii)
If f is strongly nano Iαcontinuous function, it is not necessary that\( f(A)\in {\tau}_{\acute{R}}(Y) \), for all A ∈ NIαC(U).

(iii)
If f is nano Iαirresolute continuous function, it is not necessary that f(A) ∈ NIαC(V), for all A ∈ NIαC(U).
We show this remark by using the following example.
Example 2.1.8. Consider the nano ideal topological spaces ( U, τ_{R}(X), I) and \( \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that \( U=\left\{x,y,z\right\},V=\left\{a,b,c\right\},U/R=\left\{\left\{x\right\},\left\{y\right\},\left\{z\right\}\right\},V/\acute {R}=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\} \), if we take X = {x}, Y = {b}, then \( {\tau}_R(X)=\left\{U,\varnothing, \left\{x\right\}\right\},{\tau}_{\acute{R}}(Y)=\left\{V,\varnothing, \left\{b\right\}\right\} \) and by taking I = {∅, {y}}, J = {∅, {a}, {c}, {a, c}}, so NIαO(U) = {U, ∅ , {x}, {x, y}, {x, z}}, NIαO(V) = {V, ∅ , {b}, {a, b}, {b, c}}. Define the function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that

(i)
f(x) = f(y) = f(z) = c. This function is nano Iαcontinuous. But {y, z} ∈ F_{R}(X) and f({y, z}) = {c} ∉ NIαC(V).

(ii)
f(x) = b and f(y) = f(z) = c. This function is strongly nano Iαcontinuous. But {y} ∈ NIαC(X) and \( f\left(\left\{y\right\}\right)=\left\{c\right\}\notin {F}_{\acute{R}}(Y) \).

(iii)
f(x) = f(y) = c , f(z) = b. This function is nano Iαirresolute continuous. But {z} ∈ NIαC(X) and ({z}) = {b} ∉ NIαC(V) .
Definition 2.1.9. Let \( f:\left(U,{\tau}_R(X),I\right)\to \left(V,{\tau}_{\acute{R}}(Y),J\right) \) be a function. f is said to be

(i)
Nano Iαopen [nano Iαclosed] function if f(A) ∈ NIαO(V), for all A ∈ τ_{R}(X) [f(A) ∈ NIαC(V)], for all A ∈ F_{R}(X)) respectively.

(ii)
Strongly nano Iαopen [strongly nano Iαclosed] function if \( f(A)\in {\tau}_{\acute{R}}(Y) \), for all A ∈ NIαO(U) [\( f(A)\in {F}_{\acute{R}}(Y)\Big] \), for all A ∈ NIαC(U)), respectively.

(iii)
Nano Iαalmost open (nano Iαalmost closed) function if f(A) ∈ NIαO(V), for all A ∈ NIαO(U) [f(A) ∈ NIαC(V)], for all A ∈ NIαC(U)), respectively.
Remark 2.1.10. The following implication shows the relationships between different types of nanoopen functions.
The converse of the above diagram is not reversible, in general, as shown in Examples 2.1.11 and 2.1.12.
Example 2.1.11. Consider the nano ideal topological spaces ( U, τ_{R}(X), I) and \( \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that \( U=\left\{x,y,z\right\},V=\left\{a,b,c\right\},U/R=\left\{\left\{x\right\},\left\{y\right\},\left\{z\right\}\right\},V/\acute{R}=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\} \), if we take X = {x}, Y = {b}, then \( {\tau}_R(X)=\left\{U,\varnothing, \left\{x\right\}\right\},{\tau}_{\acute{R}}(Y)=\left\{V,\varnothing, \left\{b\right\}\right\} \) and by taking I = {∅, {y}}, J = {∅, {a}, {b}, {a, b}}, so NIαO(U) = {U, ∅ , {x}, {x, y}, {x, z}}, NIαO(V) = {V, ∅ , {b}}. Define the function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that f(x) = b, f(y) = a and f(z) = c. This function is nano Iαopen and nanoopen, but it is neither nano Iαalmost open nor strongly nano Iαopen for {x, y} ∈ NIαO(U), but f({x, y}) = {a, b} ∉ NIαO(V) and \( f\left(\left\{x,y\right\}\right)=\left\{a,b\right\}\notin {\tau}_{\acute{R}}(Y) \).
Example 2.1.12. Consider the nano ideal topological spaces ( U, τ_{R}(X), I) and \( \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that \( U=\left\{x,y,z\right\},V=\left\{a,b,c\right\},U/R=\left\{\left\{x\right\},\left\{y\right\},\left\{z\right\}\right\},V/\acute{R}=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\} \), if we take X = {x}, Y = {b}, then \( {\tau}_R(X)=\left\{U,\varnothing, \left\{x\right\}\right\},{\tau}_{\acute{R}}(Y)=\left\{V,\varnothing, \left\{b\right\}\right\} \) and by taking I = {∅, {y}}, J = {∅, {a}, {c}, {a, c}}, so NIαO(U) = {U, ∅ , {x}, {x, y}, {x, z}}, NIαO(V) = {V, ∅ , {b}, {a, b}, {b, c}}. Define the function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that f(x) = b, f(y) = f(z) = a. This function is nano Iαopen and nano Iαalmost open, but it is neither strongly nano Iαopen nor nanoopen for U ∈ τ_{R}(X) ⊆ NIαO(U), but \( f(U)=\left\{a,b\right\}\notin {\tau}_{\acute{R}}(Y) \).
Remark 2.1.13. The following implication shows the relationships between different types of nano closed functions.
The converse of the above diagram is not reversible, in general, as shown in Examples2.1.14 and 2.1.15.
Example 2.1.14. Consider the nano ideal topological spaces ( U, τ_{R}(X), I) and \( \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that \( U=\left\{x,y,z\right\},V=\left\{a,b,c\right\},U/R=\left\{\left\{x\right\},\left\{y\right\},\left\{z\right\}\right\},V/\acute{R}=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\} \), if we take X = {x}, Y = {b}, then \( {\tau}_R(X)=\left\{U,\varnothing, \left\{x\right\}\right\},{\tau}_{\acute{R}}(Y)=\left\{V,\varnothing, \left\{b\right\}\right\} \) and by taking I = {∅, {y}}, J = {∅, {a}, {b}, {a, b}}, so NIαO(U) = {U, ∅ , {x}, {x, y}, {x, z}}, NIαO(V) = {V, ∅ , {b}}. Define the function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that f(x) = f(y) = a and f(z) = c. This function is nano Iαclosed and nano closed, but it is neither nano Iαalmost closed nor strongly nano Iαclosed for {y} ∈ NIαC(U), but f({y}) = {a} ∉ NIαC(V) and \( f\left(\left\{y\right\}\right)=\left\{a\right\}\notin {F}_{\acute{R}}(Y) \).
Example 2.1.15. Consider the nano ideal topological spaces ( U, τ_{R}(X), I) and \( \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that \( U=\left\{x,y,z\right\},V=\left\{a,b,c\right\},U/R=\left\{\left\{x\right\},\left\{y\right\},\left\{z\right\}\right\},V/\acute{R}=\left\{\left\{a\right\},\left\{b\right\},\left\{c\right\}\right\} \), if we take X = {x}, Y = {b}, then \( {\tau}_R(X)=\left\{U,\varnothing, \left\{x\right\}\right\},{\tau}_{\acute{R}}(Y)=\left\{V,\varnothing, \left\{b\right\}\right\} \) and by taking I = {∅, {y}}, J = {∅, {a}, {c}, {a, c}}, so NIαO(U) = {U, ∅ , {x}, {x, y}, {x, z}}, NIαO(V) = {V, ∅ , {b}, {a, b}, {b, c}}. Define the function \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) such that f(x) = a, f(y) = f(z) = c. This function is nano Iαclosed and nano Iαalmost closed, but it is neither strongly nano Iαclosed nor nano closed for {y, z} ∈ F_{R}(X) ⊆ NIαC(U) but, \( f\left(\left\{y,z\right\}\right)=\left\{c\right\}\notin {F}_{\acute{R}}(Y) \).
NIαhomeomorphism functions:
Definition 2.2.1. Let \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) be a bijective function. 푓 is said to be

(i)
Nano Iαhomeomorphism function if f and f^{−1} are both nano Iαcontinuous functions.

(ii)
Strongly nano Iαhomeomorphism function if f and f^{−1} are both strongly nano Iαcontinuous functions.

(iii)
Nano Iαirresolute homeomorphism function if f and f^{−1} are both nano Iαirresolute continuous functions.
Remark 2.2.2. Let \( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) be a bijective function. f is said to be

(i)
Nano Iαhomeomorphism function if f is both nano Iαcontinuous and nano Iαopen function.

(ii)
Strongly nano Iαhomeomorphism function if f is both strongly nano Iαcontinuous and is strongly nano Iαopen function.

(iii)
Nano Iα irresolute homeomorphism function if f is both nano Iαirresolute continuous and nano Iαalmost open function.
Proposition 2.2.3. Let\( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) and \( g:\left(\ V,{\tau}_{\acute{R}}(Y),J\right)\to \left(\ W,{\tau}_{\acute{R}}(Z),K\right) \) be two functions. Then g ∘ f is

(i)
Nano continuous function if f, g are strongly nano Iαcontinuous and nano Iαcontinuous functions.

(ii)
Nano Iαcontinuous function if f, g are nano Iαirresolute continuous and nano continuous functions.

(iii)
Strongly nano Iαcontinuous function if f, g are strongly nano Iαcontinuous and nano Iαirresolute continuous functions.
Proof:

(i)
Take C ⊆ W such that \( C\in {\tau}_{\acute{R}}(Z) \), then g^{−1}(C) ∈ NIαO(V) and f^{−1}(g^{−1}(C)) ∈ τ_{R}(X). Thus \( C\in {\tau}_{\acute{R}}(Z),{\left(g\circ f\right)}^{1}\in {\tau}_R(X) \), so g ∘ f is nano continuous function.

(ii)
Take C ⊆ W such that \( C\in {\tau}_{\acute{R}}(Z) \), then\( {g}^{1}(C)\in {\tau}_{\acute{R}}(Y)\subseteq NI\alpha O(V) \) and f^{−1}(g^{−1}(C)) ∈ NIαO(U). Thus \( C\in {\tau}_{\acute{R}}(Z),{\left(g\circ f\right)}^{1}\in NI\alpha O(U) \), so g ∘ f is nano Iαcontinuous function.

(iii)
Take C ⊆ W such that C ∈ NIαO(W), then g^{−1}(C) ∈ NIαO(V) and f^{−1}(g^{−1}(C)) ∈ τ_{R}(X). Thus C ∈ NIαO(W), (g ∘ f)^{−1} ∈ τ_{R}(X), and g ∘ f is strongly nano Iαcontinuous function.
Proposition 2.2.4. Let\( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) and \( g:\left(\ V,{\tau}_{\acute{R}}(Y),J\right)\to \left(\ W,{\tau}_{\acute{R}}(Z),K\right) \) be two functions. Then g ∘ f is nano Iαirresolute continuous function in the following cases.

(i)
If f, g are both nano Iαirresolute continuous functions.

(ii)
If f, g are nano Iαirresolute continuous and strongly nano Iαcontinuous functions, respectively.

(iii)
If f, g are nano Iαcontinuous and strongly nano Iαcontinuous functions, respectively.
Proof: Take C ⊆ W such that C ∈ IαO(W).

(i)
Since C ∈ NIαO(W) then g^{−1}(C) ∈ NIαO(V) and f^{−1}(g^{−1}(C)) ∈ NIαO(U).

(ii)
Since C ∈ NIαO(W) then \( {g}^{1}(C)\in {\tau}_{\acute{R}}(Y)\subseteq NI\alpha O(Y) \)and f^{−1}(g^{−1}(C)) ∈ NIαO(U).

(iii)
Since C ∈ NIαO(W) then \( {g}^{1}(C)\in {\tau}_{\acute{R}}(Y) \)and f^{−1}(g^{−1}(C)) ∈ NIαO(U).
Thus, we have that C ∈ NIαO(W), (g ∘ f)^{−1} ∈ NIαO(U), and g ∘ f is nano Iαirresolute continuous function.
Proposition 2.2.5. Let\( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) and \( g:\left(\ V,{\tau}_{\acute{R}}(Y),J\right)\to \left(\ W,{\tau}_{\acute{R}}(Z),K\right) \) be two functions. Then g ∘ f is nanoopen function in the following cases:

(i)
If f, g are nano Iαopen and strongly nano Iαopen functions, respectively.

(ii)
If f, g are nanoopen and strongly nano Iαopen functions, respectively.
Proof: Take A ⊆ U such that A ∈ τ_{R}(X).

(i)
Since A ∈ τ_{R}(X) then \( f(A)\in {\tau}_{\acute{R}}(Y)\subseteq NI\alpha O(V) \)and \( g\left(f(A)\right)\in {\tau}_{\acute{R}}(Z) \).

(ii)
Since A ∈ τ_{R}(X) then \( f(A)\in {\tau}_{\acute{R}}(Y) \)and \( g\left(f(A)\right)\in {\tau}_{\acute{R}}(Z) \).
Thus, in each case, we have that \( A\in {\tau}_R(X),\left(g\circ f\right)\in {\tau}_{\acute{R}}(Z) \), and g ∘ f is nanoopen function.
Proposition 2.2.6. Let\( \kern0.5em f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) and \( g:\left(\ V,{\tau}_{\acute{R}}(Y),J\right)\to \left(\ W,{\tau}_{\acute{R}}(Z),K\right) \) be two functions. Then g ∘ f is nano Iαopen function in the following cases:

(i)
If f, g are nano Iαopen and nano Iαalmost open functions, respectively.

(ii)
If f, g are nanoopen and nano Iαalmost open functions, respectively.

(iii)
If f, g are nanoopen and nano Iαopen functions, respectively.
Proof: Take A ⊆ U such that A ∈ τ_{R}(X).

(i)
Since A ∈ τ_{R}(X) then f(A) ∈ NIαO(V) and g(f(A)) ∈ NIαO(W).

(ii)
Since A ∈ τ_{R}(X) then \( f(A)\in {\tau}_{\acute{R}}(Y)\subseteq NI\alpha O(V) \)and g(f(A)) ∈ NIαO(W).

(iii)
Since A ∈ τ_{R}(X) then \( f(A)\in {\tau}_{\acute{R}}(Y) \)and g(f(A)) ∈ NIαO(W).
Thus, in each case, we have that A ∈ τ_{R}(X), (g ∘ f) ∈ NIαO(W), and g ∘ f is nano Iαopen function.
Proposition 2.2.7. Let\( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) and \( g:\left(\ V,{\tau}_{\acute{R}}(Y),J\right)\to \left(\ W,{\tau}_{\acute{R}}(Z),K\right) \) be two functions. Then g ∘ f is strongly nano Iαopen function in the following cases.

(i)
If f, g are nano Iαalmost open and strongly nano Iαopen functions, respectively.

(ii)
If f, g are both strongly nano Iαopen functions.

(iii)
If f, g are strongly nano Iαopen and nanoopen functions, respectively.
Proof: Take A ⊆ U such that A ∈ NIαO(U).

(i)
Since A ∈ NIαO(U) then f(A) ∈ NIαO(V) and \( g\left(f(A)\right)\in {\tau}_{\acute{R}}(Z) \).

(ii)
Since A ∈ NIαO(U) then \( f(A)\in {\tau}_{\acute{R}}(Y)\subseteq NI\alpha O(V) \)and \( g\left(f(A)\right)\in {\tau}_{\acute{R}}(Z) \).

(iii)
Since A ∈ NIαO(U) then \( f(A)\in {\tau}_{\acute{R}}(Y) \)and \( g\left(f(A)\right)\in {\tau}_{\acute{R}}(Z) \).
Thus, we have that \( A\in NI\alpha O(U),\left(g\circ f\right)\in {\tau}_{\acute{R}}(Z) \), and g ∘ f is strongly nano Iαopen function.
Proposition 2.3.6. Let\( f:\left(\ U,{\tau}_R(X),I\right)\to \left(\ V,{\tau}_{\acute{R}}(Y),J\right) \) and \( g:\left(\ V,{\tau}_{\acute{R}}(Y),J\right)\to \left(\ W,{\tau}_{\acute{R}}(Z),K\right) \) be two functions. Then g ∘ f is nano Iαalmost open function in the following cases:

(i)
If f, g are both nano Iαalmost open functions.

(ii)
If f, g are strongly nano Iαopen and nano Iαalmost open functions, respectively.

(iii)
If f, g are strongly nano Iαopen and nano Iαopen functions, respectively.
Proof: Take A ⊆ U such that A ∈ NIαO(U).

(i)
Since A ∈ NIαO(U) then f(A) ∈ NIαO(V) and g(f(A)) ∈ NIαO(W).

(ii)
Since A ∈ NIαO(U) then \( f(A)\in {\tau}_{\acute{R}}(Y)\subseteq NI\alpha O(V) \)and g(f(A)) ∈ NIαO(W).

(iii)
Since A ∈ NIαO(U) then \( f(A)\in {\tau}_{\acute{R}}(Y) \)and g(f(A)) ∈ NIαO(W).
Thus, we have that A ∈ NIαO(U), (g ∘ f) ∈ NIαO(W), and g ∘ f is nano Iαalmost open function.
3 Ideal expansion on topological rough sets and topological graphs
We extend both the rough sets and graphs induced by topology in Examples 3.1 and 3.2 respectively. The expansion will be used to give a decision for some diseases as flu.
Example 3.1. An example of a decision table is presented in Table 1. Four attributes [temperature, headache, nausea and cough], one decision [flu] and six cases.
Let

(i)
R_{1} = {Temperature} , the family of all equivalence classes of IND(R) is U ∕ R_{1} = {{1, 3, 4}, {2}, {5, 6}}

(ii)
R_{2} = {Temperature, Headache}, then U ∕ R_{2} = {{1, 4}, {2}, {3}, {5}, {6}}

(iii)
R_{3} = {Headache, Cough}, then U ∕ R_{3} = {{1, 4}, {2, 5}, {3}, {6}}.
If we take, X = {x : [x]_{Nausea} = no} = {1, 3, 5} then

(i)
\( {L}_{R_1}(X)=\varnothing \), \( {U}_{R_1}(X)=\left\{1,3,4,5,6\right\} \)and \( {B}_{R_1}(X)=\left\{1,3,4,5,6\right\} \). Thus \( {\tau}_{R_1}(X)=\left\{U,\varnothing, \left\{1,3,4,5,6\right\}\right\} \).

(ii)
\( {L}_{R_2}(X)=\left\{3,5\right\} \), \( {U}_{R_2}(X)=\left\{1,3,4,5\right\} \), and \( {B}_{R_2}(X)=\left\{1,4\right\} \). Thus \( {\tau}_{R_2}(X)=\left\{X,\varnothing, \left\{1,4\right\},\left\{3,5\right\},\left\{1,3,4,5\right\}\right\} \).

(iii)
\( {L}_{R_3}(X)=\left\{3\right\} \), \( {U}_{R_3}(X)=\left\{1,2,3,4,5\right\} \)and \( {B}_{R_3}(X)=\left\{1,2,4,5\right\} \). Thus \( {\tau}_{R_3}(X)=\left\{X,\varnothing, \left\{3\right\},\left\{1,2,4,5\right\},\left\{1,2,3,4,5\right\}\right\} \).
If we take, I = {∅, {2}, {4}, {2, 4}} then

(i)
$$ {\left( NI\alpha O(U)\right)}_1=\left\{U,\varnothing, \left\{1,3,4,5,6\right\}\right\}. $$

(ii)
$$ {\left( NI\alpha O(U)\right)}_2=\left\{U,\varnothing, \left\{1,4\right\},\left\{3,5\right\},\left\{1,3,4,5\right\},\left\{1,2,3,4,5\right\},\left\{1,3,4,5,6\right\}\right\}. $$

(iii)
$$ {\left( NI\alpha O(U)\right)}_3=\left\{U,\varnothing, \left\{3\right\},\left\{1,2,4,5\right\},\left\{1,2,3,4,5\right\}\right\}. $$
Define a function\( f:\left(\ U,{\tau}_{R_3}(X),I\right)\to \left(\ V,{\tau}_{R_2}(X),I\right) \) such that f(1) = 1, f(2) = 4, f(3) = 2, f(4) = 1, f(5) = 4 and f(6) = 6. This function is nano Iαcontinuous and nano continuous, but it is neither nano Iα irresolute continuous nor strongly nano Iαcontinuous for, {1, 3, 4, 5, 6} ∈ (NIαO(U))_{2}, but f^{−1}({1, 3, 4, 5, 6}) = {1, 2, 4, 5, 6} ∉ (NIαO(U))_{3}.
Define a function\( f:\left(\ U,{\tau}_{R_2}(X),I\right)\to \left(\ V,{\tau}_{R_3}(X),I\right) \) such that f(1) = 1, f(2) = 6, f(3) = 2, f(4) = 4, f(5) = 5 and f(6) = 2. This function is nano Iαcontinuous and nano Iα irresolute continuous, but it is neither nano continuous nor strongly nano Iαcontinuous for, \( \left\{1,2,4,5\right\}\in {\tau}_{R_3}(X)\subseteq {\left( NI\alpha O(U)\right)}_3 \), but \( {f}^{1}\left(\left\{1,2,4,5\right\}\right)=\left\{1,3,4,5,6\right\}\notin {\tau}_{R_2}(X) \).
Example 3.2. A nano topology will be induced by a general graph. Figure 1 shows two different simple directed graphs G and H, where V(G) = {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}} and V(H) = {w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6}}.
From the previous figure N(v_{1}) = {v_{1}, v_{2}, v_{4}, v_{5}}, N(v_{2}) = {v_{2}, v_{3}, v_{6}}, N(v_{3}) = {v_{3}, v_{4}, v_{5}}, N(v_{4}) = {v_{4}, v_{6}}, N(v_{5}) = {v_{5}} and N(v_{6}) = {v_{5}, v_{6}}. Let X = {v_{5}}, then L(X) = {v_{5}}, U(X) = {v_{1}, v_{3}, v_{5}, v_{6}} and b(X) = {v_{1}, v_{3}, v_{6}}, which mean that τ_{R} = {V(G), ∅ , {v_{5}}, {v_{1}, v_{3}, v_{6}}, {v_{1}, v_{3}, v_{5}, v_{6}} }. take I = {∅, {v_{1}}} then NIαO(V(G)) = {V(G), ∅ , {v_{5}}, {v_{1}, v_{3}, v_{6}}, {v_{1}, v_{3}, v_{5}, v_{6}}, {v_{1}, v_{2}, v_{3}, v_{5}, v_{6}}, {v_{1}, v_{3}, v_{4}, v_{5}, v_{6}} }.
Similarly N(w_{1}) = {w_{1}, w_{4}, w_{5}, w_{6}}, N(w_{2}) = {w_{2}, w_{5}}, N(w_{3}) = {w_{3}, w_{5}, w_{6}}, N(w_{4}) = {w_{2}, w_{3}, w_{4}},
N(w_{5}) = {w_{5}} and N(w_{6}) = {w_{2}, w_{6}}. Let Y = {w_{5}}, then L(Y) = {w_{5}}, U(Y) = {w_{1}, w_{2}, w_{3}, w_{5}} and b(Y) = {w_{1}, w_{2}, w_{3}}, which mean that \( {\tau}_{\acute{R}}=\left\{V(H),\varnothing, \left\{{w}_5\right\},\left\{{w}_1,{w}_2,{w}_3\right\},\left\{{w}_1,{w}_2,{w}_3,{w}_5\right\}\ \right\} \). Take J = {∅, {w_{1}}} then NIαO(V(H)) = {V(H), ∅ , {w_{5}}, {w_{1}, w_{2}, w_{3}}, {w_{1}, w_{2}, w_{3}, w_{5}}, {w_{1}, w_{2}, w_{3}, w_{4}, w_{5}}, {w_{1}, w_{2}, w_{3}, w_{5}, w_{6}} }.
Define a function\( f:\left(\ V(G),{\tau}_R(X),I\right)\to \left(\ V(H),{\tau}_{\acute{R}}(Y),J\right) \) such that f(v_{1}) = w_{1}, f(v_{2}) = w_{4}, f(v_{3}) = w_{3}, f(v_{4}) = w_{6}, f(v_{5}) = w_{5} and f(v_{6}) = w_{6}. This function is nano continuous, nano Iαcontinuous and nano Iα – irresolute, but it is not strongly nano Iαcontinuous for, {w_{1}, w_{2}, w_{3}, w_{4}, w_{5}} ∈ NIαO(V(H)) but f^{−1}({w_{1}, w_{2}, w_{3}, w_{4}, w_{5}}) = {v_{1}, v_{2}, v_{3}, v_{5}, v_{6}} ∉ τ_{R}(X), and this function is nano open, nano Iαopen and nano Iαalmost open, but it is not strongly nano Iαopen for, {v_{1}, v_{3}, v_{4}, v_{5}, v_{6}} ∈ NIαO(V(G)), but \( f\left(\left\{{v}_1,{v}_3,{v}_4,{v}_5,{v}_6\right\}\right)=\left\{{w}_1,{w}_2,{w}_3,{w}_5,{w}_6\right\}\notin {\tau}_{\acute{R}}(Y) \), also this function is one to one and onto, therefore it is nano homeomorphism, nano Iαhomeomorphism and nano Iαirresolute homeomorphism, but it is not strongly nano Iαhomeomorphism.
4 Topological models in terms of graphs and nano topology
In this section, we apply these new types of functions on some reallife problems, especially, in medicine and physics.
4.1 The foetal circulation
In this section, we apply some of the graphs, nano topology and NIαopen sets on some of the medical application such as the blood circulation in the foetus. [D1, D2] Foetal circulation differs from adult circulation in a variety of ways to support the unique physiologic needs of a developing foetus. Once there is adequate foetalplacental circulation established, blood transports between foetus and placenta through the umbilical cord containing two umbilical arteries and one umbilical vein. The umbilical arteries carry deoxygenated foetal blood to the placenta for replenishment, and the umbilical vein carries newly oxygenated and nutrientrich blood back to the foetus. When delivering oxygenated blood throughout the developing foetus, there are unique physiologic needs, supported by specific structures unique to the foetus which facilitate these needs.
Through the medical application, we can mention a new topological model. From it, we can know each vertex in foetal circulation and what are the regions that send and receive the blood by dividing the foetal circulation into groups of vertices and edges and forming the graph on it (Fig. 2) [19]. Also, we can conclude the nano topology and NIαopen sets on it. In the graph, we consider the foetal circulation as a graph G = (V, E) by working to divide it into a set of vertices and a set of edges. The vertices represent the regions where the blood flows on it. Also, the edges represent the pathway of blood through the foetal circulation (Fig. 3) [19]. The vertices v_{1}, v_{2}, v_{3} and v_{4} (high oxygen content) represent placenta, umbilical vein, liver and ductus venosus respectively; the vertices v_{6}, v_{7}, v_{8}, v_{9}, v_{10}, v_{14}, v_{15}, v_{16} and v_{17} (medium oxygen content) represent right atrium, right ventricle, foramen ovale, pulmonary trunk, lung, ductus arteriosus, aorta, systemic circulation and umbilical arteries respectively. Also, the vertices v_{5}, v_{11}, v_{12} and v_{13} (low oxygen content) represent inferior vena cava, left atrium and left ventricle respectively.
From the previous figures, we can construct the graph of the foetal circulation as shown in Fig. 4. It is easy to generate the nano topology τ_{R} on it by using the neighbourhood of each vertex.
Define a function \( f:\left(V(G),{\tau}_R\left(V(A)\right),I\right)\to \left(V(G),{\tau}_{\acute{R}}\left(V(B)\right),J\right) \), such that f(v_{1}) = {v_{6}},

$$ f\left({v}_2\right)=\left\{{v}_7\right\},f\left({v}_3\right)=\left\{{v}_9\right\},f\left({v}_4\right)=\left\{{v}_{14}\right\},f\left({v}_5\right)=\left\{{v}_5\right\},f\left({v}_6\right)=\left\{{v}_1\right\},f\left({v}_7\right)=\left\{{v}_2\right\},f\left({v}_8\right)=\left\{{v}_8\right\}, $$

$$ f\left({v}_9\right)=\left\{{v}_3\right\},f\left({v}_{10}\right)=\left\{{v}_{10}\right\},f\left({v}_{11}\right)=\left\{{v}_{15}\right\},f\left({v}_{12}\right)=\left\{{v}_{16}\right\},f\left({v}_{13}\right)=\left\{{v}_{13}\right\},f\left({v}_{14}\right)=\left\{{v}_4\right\}, $$
f(v_{15}) = {v_{11}}, f(v_{16}) = {v_{12}} and f(v_{17}) = {v_{17}}. This function is nanocontinuous, NIαcontinuous and NIαirresolute continuous, but it is not strongly NIαcontinuous for {v_{1}, v_{5}, v_{6}, v_{7}, v_{8}, v_{9}, v_{13}, v_{14}, v_{15}, v_{16}, v_{17}} ∈ NIαO(V(B)), but f^{−1}({v_{1}, v_{5}, v_{6}, v_{7}, v_{8}, v_{9}, v_{13}, v_{14}, v_{15}, v_{16}, v_{17}}) = {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{8}, v_{11}, v_{12}, v_{13}, v_{17}} ∉ τ_{R}(V(A)).
Also, this function is nanoopen, NIαopen and NIαalmost open, but it is not strongly NIαopen for {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{8}, v_{11}, v_{12}, v_{13}, v_{17}} ∈ NIαO(V(A)), but \( f\left(\left\{{v}_1,{v}_2,{v}_3,{v}_4,{v}_5,{v}_6,{v}_8,{v}_{11},{v}_{12},{v}_{13},{v}_{17}\right\}\right)=\left\{{v}_1,{v}_5,{v}_6,{v}_7,{v}_8,{v}_9,{v}_{13},{v}_{14},{v}_{15},{v}_{16},{v}_{17}\right\}\notin {\tau}_{\acute{R}}\left(V(B)\right) \).
Clearly, this function is bijective; thus, from the previous properties, f is nanohomeomorphism, NIαhomeomorphism and NIαirresolute homeomorphism. Finally, by studying one part of this function, say A and by making new results, this function that satisfies NIαirresolute homeomorphism makes the examination of foetal circulation simplest, and by NIαirresolute homeomorphism that preserve all the topological properties of a given space, this new results will be used for the other part of this function, which is B. Therefore, there is no need to study all the foetal circulation.
4.2 Electric circuit
In this section, we study an application in physics such as an electrical circuit using graphs, nanotopology andNIαopen sets. Take two different electrical circuits and transform them into graphs that simply display different graphs. However, we can prove that these circuits have the same electrical properties with ideal nano topology on these graphs.
In Figs. 5 and 6 [20], there are two different electrical circuits C_{1} and C_{2} with two different graphs G_{1} and G_{2}, respectively. So, by taking V(A) ⊆ V(G_{1}) and V(B) ⊆ V(G_{2}), we can construct a nano topology on them.
The neighborhood of each vertex of V(G_{1}) : N_{1} = {1, 2}, N_{2} = {2, 5}, N_{3} = {1, 2, 3}, N_{4} = {3, 4} and N_{5} = {1, 4, 5}. So, by taking V(A) = {3, 4}, we get L(V(A)) = {4}, U(V(A)) = {3, 4, 5} and b(V(A)) = {3, 5} . Therefore τ_{R}(V(A)) = {V(G_{1}), ∅ , {4}, {3, 5}, {3, 4, 5}}. Let ={∅, {1}}. Then NIαO(V(A)) = {V(G_{1}), ∅ , {4}, {3, 5}, {3, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}}.
The neighbourhood of each vertex of V(G_{2}) : N_{a} = {a, c}, N_{b} = {a, b, e}, N_{c} = {c, d, e}, N_{d} = {b, d} and N_{e} = {d, e}. So by taking V(B) = {a, c}, we get L(V(B)) = {a}, U(V(B)) = {a, b, c} and b(V(B)) = {b, c}. Therefore \( {\tau}_{\acute{R}}\left(V(B)\right)=\left\{V\left({G}_2\right),\varnothing, \left\{a\right\},\left\{b,c\right\},\left\{a,b,c\right\}\right\} \). Let J = {∅, {e}}, then NIαO(V(B)) = {V(G_{2}), ∅ , {a}, {b, c}, {a, b, c}, {a, b, c, d}, {a, b, c, d}}.
Define a function\( f:\left(V\left({G}_1\right),{\tau}_R\left(V(A)\right),I\right)\to \left(V\left({G}_2\right),{\tau}_{\acute{R}}\left(V(B)\right),J\right) \), such that f(1) = {e}, f(2) = {d}, f(3) = {c}, f(4) = {a} and f(5) = {b}. This function is nanocontinuous, NIαcontinuous and NIαirresolute continuous, but it is not strongly NIαcontinuous for {a, b, c, d} ∈ NIαO(V(B)), but f^{−1}({a, b, c, d}) = {2, 3, 4, 5} ∉ τ_{R}(V(A)). Also, this function is nanoopen, NIαopen and NIαalmost open, but it is not strongly NIαopen for {2, 3, 4, 5} ∈ NIαO(V(A)), but \( f\left(\left\{2,3,4,5\right\}\right)=\left\{a,b,c,d\right\}\notin {\tau}_{\acute{R}}\left(V(B)\right) \). Clearly, this function is bijective and from the previous properties f is nanohomeomorphism, NIαhomeomorphism and NIαirresolute homeomorphism. Finally, this function which satisfies the NIαirresolute homeomorphism will make the study of the electrical circuit is easier by study one part of this function and made new results on it, then by homeomorphism, these new results can be applied to the other part of this equation.
Another application of NIαirresolute homeomorphism is to prove that two different circuits are identical in their electrical properties. To prove that we define the previous function, \( f:\left(V\left({G}_1\right),{\tau}_R\left(V(A)\right),I\right)\to \left(V\left({G}_2\right),{\tau}_{\acute{R}}\left(V(B)\right),J\right) \). Clearly, f is an isomorphism. Since G_{2} can be obtained by relabeling the vertices of G_{1}, that is, f is a bijection between the vertices of G_{1} and those of G_{2}, such that the arcs joining each pair of vertices in G_{1} accepted in both numbers and direction with the arcs joining the corresponding pair of vertices in G_{2}.
We also have \( f:\left(V\left({G}_1\right),{\tau}_R\left(V(A)\right),I\right)\to \left(V\left({G}_2\right),{\tau}_{\acute{R}}\left(V(B)\right),J\right) \) is NIαirresolute homeomorphism for every subgraph A of G_{1}, which will be studied in Table 2.
It is clear that from Table 2, the two circuits are NIαirresolute homeomorphism for every subgraph A of G_{1}, and using the previous structural equivalence technique we checked that the two circuits are equivalent.
Conclusion
In this paper, different types of NIαcontinuous, NIαopen, NIαclosed and NIαhomeomorphism are introduced and studied. Some applications on them are given in some reallife branches such as medicine and physics. We give some examples of electric circuits and study its relationship with graph theory.
Availability of data and materials
The datasets used during the current study are available in the
[D1] https://www.chop.edu/conditionsdiseases/bloodcirculationfetusandnewborn
[D3] https://www.stanfordchildrens.org/en/topic/default?id=fetalcirculation90P01790
The datasets generated during the current study are available from the corresponding author on reasonable request.
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ElAtik, A.EF.A., Hassan, H.Z. Some nano topological structures via ideals and graphs. J Egypt Math Soc 28, 41 (2020). https://doi.org/10.1186/s42787020000935
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DOI: https://doi.org/10.1186/s42787020000935
Keywords
 Ideals
 Nano topology
 NIαopen sets
 NIαcontinuous functions
 NIαhomeomorphism functions
 Directed graphs
 Foetal circulation
 Electric circuits
Mathematical Subject Classification 2010
 54C60
 54E55
 90D42
 03F55