Some nano topological structures via ideals and graphs

In this paper, new forms of nano continuous functions in terms of the notion of nano Iα-open sets called nano Iα-continuous functions, strongly nano Iα-continuous functions and nano Iα-irresolute functions will be introduced and studied. We establish new types of nano Iα-open functions, nano Iα-closed functions and nano Iα-homeomorphisms. 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 real-life 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 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 ðXÞ ¼ ⋃ x∈U fR x : R x ∩X≠∅g , 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 ∅∈τ R X ð Þ; (ii) The union of elements of any sub-collection of τ R (X) is in τ R (X); (iii)The intersection of the elements of any finite sub-collection 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. Nano-open 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 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 [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 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: (ii) u is outvertex of v if vu ! ∈EðGÞ.
(iii)The neighborhood of v is denoted by N(v), and given by NðvÞ ¼ fvg∪fu∈V ðGÞ : vu ! ∈EðGÞg 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 ðV ðHÞÞ ¼ ⋃ v∈V ðGÞ fv : NðvÞ ⊆V ðHÞg; (ii) [7] The upper approximation U : P(V(G)) P(V(G)) is U N ðV ðHÞÞ ¼ ⋃ 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 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, τ R (X)) and ðV ; τ Ŕ ðY ÞÞ be nano topological spaces. 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: (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 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, Proof: . Therefore, f −1 (B ) ∈NIαC(U). Thus, the inverse image of every nano closed set in That is, f is nano Iα-continuous on U.
is NIα-closed in U for every nano closed set B in V. Therefore, f is nano Iα-continuous on U. (iv) Necessity: let f be nano Iα-continuous and B ⊆ V. Then NIntðBÞ∈τ Ŕ ðY Þ. Therefore, . .
continuous function if and only if one of the following is satisfied; Proof: (i) Necessity: let f be strongly nano Iα-continuous and B ∈ NIαC(V). That is, That is, f is strongly nano Iαcontinuous on U.
for every NIα-closed set B in V. Therefore, f is strongly nano Iα-continuous on U. (iv) Necessity: let f be strongly nano Iα-continuous and B ⊆ V. Then NIα-int(B) ∈ NIαO(V). Therefore, NIα-open set B in V. Therefore, f is strongly nano Iα-continuous.
continuous function if and only if one of the following is satisfied; Proof: . Therefore, Therefore, f is nano Iα-irresolute continuous on U. (iv) Necessity: let f be nano Iα-irresolute continuous and B ⊆ V. Then NIα-int(B) ∈ NIαO(V). Therefore,  ( We show this remark by using the following example. Proof: (i) Take C ⊆ W such that C∈τ Ŕ ðZÞ, then g −1 (C) ∈ NIαO(V) and f −1 (g −1 (C)) ∈ τ R (X).
, and g ∘ f is strongly nano Iα-continuous function.  (iii)If f, g are nano Iα-continuous and strongly nano Iα-continuous functions, respectively.
Proof: Take C ⊆ W such that C ∈ IαO(W).
Thus, in each case, we have that A ∈ τ R (X), (g ∘ f) ∈ NIαO(W), and g ∘ f is nano Iα-open function.  Proof: Take A ⊆ U such that A ∈ NIαO(U).

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.

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.

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 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 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 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. 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.

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 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  A)), but f ðf2; 3; 4; 5gÞ ¼ fa; b; c; dg∉τ Ŕ ðV ðBÞÞ . 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 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 : ðV ðG 1 Þ; τ R ðV ðAÞÞ; IÞ→ðV ðG 2 Þ; τ Ŕ ðV ðBÞÞ; JÞ is NIα-irresolute homeomorphism for every subgraph A of G 1 , which will be studied in Table 2. Table 2 Comparison between NIα-irresolute homeomorphisms  It is clear that from Table 2, the two circuits are NIα-irresolute 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 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.