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.

  1. (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 LR(X). That is \( {L}_R(X)=\bigcup \limits_{x\in U}\left\{{R}_x:{R}_x\subseteq X\right\} \), where Rx denotes to the equivalence class determined by x.

  2. (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 UR(X) . That is \( {U}_R(X)=\bigcup \limits_{x\in U}\left\{{R}_x:{R}_x\cap X\ne \varnothing \right\} \), where Rx denotes to the equivalence class determined by x.

  3. (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 BR(X). That is BR(X) = UR(X) − LR(X), where Rx denotes the equivalence class determined by x.

According to Pawlak’s definition, X is called a rough set if UR(X) ≠ LR(X).

Definition 1.2 [3, 4]. Let U be the universe and R be an equivalence relation on U and τR(X) = {U,  ∅ , LR(X), UR(X), BR(X)}, where X ⊆ U and τR(X ) satisfies the following axioms:

  1. (i)
    $$ U\ and\varnothing \in {\tau}_R(X); $$
  2. (ii)

    The union of elements of any sub-collection of τR(X) is in τR(X);

  3. (iii)

    The intersection of the elements of any finite sub-collection of τR(X) in τR(X).

That is τR(X) forms a topology on U. (U, τR(X)) is called a nano topological space. Nano-open sets are the elements of (U, τR(X)). It originates from the Greek word ‘nanos’ which means ‘dwarf’ in its modern scientific sense, an order to magnitude-one 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 [τR(X)]c = FR(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 non-empty 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:

  1. (i)

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

  2. (ii)

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

  3. (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

  1. (i)

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

  2. (ii)

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

  3. (iii)

    [5] The boundary is BN(V(H)) = UN(V(H)) − LN(V(H)).

Let G be a graph, N(v) be a neighbourhood of v in V and H be a subgraph of G. τN(V(H)) = {V(G),  ∅, LN(V(H)), UN(V(H)), BN(V(H))} forms a topology on V(G) called the nano topology on V(G) with respect to V(H). (V(G), τN(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, τR(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 nano-open subsets of A and it is denoted by NInt(A). That is, NInt(A) is the largest nano-open 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 nano-open maps, nano closed maps and nano homeomorphisms and their representations in terms of nano closure and nano interior.

Definition 1.7 [4]. Let (U, τR(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 nano-open set in V is nano-open 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 nano-open map if the image of every nano-open 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

  1. (i)

    f is 1-1 and onto,

  2. (ii)

    f is nano continuous and

  3. (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:

  1. (i)

    A ∈ I and B ⊆ A implies B ∈ I,

  2. (ii)

    A ∈ I and B ∈ I implies A ∪ B ∈ I.

  3. (iii)

    The concept of a set operator ()α∗ : Ρ(X) → Ρ(X) was introduced by Nasef [14] in 1992, which is called an α-local function of I with respect to τ. In 2013, the notion of Iα-open set was introduced by Abd El-Monsef 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 Iα-open if it satisfies that A ⊆  int (clα∗[int(A)]). The family of all Ia-open sets in ideal topological space (X, τ, I) is denoted by IαO(X).

It was made clear that each open set is Iα-open, but the converse may not be true, in general [16]. Radwan et al. have shown that the family of all Iα-open sets is a supra topology. In [18], the method of generating nano Iα-open 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 Iα-open if it satisfies that A ⊆ NInt(NClα∗[NInt(A)]). The family of all nano Iα-open sets in nano ideal topological space (U, τ(X), I) is denoted by NIαO(U). The elements of [NIαO(U)]c are nano Iα-closed sets in nano ideal topological space (U, τ(X), I) and denoted by NIαC(U).

Also, discussions of various properties of nano Iα-open sets are given, such as nano Iα-closure and nano Iα-interior of a set.

Definition 1.15 [18]. Let (U, τ(X), I) be a nano ideal topological space and A ⊆ U. The nano Iα-interior of A is defined as the union of all nano Iα-open subsets of A and it is denoted by NIα-Int(A). That is, NIα-Int(A) is the largest nano Iα-open subset of A. The nano Iα-closure of A is defined as the intersection of all nano Iα-closed sets containing A and it is denoted by NIα-Cl(A). That is, NIα-Cl(A) is the smallest nano Iα-closed set containing A.

2 NIα-continuous functions and NIα-homeomorphims

We define some new functions in this section, say, nano Iα-continuous, nano Iα-open (closed), nano Iα-homeomorphism and other functions. Also, study the relationship between these functions, one to other and between them and nano continuous function, nano-open, nano closed and nano homeomorphism functions.

2.1 New types of NIα-continuous 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

  1. (i)

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

  2. (ii)

    Strongly nano Iα-continuous function if f−1(B) ∈ τR(X), for all B ∈ NIαO(V).

  3. (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 Iα-continuous function if and only if one of the following is satisfied;

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

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

  3. (iii)

    NIα-cl [f−1 (B)] ⊆ f−1 [NCl(B)] , for all B ⊆ V.

  4. (iv)

    f−1 [NInt (B)] ⊆ NIα-int [f−1(B)] , for all B ⊆ V.

Proof:

  1. (i)

    Necessity: let f be nano Iα-continuous and \( B\in {F}_{\acute{R}}(Y) \). That is, \( \left(V-B\right)\in {\tau}_{\acute{R}}(Y) \). Since f is nano Iα-continuous, 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 NIα-closed in U, if f is nano Iα-continuous 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 \( \left(V-B\right)\in {F}_{\acute{R}}(Y) \) 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 nano-open set in V is NIα-open in U. That is, f is nano Iα-continuous on U.

  2. (ii)

    Necessity: let f be nano Iα-continuous on U. Let \( B\in \overset{\hbox{'}}{B} \). Then \( B\in {\tau}_{\acute{R}}(Y) \). Since f is nano Iα-continuous, f−1(B) ∈ NIαO(U). That is, the inverse image of every member of \( \overset{\hbox{'}}{B} \)is NIα-open set in U. Sufficiency: let the inverse image of every member of \( \overset{\hbox{'}}{B} \)be NIα-open set in U. Let G be a nano-open set in V. Then G =  ∪ {B : B ∈ B1}, where \( {B}_1\in \overset{\hbox{'}}{B} \). Then f−1(G) = f−1 (∪{B : B ∈ B1}) =  ∪ {f−1(B) : B ∈ B1} , where each f−1(B) ∈ NIαO(U) and hence their union, which is f−1(G) is NIα-open in U. Thus f is nano Iα-continuous on U.

  3. (iii)

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

  4. (iv)

    Necessity: let f be nano Iα-continuous 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 NIα-int(f−1(NInt(B))) ⊆ NIα-int(f−1(B)). Therefore f−1(NInt(B)) = NIα-int(f−1(NInt(B))) ⊆ NIα-int(f−1(B)). That is, f−1(NInt(B)) ⊆ NIα-int(f−1(B)) . Sufficiency: let f−1(NInt(B)) ⊆ NIα-int(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)) ⊆ NIα-int(f−1(B)). That is, f−1(B) = f−1(NInt(B)) ⊆ NIα-int(f−1(B)). Therefore, f−1(B) = NIα-int(f−1(B)). Thus, f−1(B) is NIα-open in U for every nano-open set B in V. Therefore, f is nano Iα-continuous.

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 Iα-continuous function if and only if one of the following is satisfied;

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

    The inverse image of every member of the basis \( \overset{\hbox{'}}{B} \)of NIα-open set of V is nano-open set in U.

  3. (iii)

    NCl [f−1 (B)] ⊆ f−1[NIα-cl(B)], for all B ⊆ V.

  4. (iv)

    f−1[NIα-int(B)] ⊆ NInt[f−1(B)] , for all B ⊆ V.

Proof:

  1. (i)

    Necessity: let f be strongly nano Iα-continuous and B ∈ NIαC(V). That is, (V − B) ∈ NIαO(V), since f is strongly nano Iα-continuous, f−1(V − B) ∈ τR(X), and (U − f−1(B )) ∈ τR(X). Therefore, f−1(B ) ∈ FR(X). Thus, f−1(B) ∈ FR(X), for all B ∈ NIαC(V) , if f is strongly nano Iα-continuous on U. Sufficiency: let f−1(B) ∈ FR(X), for all B ∈ NIαC(V). Let B ∈ NIαO(V). Then (V − B) ∈ NIαC(V). Then, f−1(V − B) ∈ FR(X) that is, (U − f−1(B)) ∈ FR(X). Therefore, f−1(B) ∈ τR(X). Thus, the inverse image of every NIα-open set in V is nano-open in U. That is, f is strongly nano Iα-continuous on U.

  2. (ii)

    Necessity: let f be strongly nano Iα-continuous on U. Let \( B\in \overset{\hbox{'}}{B} \). Then B ∈ NIαO(V). Since f is strongly nano Iα-continuous, f−1(B) ∈ NIαO(U). That is, the inverse image of every member of \( \overset{\hbox{'}}{B} \)is nano-open set in U. Sufficiency: let the inverse image of every member of \( \overset{\hbox{'}}{B} \)be nano-open set in U. Let G be NIα-open set in V. Then G =  ∪ {B : B ∈ B1}, where \( {B}_1\in \overset{\hbox{'}}{B} \). Then f−1(G) = f−1 (∪{B : B ∈ B1}) =  ∪ {f−1(B) : B ∈ B1}, where each f−1(B) ∈ τR(X) and hence their union, which is f−1(G) is nano-open in U. Thus f is strongly nano Iα-continuous on U.

  3. (iii)

    Necessity: if f is strongly nano Iα-continuous and B ⊆ V , NIα-cl(B) ∈ NIαC(V) and from (i) f−1(NIα-cl(B)) ∈ FR(X). Therefore, NCl(f−1(NIα- cl(B))) = f−1(NIα- cl(B)). Since B ⊆ NIα-cl(B) , f−1(B) ⊆ f−1(NIα- cl(B)). Therefore, NCl(f−1(B)) ⊆ NCl(f−1(NIα- cl(B))) = (f−1(NIα-cl(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 NIα-cl (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) ∈ FR(X) for every NIα-closed set B in V. Therefore, f is strongly nano Iα-continuous on U.

  4. (iv)

    Necessity: let f be strongly nano Iα-continuous and B ⊆ V. Then NIα-int(B) ∈ NIαO(V). Therefore, (f−1(NIα-int(B))) ∈ τR(X). That is, f−1(NIα-int(B)) = NInt(f−1(NIα-int(B))) . Also, NIα-int(B) ⊆ B implies that NInt(f−1(NIα-int(B))) ⊆ NInt(f−1(B)). Therefore f−1(NIα-int(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(NIα-int(B))) ⊆ NInt(f−1(B)) for every subset B of V. If B is NIα-open set in V, B = (NIα-int(B)). Also,f−1(B) = f−1(NIα-int(B)), but f−1(NIα-int(B)) ⊆ NInt(f−1(B)). That is, f−1(B) = f−1(NIα-int(B)) ⊆ NInt(f−1(B)). Therefore, f−1(B) = NInt(f−1(B)). Thus, f−1(B) is nano-open in U for every NIα-open set B in V. Therefore, f is strongly nano Iα-continuous.

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 Iα-irresolute continuous function if and only if one of the following is satisfied;

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

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

  3. (iii)

    NIα-cl [f−1 (B)] ⊆ f−1[NIα-cl(B)], for all B ⊆ V.

  4. (iv)

    f−1[NIα-int(B)] ⊆ NIα-int [f−1(B)], for all B ⊆ V.

Proof:

  1. (i)

    Necessity: let f be nano Iα-irresolute continuous and B ∈ NIαC(V). That is, (V − B) ∈ NIαO(V). Since f is nano Iα-irresolute 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 Iα-irresolute 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 Iα-irresolute continuous on U.

  2. (ii)

    Necessity: let f be nano Iα-irresolute continuous on U. Let \( B\in \overset{\hbox{'}}{B} \). Then B ∈ NIαO(V). Since f is nano Iα-irresolute 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 NIα-open set in U. Let G ∈ NIαO(V). Then G =  ∪ {B : B ∈ B1} , where \( {B}_1\in \overset{\hbox{'}}{B} \). Then f−1(G) = f−1 (∪{B : B ∈ B1}) =  ∪ {f−1(B) : B ∈ B1} , where each f−1(B) ∈ NIαO(U) and hence their union, which is f−1(G). Thus f is nano Iα-irresolute continuous on U.

  3. (iii)

    Necessity: if f is nano Iα-irresolute continuous and B ⊆ V , NIα-cl(B) ∈ NIαC(V) and from (i) f−1(NIα-cl(B))∈ NIαC(U). Therefore, NIα-cl(f−1(NIα- cl(B))) = f−1(NIα-cl(B)). Since B ⊆ NIα-cl(B) , f−1(B) ⊆ f−1(NIα-cl(B)). Therefore, NIα-cl(f−1(B)) ⊆ NIα-cl(f−1(NIα-cl(B))) = (f−1(NIα-cl(B))). That is, NIα-cl(f−1(B)) ⊆ (f−1(NIα-cl(B))). Sufficiency: let NIα-cl(f−1(B)) ⊆ (f−1(NIα-cl(B))) for every B ⊆ V. Let B ∈ NIαC(V). Then NIα-cl (B) = B . By assumption, NIα-cl(f−1(B)) ⊆ (f−1(NIα-cl(B))) = f−1 (B). Thus, NIα-cl(f−1(B)) ⊆ f−1 (B). But f−1 (B) ⊆ NIα-cl(f−1(B)) . Therefore, NIα-cl(f−1(B)) = f−1(B) . That is, f−1(B) is NIα-closed in U for every NIα-closed set B in V. Therefore, f is nano Iα-irresolute continuous on U.

  4. (iv)

    Necessity: let f be nano Iα-irresolute continuous and B ⊆ V. Then NIα-int(B) ∈ NIαO(V). Therefore, (f−1(NIα-int(B)))∈ NIαO(U). That is, NIα-int( f−1(NIα-int(B))) = (f−1(NIα-int(B))) . Also, NIα-int(B) ⊆ B implies that NIα-int(f−1(NIα-int(B))) ⊆ NIα-int(f−1(B)). Therefore f−1(NIα-int(B)) = NIα-int(f−1(NIα-int(B))) ⊆ NIα-int(f−1(B)) . That is, f−1(NIα-int(B))) ⊆ NIα-int(f−1(B)). Sufficiency: let f−1(NIα-int(B))) ⊆ NIα-int(f−1(B)) for every subset B of V. If B ∈ NIαO(V), B = (NIα-int(B)). Also, f−1(B) = f−1(NIα-int(B)) but, f−1(Iα-int(B)) ⊆ NIα-int(f−1(B)). That is, f−1(B) = f−1(NIα-int(B)) ⊆ NIα-int(f−1(B)). Therefore, f−1(B) = NIα-int(f−1(B)). Thus, f−1(B) is NIα-open in U for every NIα-open set B in V . Therefore, f is nano NIα-irresolute continuous.

Remark 2.1.5. The following implication shows the relationships between different types of nano continuous functions.

figure a

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, τR(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

  1. (i)

    f(x) = f(y) = a, f(z) = c. This function is nano Iα-continuous and nano continuous, but it is not nano Iα-irresolute continuous for {b, c} ∈ NIαO(V), but f−1({b, c}) = {z} ∉ NIαO(U) . It is not strongly nano Iα-continuous since {a, b} ∈ NIαO(V), but f−1({a, b}) = {x, y} ∉ τR(X).

  2. (ii)

    f(x) = f(z) = b and f(y) = a. This function is nano Iα-irresolute continuous and nano Iα-continuous but neither strongly nano Iα-continuous nor nano continuous function for \( \left\{b\right\}\in {\tau}_{\acute{R}}(Y)\subseteq NI\alpha O(V) \), but f−1({b}) = {x, z} ∉ τR(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.

  1. (i)

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

  2. (ii)

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

  3. (iii)

    If f is nano Iα-irresolute 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, τR(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

  1. (i)

    f(x) = f(y) = f(z) = c. This function is nano Iα-continuous. But {y, z} ∈ FR(X) and f({y, z}) = {c} ∉ NIαC(V).

  2. (ii)

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

  3. (iii)

    f(x) = f(y) = c , f(z) = b. This function is nano Iα-irresolute 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

  1. (i)

    Nano Iα-open [nano Iα-closed] function if f(A) ∈ NIαO(V), for all A ∈ τR(X) [f(A) ∈ NIαC(V)], for all A ∈ FR(X)) respectively.

  2. (ii)

    Strongly nano Iα-open [strongly nano Iα-closed] 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.

  3. (iii)

    Nano Iα-almost open (nano Iα-almost 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 nano-open functions.

figure b

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, τR(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 Iα-open and nano-open, but it is neither nano Iα-almost open nor strongly nano Iα-open 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, τR(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 Iα-open and nano Iα-almost open, but it is neither strongly nano Iα-open nor nano-open for U ∈ τR(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.

figure c

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, τR(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 Iα-closed and nano closed, but it is neither nano Iα-almost closed nor strongly nano Iα-closed 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, τR(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 Iα-closed and nano Iα-almost closed, but it is neither strongly nano Iα-closed nor nano closed for {y, z} ∈ FR(X) ⊆ NIαC(U) but, \( f\left(\left\{y,z\right\}\right)=\left\{c\right\}\notin {F}_{\acute{R}}(Y) \).

NIα-homeomorphism 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

  1. (i)

    Nano Iα-homeomorphism function if f and f−1 are both nano Iα-continuous functions.

  2. (ii)

    Strongly nano Iα-homeomorphism function if f and f−1 are both strongly nano Iα-continuous functions.

  3. (iii)

    Nano Iα-irresolute homeomorphism function if f and f−1 are both nano Iα-irresolute 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

  1. (i)

    Nano Iα-homeomorphism function if f is both nano Iα-continuous and nano Iα-open function.

  2. (ii)

    Strongly nano Iα-homeomorphism function if f is both strongly nano Iα-continuous and is strongly nano Iα-open function.

  3. (iii)

    Nano Iα -irresolute homeomorphism function if f is both nano Iα-irresolute continuous and nano Iα-almost 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

  1. (i)

    Nano continuous function if f, g are strongly nano Iα-continuous and nano Iα-continuous functions.

  2. (ii)

    Nano Iα-continuous function if f, g are nano Iα-irresolute continuous and nano continuous functions.

  3. (iii)

    Strongly nano Iα-continuous function if f, g are strongly nano Iα-continuous and nano Iα-irresolute continuous functions.

Proof:

  1. (i)

    Take C ⊆ W such that \( C\in {\tau}_{\acute{R}}(Z) \), then g−1(C) ∈ NIαO(V) and f−1(g−1(C)) ∈ τR(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.

  2. (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 Iα-continuous function.

  3. (iii)

    Take C ⊆ W such that C ∈ NIαO(W), then g−1(C) ∈ NIαO(V) and f−1(g−1(C)) ∈ τR(X). Thus C ∈ NIαO(W), (g ∘ f)−1 ∈ τR(X), and g ∘ f is strongly nano Iα-continuous 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 Iα-irresolute continuous function in the following cases.

  1. (i)

    If f, g are both nano Iα-irresolute continuous functions.

  2. (ii)

    If f, g are nano Iα-irresolute continuous and strongly nano Iα-continuous functions, respectively.

  3. (iii)

    If f, g are nano Iα-continuous and strongly nano Iα-continuous functions, respectively.

Proof: Take C ⊆ W such that C ∈ IαO(W).

  1. (i)

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

  2. (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).

  3. (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 Iα-irresolute 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 nano-open function in the following cases:

  1. (i)

    If f, g are nano Iα-open and strongly nano Iα-open functions, respectively.

  2. (ii)

    If f, g are nano-open and strongly nano Iα-open functions, respectively.

Proof: Take A ⊆ U such that A ∈ τR(X).

  1. (i)

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

  2. (ii)

    Since A ∈ τR(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 nano-open 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 Iα-open function in the following cases:

  1. (i)

    If f, g are nano Iα-open and nano Iα-almost open functions, respectively.

  2. (ii)

    If f, g are nano-open and nano Iα-almost open functions, respectively.

  3. (iii)

    If f, g are nano-open and nano Iα-open functions, respectively.

Proof: Take A ⊆ U such that A ∈ τR(X).

  1. (i)

    Since A ∈ τR(X) then f(A) ∈ NIαO(V) and g(f(A)) ∈ NIαO(W).

  2. (ii)

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

  3. (iii)

    Since A ∈ τR(X) then \( f(A)\in {\tau}_{\acute{R}}(Y) \)and g(f(A)) ∈ NIαO(W).

Thus, in each case, we have that A ∈ τR(X), (g ∘ f) ∈ NIαO(W), and g ∘ f is nano Iα-open 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 Iα-open function in the following cases.

  1. (i)

    If f, g are nano Iα-almost open and strongly nano Iα-open functions, respectively.

  2. (ii)

    If f, g are both strongly nano Iα-open functions.

  3. (iii)

    If f, g are strongly nano Iα-open and nano-open functions, respectively.

Proof: Take A ⊆ U such that A ∈ NIαO(U).

  1. (i)

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

  2. (ii)

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

  3. (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 Iα-open 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 Iα-almost open function in the following cases:

  1. (i)

    If f, g are both nano Iα-almost open functions.

  2. (ii)

    If f, g are strongly nano Iα-open and nano Iα-almost open functions, respectively.

  3. (iii)

    If f, g are strongly nano Iα-open and nano Iα-open functions, respectively.

Proof: Take A ⊆ U such that A ∈ NIαO(U).

  1. (i)

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

  2. (ii)

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

  3. (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 Iα-almost 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.

Table 1 A decision table for flu
Full size table

Let

  1. (i)

    R1 = {Temperature} , the family of all equivalence classes of IND(R) is U ∕ R1 = {{1, 3, 4}, {2}, {5, 6}}

  2. (ii)

    R2 = {Temperature, Headache}, then U ∕ R2 = {{1, 4}, {2}, {3}, {5}, {6}}

  3. (iii)

    R3 = {Headache, Cough}, then U ∕ R3 = {{1, 4}, {2, 5}, {3}, {6}}.

If we take, X = {x : [x]Nausea = no} = {1, 3, 5} then

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

  2. (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 \( {\tau}_{R_2}(X)=\left\{X,\varnothing, \left\{1,4\right\},\left\{3,5\right\},\left\{1,3,4,5\right\}\right\} \).

  3. (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

  1. (i)
    $$ {\left( NI\alpha O(U)\right)}_1=\left\{U,\varnothing, \left\{1,3,4,5,6\right\}\right\}. $$
  2. (ii)
    $$ {\left( NI\alpha O(U)\right)}_2=\left\{U,\varnothing, \left\{1,4\right\},\left\{3,5\right\},\left\{1,3,4,5\right\},\left\{1,2,3,4,5\right\},\left\{1,3,4,5,6\right\}\right\}. $$
  3. (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 Iα-continuous and nano continuous, but it is neither nano Iα- irresolute continuous nor strongly nano Iα-continuous 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 Iα-continuous and nano Iα- irresolute continuous, but it is neither nano continuous nor strongly nano Iα-continuous 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) = {v1, v2, v3, v4, v5, v6} and V(H) = {w1, w2, w3, w4, w5, w6}.

Fig. 1
figure 1

Simple directed graphs

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From the previous figure N(v1) = {v1, v2, v4, v5}, N(v2) = {v2, v3, v6}, N(v3) = {v3, v4, v5}, N(v4) = {v4, v6}, N(v5) = {v5} and N(v6) = {v5, v6}. Let X = {v5}, then L(X) = {v5}, U(X) = {v1, v3, v5, v6} and b(X) = {v1, v3, v6}, which mean that τR = {V(G),  ∅ , {v5}, {v1, v3, v6}, {v1, v3, v5, v6} }. take I = {∅, {v1}} then NIαO(V(G)) = {V(G),  ∅ , {v5}, {v1, v3, v6}, {v1, v3, v5, v6}, {v1, v2, v3, v5, v6}, {v1, v3, v4, v5, v6} }.

Similarly N(w1) = {w1, w4, w5, w6}, N(w2) = {w2, w5}, N(w3) = {w3, w5, w6}, N(w4) = {w2, w3, w4},

N(w5) = {w5} and N(w6) = {w2, w6}. Let Y = {w5}, then L(Y) = {w5}, U(Y) = {w1, w2, w3, w5} and b(Y) = {w1, w2, w3}, 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 = {∅, {w1}} then NIαO(V(H)) = {V(H),  ∅ , {w5}, {w1, w2, w3}, {w1, w2, w3, w5}, {w1, w2, w3, w4, w5}, {w1, w2, w3, w5, w6} }.

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(v1) = w1, f(v2) = w4, f(v3) = w3, f(v4) = w6, f(v5) = w5 and f(v6) = w6. This function is nano continuous, nano Iα-continuous and nano Iα – irresolute, but it is not strongly nano Iα-continuous for, {w1, w2, w3, w4, w5} ∈ NIαO(V(H)) but f−1({w1, w2, w3, w4, w5}) = {v1, v2, v3, v5, v6} ∉ τR(X), and this function is nano open, nano Iα-open and nano Iα-almost open, but it is not strongly nano Iα-open for, {v1, v3, v4, v5, v6} ∈ 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 Iα-homeomorphism and nano Iα-irresolute homeomorphism, but it is not strongly nano Iα-homeomorphism.

4 Topological models in terms of graphs and nano topology

In this section, we apply these new types of functions on some real-life problems, especially, in medicine and physics.

4.1 The foetal circulation

In this section, we apply some of the graphs, nano topology and NIα-open 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 foetal-placental 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 nutrient-rich 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 NIα-open 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 v1, v2, v3 and v4 (high oxygen content) represent placenta, umbilical vein, liver and ductus venosus respectively; the vertices v6, v7, v8, v9, v10, v14, v15, v16 and v17 (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 v5, v11, v12 and v13 (low oxygen content) represent inferior vena cava, left atrium and left ventricle respectively.

Fig. 2
figure 2

Foetal circulation

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Fig. 3
figure 3

Foetal circulation step by step

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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 τR on it by using the neighbourhood of each vertex.

Fig. 4
figure 4

Graph of foetal circulation

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figure d

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(v1) = {v6},

  • $$ f\left({v}_2\right)=\left\{{v}_7\right\},f\left({v}_3\right)=\left\{{v}_9\right\},f\left({v}_4\right)=\left\{{v}_{14}\right\},f\left({v}_5\right)=\left\{{v}_5\right\},f\left({v}_6\right)=\left\{{v}_1\right\},f\left({v}_7\right)=\left\{{v}_2\right\},f\left({v}_8\right)=\left\{{v}_8\right\}, $$
  • $$ f\left({v}_9\right)=\left\{{v}_3\right\},f\left({v}_{10}\right)=\left\{{v}_{10}\right\},f\left({v}_{11}\right)=\left\{{v}_{15}\right\},f\left({v}_{12}\right)=\left\{{v}_{16}\right\},f\left({v}_{13}\right)=\left\{{v}_{13}\right\},f\left({v}_{14}\right)=\left\{{v}_4\right\}, $$

f(v15) = {v11}, f(v16) = {v12} and f(v17) = {v17}. This function is nano-continuous, NIα-continuous and NIα-irresolute continuous, but it is not strongly NIα-continuous for {v1, v5, v6, v7, v8, v9, v13, v14, v15, v16, v17} ∈ NIαO(V(B)), but f−1({v1, v5, v6, v7, v8, v9, v13, v14, v15, v16, v17}) = {v1, v2, v3, v4, v5, v6, v8, v11, v12, v13, v17} ∉ τR(V(A)).

Also, this function is nano-open, NIα-open and NIα-almost open, but it is not strongly NIα-open for {v1, v2, v3, v4, v5, v6, v8, v11, v12, v13, v17} ∈ 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 nano-homeomorphism, NIα-homeomorphism and NIα-irresolute homeomorphism. Finally, by studying one part of this function, say A and by making new results, this function that satisfies NIα-irresolute homeomorphism makes the examination of foetal circulation simplest, and by NIα-irresolute 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, nano-topology andNIα-open 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 C1 and C2 with two different graphs G1 and G2, respectively. So, by taking V(A) ⊆ V(G1) and V(B) ⊆ V(G2), we can construct a nano topology on them.

Fig. 5
figure 5

Electrical circuits

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Fig. 6
figure 6

The corresponding graphs

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The neighborhood of each vertex of V(G1) : N1 = {1, 2}, N2 = {2, 5}, N3 = {1, 2, 3}, N4 = {3, 4} and N5 = {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 τR(V(A)) = {V(G1),  ∅ , {4}, {3, 5}, {3, 4, 5}}. Let ={∅, {1}}. Then NIαO(V(A)) = {V(G1),  ∅ , {4}, {3, 5}, {3, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}}.

The neighbourhood of each vertex of V(G2) : Na = {a, c}, Nb = {a, b, e}, Nc = {c, d, e}, Nd = {b, d} and Ne = {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(G2),  ∅ , {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 nano-continuous, NIα-continuous and NIα-irresolute continuous, but it is not strongly NIα-continuous for {a, b, c, d} ∈ NIαO(V(B)), but f−1({a, b, c, d}) = {2, 3, 4, 5} ∉ τR(V(A)). Also, this function is nano-open, NIα-open and NIα-almost open, but it is not strongly NIα-open 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 nano-homeomorphism, NIα-homeomorphism and NIα-irresolute homeomorphism. Finally, this function which satisfies the NIα-irresolute 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 NIα-irresolute 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 G2 can be obtained by relabeling the vertices of G1, that is, f is a bijection between the vertices of G1 and those of G2, such that the arcs joining each pair of vertices in G1 accepted in both numbers and direction with the arcs joining the corresponding pair of vertices in G2.

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 NIα-irresolute homeomorphism for every subgraph A of G1, which will be studied in Table 2.

Table 2 Comparison between NIα-irresolute homeomorphisms
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It is clear that from Table 2, the two circuits are NIα-irresolute homeomorphism for every subgraph A of G1, and using the previous structural equivalence technique we checked that the two circuits are equivalent.

Conclusion

In this paper, different types of NIα-continuous, NIα-open, NIα-closed and NIα-homeomorphism are introduced and studied. Some applications on them are given in some real-life branches such as medicine and physics. We give some examples of electric circuits and study its relationship with graph theory.