II-Bitopology, II-Induced Topology and II-Separation Axioms on Locally Finite Graphs

Basic definitions and introductions to bitopology and graph theory are covered in this part. These ideas are all often utilized and can be found in references [1, 2, 19].

Typically, a graph consists of two sets, $\mathfrak{S}=(\Lambda, \mathfrak{L}$, where $\Lambda$ represents the set of vertices and $\mathfrak{L}$ represents the set of edges, an edge of the form $\vartheta=(\varpi, \varpi)$ is loop. Parallel edges are those with the identical end vertices. If a graph contains no parallel edges or loop, it is considered simple. If the vertices $\varpi$ and $\rho$ are linked via the same edge then they are said to be adjacent vertices. All of these ideas are well-known and are available in books mentioned above.

We use the symbols $K_n$ for the complete graph with $n$ vertices and $C_n$ is cycle graph on $n$ vertices and $P_m$ is a path on m edges and $K_{n l, n 2}$ is a whole bipartite graph of size partite $n 1$ and $n 2$.

The family of open subsets of the any non-empty set $\mathcal{X}$ is named a topology when the following conditions are hold: $\mathcal{X}, \emptyset \in \mathcal{T}$, for every $\mathrm{H}, \mathrm{G} \in \mathcal{T}, \mathrm{H} \cap \mathrm{G} \in \mathcal{T}$ and $\mathrm{U}_{\mathrm{i} \in \Delta} \mathrm{G}_{\mathrm{i}} \in \mathcal{T}$ for all sub-combination $\mathrm{G}_{\mathrm{i}}$ of $\mathcal{T}$, then $(\mathcal{X}, \mathcal{T})$ is named a topological space, an open set is a sub-set of $\mathcal{X}$ which is satisfied the conditions. Indiscrete topology is defined as $\mathcal{T}$ $=\{\emptyset, \mathcal{X}\}$ on $\mathcal{X}$ while discrete topology is def. $\mathcal{T}=\mathrm{P}(\mathcal{X})$ on $\mathcal{X}$.

A bi-topological space is any non-empty set $\mathcal{X}$ combined with two topologies $\mathcal{T}_1$ and $\mathcal{T}_2$ on $\mathcal{X}$, written as $\left(X, \mathcal{T}_1, \mathcal{T}_2\right)$. The intersection of two topologies is induced a new topology.

In this work we are depended on two well-known topological spaces associated with un-directed graphs $\mathfrak{S}=$ $(\Lambda, \mathfrak{L})$, the Independent and Incidence, which are topologies defined as follows:

The graph $\mathfrak{S}$ may contain one or more isolated vertex, let $\mathrm{S}_{\mathrm{I} d}=\left\{\mathrm{I}_{\varpi}: \varpi \in \Lambda\right\}$ s. t. $\mathrm{I}_{\varpi}$ is represent the set of all not adjacent vertices with a vertex $\varpi$. we have $\Lambda=U_{\varpi \in \Lambda} \mathrm{I}_{\varpi}$. Hence $\mathrm{S}_{\mathrm{I} d}$ represents a sub-basis for the topology $\mathcal{T}_{\mathrm{I} d}$ on $\Lambda$, so $\mathcal{T}_{\mathrm{I} d}$ named independent topology.

Now, let $I_{\vartheta}$ be the set of incidence vertices of the edge $\vartheta$. And $S_{\text {I } c}$ is defined as follows:

$S_{\mathrm{I} c}=\left\{I_{\vartheta} \mid \vartheta \in \mathfrak{L}\right\}$, where $\mathfrak{S}$ is a locally finite graph, we have $\Lambda=\mathrm{U}_{\vartheta \in \mathcal{R}} I_{\vartheta}$. Hence $S_{\mathrm{I} c}$ represents a sub-basis of a topology $\mathcal{T}_{\text {I } c}$ on $\mathcal{V}$, so $\mathcal{T}_{\mathrm{I} c}$ is named Incidence topology.

In this part of our paper, we will define and study a new bi-topological space, that is by using the two well-known topological spaces mentioned above as follows:

Definition 4.1

$\left(\Lambda, \mathcal{T}_{\mathrm{I} d}, \mathcal{T}_{\mathrm{I} c}\right)$ is a bi-topological space, then $\left(\Lambda, \mathcal{T}_{\mathrm{II}}\right)$ represents a topological space, where $\mathcal{T}_{\text {II }}=\mathcal{J}_{\text {I } d} \cap \mathcal{T}_{\text {I } c}$ is called the II-induced topological space from the bi-topological space $\left(\Lambda, \mathcal{T}_{\mathrm{I} d}, \mathcal{T}_{\mathrm{I} c}\right)$ on locally finite graphs.

Definition 4.2

$\left(\Lambda, \mathcal{T}_{\mathrm{I} d}, \mathcal{T}_{\mathrm{Ic}}\right)$ is a bi-topological space, and the set $\gamma \subseteq \Lambda. \gamma$ is named II-bi-open set iff $\gamma$ is open in II-induced topology $(\gamma \in$ $\left.\mathcal{T}_{\mathrm{I} d} \cap \mathcal{T}_{\mathrm{I} c}\right)$.

Definition 4.3

$\left(\Lambda, \mathcal{T}_{\mathrm{Id}}, \mathcal{T}_{\mathrm{Ic}}\right)$ is a bi-topological space. and the set $\mathbb{M} \subseteq \Lambda$ is named II-bi-closed set iff $M$ is closed in II-induced topology (the complement of $M \in \mathcal{T}_{\mathrm{I} d} \cap \mathcal{T}_{\mathrm{I} c}$).

Example 4.4

$\mathfrak{S}=(\Lambda, \mathfrak{L})$ is a locally finite graph as in Figure 2, such that: $\Lambda=\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_4, \varpi_5\right\}, \mathfrak{L}=\left\{\vartheta_1, \vartheta_2, \vartheta_3\right\}$. Then,

2.png

Figure 2. Locally finite graph

Independent topology is:

$\begin{aligned} & \mathcal{T}_{\mathrm{I} d}=\left\{\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_3\right\},\left\{\varpi_3, \varpi_4, \varpi_5\right\},\left\{\varpi_4, \varpi_5\right\},\right. \left.\left\{\varpi_1, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3, \varpi_4, \varpi_5\right\}\right\} \end{aligned}$

And incident topology is:

$\begin{aligned} & \mathcal{T}_{\text {Ic }}=\left\{\emptyset, \Lambda,\left\{\varpi_2\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_2, \varpi_4, \varpi_5\right\},\right.  \left.\quad\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_2, \varpi_4, \varpi_5\right\},\left\{\varpi_2, \varpi_3, \varpi_4, \varpi_5\right\}\right\} . \end{aligned}$

Since $\mathcal{T}_{\mathrm{II}}=\mathcal{T}_{\mathrm{I} d} \cap \mathcal{T}_{\mathrm{I} c}$, then II-induced topology: $\mathcal{T}_{\mathrm{II}}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\}\right\}$.

Then, $\emptyset, \Lambda,\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\}$ are II-bi-open sets, and $\Lambda, \emptyset,\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_4, \varpi_5\right\}$ are II-bi-closed sets (resp.).

Therefore $\mathcal{T}_{\mathrm{I} d}$ and $\mathcal{T}_{\mathrm{I} c}$ on $\Lambda$ give us a new bi-topological space $\left(\Lambda, \mathcal{T}_{\mathrm{I} d}, \mathcal{T}_{\mathrm{I} c}\right)$.

Clear that $\mathcal{T}_{\text {I } d}$ and $\mathcal{T}_{\text {I } c}$ are non-similar, and they induce $\mathcal{T}_{\text {II }}$ topology on vertices set $\Lambda$ of a graph $\mathfrak{S}=(\Lambda, \mathfrak{L})$.

Remark 4.5

Independent topology $\mathcal{T}_{\mathrm{I} d}$ of $\mathfrak{S}=(\Lambda, \mathfrak{L})$ is discrete iff $I_{\varpi} \nsubseteq$ $I_\psi \wedge I_\psi \nsubseteq I_{\varpi}$ for all different vertices $\varpi, \psi \in \Lambda$ [12], and the incidence topology $\mathcal{J}_{\mathrm{I} c}$ of a graph $\mathfrak{\Im}$ will be discrete if $d(\varpi)>$ 2 for all $\varpi \in \Lambda$ [6].

Thus, if a graph $\mathfrak{S}$ satisfies the above terms, then it is surely that $\mathcal{T}_{\mathrm{I} d}$ and $\mathcal{T}_{\mathrm{I} c}$ are identical (both are discrete).

Remark 4.6

Clear that II-induced topological space $\left(\Lambda, \mathcal{T}_{\mathrm{II}}\right)$ to a cycle graph $C_n, n \geq 4$ is discrete.

That is because the incidence topology $\mathcal{T}_{\mathrm{I} c}$ is discrete when $\mathrm{n} \geq 3$ [11] and the independent topology $\mathcal{T}_{\mathrm{I} d}$ is discrete when $\mathrm{n} \geq 4$ [12].

Proof: Since the II-induced topology $\mathcal{T}_{\text {II }}$ arises from the intersection of the topologies Independent $\mathcal{T}_{\mathrm{I} d}$ and incidence $\mathcal{T}_{\mathrm{I} c}$, then all the open sets will exist in the intersection (in the II-induced topological space $\left(\Lambda, \mathcal{T}_{\mathrm{II}}\right)$, thus the proof of above remark will be verified directly.

Remark 4.7

The II-induced topology $\mathcal{T}_{\text {II }}$ of $P_m$ path on $m$ edges is not discrete topology.

That is because, Incidence topology $\mathcal{T}_{\mathrm{I} c}$ on $P_m$ is not discrete since $P_m$ consists of two vertices incident with one edge which is not open [11].

And independent topology $\mathcal{T}_{\mathrm{I} d}$ on $P_m$ doesn’t discrete since the set consists of only two vertices of degree 1 is open [12].

As we will show in the following example.

Example 4.8

$\mathfrak{S}=(\Lambda, \mathscr{L})$ is a $P_3$ (path on 3 edges) as in Figure 3, such that $\Lambda=\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_4\right\}, \mathfrak{L}=\left\{\vartheta_1, \vartheta_2, \vartheta_3\right\}$. Then,

3.png

Figure 3. P₃ (path on 3 edges)

The set of all not adjacent vertices with a vertex $\varpi_1$ is $\left\{\varpi_3, \varpi_4\right\}$, and the set of all not adjacent vertices with a vertex $\varpi_2$ is $\left\{\varpi_4\right\}$ and the set of all not adjacent vertices with a vertex $\varpi_3$ is $\left\{\varpi_1\right\}$ and and the set of all not adjacent vertices with a vertex $\left\{\varpi_4\right\}$ is $\left\{\varpi_1, \varpi_2\right\}$, then the sub-base of Independent Topology is $\mathrm{S}_{\mathrm{I} d}=\left\{\left\{\varpi_1\right\},\left\{\varpi_4\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_3, \varpi_4\right\}\right\}$, and by taking finite intersection, the base $\beta_{I d}$ is produced $\left\{\left\{\varpi_1\right\},\left\{\varpi_4\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_3, \varpi_4\right\}, \emptyset\right\}$. Then, utilizing each union generate The independent topology $\mathcal{T}_{\mathrm{I} d}$ as the following $\mathcal{T}_{\mathrm{I} d}=\left\{\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_4\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_1, \varpi_4\right\},\left\{\varpi_3, \varpi_4\right\}\right.$, $\left\{\varpi_1, \varpi_2, \varpi_4\right\},\left\{\varpi_1, \varpi_3, \varpi_4\right\}$, clearly $\left(\Lambda, \mathcal{T}_{I d}\right)$ represents a topological space.

Now, the set of incidence vertices of the edge $\vartheta_1$ is $\left\{\varpi_1, \varpi_2\right\}$, and the set of incidence vertices of the edge $\vartheta_2$ is $\left\{\varpi_2, \varpi_3\right\}$, and the set of incidence vertices of the edge $\vartheta_3$ is $\left\{\varpi_3, \varpi_4\right\}$.

Then the sub-base of incidence topology is $\mathrm{S}_{\text {Ic }}=$ $\left\{\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_3, \varpi_4\right\}\right\}$, and by taking finite intersection, the base $\beta_{\text {Ic }}$ is produced $\left\{\left\{\varpi_2\right\},\left\{\varpi_3\right\}\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_3, \varpi_4\right\}, \varnothing\right\}$.

Then, utilizing each union generate the incidence topology $\mathcal{T}_{\text {Ic }}$ as $\mathcal{T}_{\text {Ic }}=\left\{\emptyset, \Lambda,\left\{\varpi_2\right\},\left\{\varpi_3\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_3, \varpi_4\right\}\right.$, $\left.\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_2,, \varpi_3, \varpi_4\right\}\right\}$, clearly $\left(\Lambda, \mathcal{T}_{\text {IC }}\right)$ represents a topological space.

Then, $\mathcal{T}_{\text {II }}=\mathcal{T}_{\text {Id }} \cap \mathcal{T}_{\text {Ic }}$ and the II-induced topology $\mathcal{T}_{\text {II }}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_3, \varpi_4\right\}\right\}$.

It is clear that the II -induced topology $\mathcal{T}_{\text {II }}$ of $P_m$ (path on $m$ edges) is not discrete topology.

Remark 4.9

The II-induced topology $\mathcal{T}_{\text {II }}$ of a complete bipartite graph $K_{n 1, n 2}$ is discrete topology. That is because, independent topology $\mathcal{T}_{\mathrm{I} d}$ and the incidence topology $\mathcal{T}_{\mathrm{I} c}$ are discrete at the complete bipartite graph, so that the intersection between them will produce a discrete topology [11, 12].

Proof: Since the II-induced topology $\mathcal{T}_{\text {II }}$ arises from the intersection of the topologies independent $\mathcal{T}_{\mathrm{I} d}$ and incidence $\mathcal{T}_{\mathrm{I} c}$, then the proof of above remark will be verified directly.

Remark 4.10

The II-induced topology $\mathcal{T}_{\text {II }}$ of uncomplete bipartite graph $K_{n 1, n 2}$ is not discrete topology. That is because, the independent topology $\mathcal{T}_{\mathrm{I} d}$ and the Incidence topology $\mathcal{T}_{\mathrm{I} c}$ are not discrete topology at the un-complete bipartite graph, so that the intersection between them will not produce a discrete topology [11, 12].

Proof: Since the II-induced topology $\mathcal{T}_{\text {II }}$ arises from the intersection of the topologies Independent $\mathcal{T}_{\mathrm{I} d}$ and Incidence $\mathcal{T}_{\mathrm{I} c}$ , then the proof of above remark will be verified directly.

Remark 4.11

The II-induced topology $\mathcal{T}_{\text {II }}$ of a tree graph (locally finite, i.e., a graph for which every vertex has finite degree) is not discrete topology.

That is because, the Independent $\mathcal{T}_{\mathrm{I} d}$ and incidence $\mathcal{T}_{\mathrm{I} c}$ are not discrete topology at a graph of tree [11, 12].

As we will show in the following example:

Example 4.12

$\mathfrak{S}=(\Lambda, \mathfrak{L})$ is a tree graph as in Figure 4 below, such that: $\Lambda=\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_4, \varpi_5, \varpi_6\right\}, \mathcal{L}=\left\{\vartheta_1, \vartheta_2, \vartheta_3, \vartheta_4, \vartheta_5\right\}$. Then,

4.png

Figure 4. Tree graph

The sets of all not adjacent vertices with a vertex $\Lambda=$ $\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_4, \varpi_5, \varpi_6\right\}$ forms the sub-base, then the subbase of Independent Topology is

$\begin{aligned} & \quad S_{I d}=\left\{\left\{\varpi_3, \varpi_4, \varpi_5, \varpi_6\right\},\left\{\varpi_5, \varpi_6\right\},\left\{\varpi_1, \varpi_4, v_5, \varpi_6\right\},\right. \left.\left\{\varpi_1, \varpi_3\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_6\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_5\right\}\right\} . \end{aligned}$

By taking the finitely intersection the base $\beta_{I d}$ is obtained as follows:

$\left\{\emptyset,\left\{\varpi_3, \varpi_4, \varpi_5, \varpi_6\right\},\left\{\varpi_5, \varpi_6\right\},\left\{\varpi_1, \varpi_4, v_5, \varpi_6\right\},,\left\{\varpi_1\right\},\right.$

$\left\{\varpi_1, \varpi_3\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_6\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_5\right\},\left\{\varpi_3\right\},\left\{\varpi_6\right\},\left\{\varpi_5\right\},$

$\left.\left\{\varpi_1, \varpi_5\right\},\left\{\varpi_4, \varpi_5, \varpi_6\right\},\left\{\varpi_3, \varpi_6\right\},\left\{\varpi_3, \varpi_5\right\},\left\{\varpi_1, \varpi_6\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\}\right\}.$

Utilizing each union generate the independent topology $\mathcal{T}_{\text {Id }}$ as the following:

$\mathcal{T}_{\text {Id }}=\left\{\emptyset,\left\{\varpi_3, \varpi_4, \varpi_5, \varpi_6\right\},\left\{\varpi_5, \varpi_6\right\},\left\{\varpi_1, \varpi_4, v_5, \varpi_6\right\}\right.$,

$\left\{\varpi_1\right\},\left\{\varpi_1, \varpi_3\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_6\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_5\right\},\left\{\varpi_3\right\},\left\{\varpi_6\right\}$,

$\left\{\varpi_5\right\},\left\{\varpi_1, \varpi_5\right\},\left\{\varpi_4, \varpi_5, \varpi_6\right\},\left\{\varpi_3, \varpi_6\right\},\left\{\varpi_3, \varpi_5\right\},\left\{\varpi_1, \varpi_6\right\}$,

$\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3, \varpi_4, \varpi_5, \varpi_6\right\},\left\{\varpi_1, \varpi_3, \varpi_5, \varpi_6\right\}$,

$\left.\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_5, \varpi_6\right\},\left\{\varpi_1, v_5, \varpi_6\right\}\right\}$, clearly $\left(\Lambda, \mathcal{T}_{\mathrm{Id}}\right)$ represents a topological space.

Now, the sets of incidence vertices of the all edges $\mathfrak{L}=$ $\left\{\vartheta_1, \vartheta_2, \vartheta_3, \vartheta_4, \vartheta_5\right\}$ forms the sub-base, then the sub-base of incidence topology is $\mathrm{S}_{\text {IC}}$=$\left\{\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_2, \varpi_4\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_4, \varpi_6\right\}\right\}$. And by taking finite intersection, the base $\beta_{\text {Ic}}$ is produced:

$\left\{\left\{\varpi_2\right\},\left\{\varpi_4\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_2, \varpi_4\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_4, \varpi_6\right\}, \varphi\right\}$

Then, utilizing each union generate the incidence topology $\mathcal{T}_{\text {Ic }}$ as the following:

$\mathcal{T}_{\text {Ic }}=\left\{\emptyset, \Lambda,\left\{\varpi_2\right\},\left\{\varpi_4\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_2, \varpi_4\right\}\right.$,

$\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_4, \varpi_6\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_2, \varpi_3, \varpi_4\right\}$,

$\left\{\varpi_2, \varpi_3, \varpi_4, \varpi_5\right\},\left\{\varpi_2, \varpi_4, \varpi_5\right\},\left\{\varpi_2, \varpi_4, \varpi_6\right\},\left\{\varpi_1, \varpi_2, \varpi_4, \varpi_5\right\}$,

$\left\{\varpi_1, \varpi_2, \varpi_4, \varpi_6\right\},\left\{\varpi_1, \varpi_2, \varpi_4\right\},\left\{\varpi_2, \varpi_3, \varpi_4, \varpi_6\right\},\left\{\varpi_4, \varpi_5, \varpi_6\right\}$,

$\left.\left\{\varpi_2, \varpi_4, \varpi_5, \varpi_6\right\}\right\}$, clearly $\left(\Lambda, \mathcal{T}_{\text {IC }}\right)$ represents a topological space.

Then, $\mathcal{T}_{\text {II }}=\mathcal{T}_{\text {Id }} \cap \mathcal{T}_{\text {IC }}$, and the II-induced topology: $\mathcal{T}_{\text {II }}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_1, \varpi_2, \varpi_3\right\}\right\}$.

Now, it is clear that the II-induced topology $\mathcal{T}_{\text {II }}$ of a tree graph is not discrete topology.

Remark 4.13

The II-induced topology $\mathcal{T}_{\mathrm{II}}$ on a graph which has two or more components does not represent discrete topology. That is because, independent $\mathcal{T}_{\mathrm{I} d}$ and incidence $\mathcal{T}_{\mathrm{I} c}$ do not represent discrete topology at the graph of two or more components [11, 12].

The II-induced topology of two components graph is equal to the as in the Independent Topology sub-base. i.e., as the following:

$\mathcal{T}_{\text {II }}=\{\emptyset, \Lambda,\{ set \ of \ vertices \ of\  first\  component \}, \{set\ of\ vertices\ of\ second\ component \}\}$

For example, the reader can see the Example 4.4. above, which is two components graph and $\mathcal{T}_{\mathrm{I} d}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_3\right\},\left\{\varpi_3, \varpi_4, \varpi_5\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_4, \varpi_5\right\}\right.$, $\left.\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3, \varpi_4, \varpi_5\right\}\right\}$, clearly not discrete.

And incident topology: $\mathcal{T}_{\mathrm{I} c}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_2\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_2, \varpi_4, \varpi_5\right\}\right.$,

$\left.\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_2, \varpi_4, \varpi_5\right\},\left\{\varpi_2, \varpi_3, \varpi_4, \varpi_5\right\}\right\}$, clearly not discrete.

Since$\ \mathcal{T}_{\mathrm{II}}=\mathcal{T}_{\mathrm{I} d} \cap \mathcal{T}_{\mathrm{II} c}, \mathcal{T}_{\mathrm{II}}=\left\{\emptyset, \Lambda,\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\}\right\},$ then II-induced topology also clearly not discrete.

And the II-induced topology of three components graph is equal to {\Lambda, the sets of vertices of the components and the arbitrary unions of these sets (vertices of components sets), i.e. The -induced topology of three components graph is as the following, i.e., $\mathcal{T}_{\mathrm{II}}=\left\{\emptyset, \Lambda, C o_i, \mathrm{U}_{i=1}^n C o_i\right\}$, where $C o_i$ is abbreviation of the word of component, and i=1, 2, …, n such that n≥3.

After the view of all above remarks and examples we conclude that:

The potential advantages of using the II-induced topology over existing topologies are that in most cases is discrete (in certain types of graphs), and disadvantages of using the II-induced topology are that it takes the characteristics of a more specific (less general) of existing topologies.

In this part, definitions of the separation axioms of II-induced topology from bi-topology on un-directed graphs (locally finite graphs) are introduced. And their basic properties are investigated with respect to two well-known topologies $\mathcal{T}_{\mathrm{I} d}$ and $\mathcal{T}_{\mathrm{I} c}$, the II-T_i; (i=0, 1, 2, 3, 4) spaces and their notions of II-normal and II-regular are defined and discussed, and we will introduce some illustrative examples. Also we will give a fundamental properties of locally finite graphs by their corresponding II-separation axioms of bi-topological spaces.

6.1 II-T_i: (i=0,1,2) spaces of bi-topological spaces on locally finite graphs

Generally, the separation axioms expressing how rich the population of open set is. And graph properties connect to fulfillment of separation axioms in the topology via the measure of tightness is the extent to which envelope can separate the subset of vertices connection from other subsets of vertices connection.

Definition 6.1.1 Let $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ be an II-induced topological space from bi-topological space $\left(\Lambda, \mathcal{T}_{\mathrm{I} d}, \mathcal{T}_{\mathrm{I} c}\right)$ over $\Lambda$, then $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ is said to be II $-\mathrm{T}_0$-space if for each pair of distinct vertices $\varpi_1, \varpi_2 \in \Lambda$, there is II -bi-open sets $W_1, W_2$ s. t. '$\varpi_1 \in$ $W_1, \varpi_2 \notin W_1$' or '$\varpi_1 \notin W_2, \varpi_2 \in W_2$'.

Example 6.1.2 $\mathfrak{S}=(\Lambda, \mathfrak{L})$ is a graph as in Figure 5, such that: $\Lambda=\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_4, \varpi_5\right\}, \mathfrak{L}=\left\{\vartheta_1, \vartheta_2, \vartheta_3, \vartheta_4, \vartheta_5, \vartheta_6\right\}$. Then,

5.png

Figure 5. Undirected graph

The sub-base of independent topology is the following set: $S_{\mathrm{I} d}=\left\{\left\{\varpi_4\right\},\left\{\varpi_5\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_1, \varpi_3\right\}\right\}$, taking the finitely intersection the base $\beta_{\mathrm{I} d}$ obtained: $\left\{\emptyset,\left\{\varpi_1\right\},\left\{\varpi_4\right\},\left\{\varpi_5\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_1, \varpi_3\right\}\right\}$.

Then via taking the unions:

$\mathcal{T}_{\mathrm{I} d}=\left\{\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_4\right\},\left\{\varpi_5\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_1, \varpi_3\right\}\right.$,

$\left\{\varpi_1, \varpi_4\right\},\left\{\varpi_1, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_4\right\},\left\{\varpi_1, \varpi_2, \varpi_5\right\},\left\{\varpi_1, \varpi_3, \varpi_4\right\}$,

$\left\{\varpi_1, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_4\right\}$,

$\left.\left\{\varpi_1, \varpi_2, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_3, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_5\right\}\right\}$

$\varpi_1, \varpi_2 \in \Lambda, \varpi_1 \neq \varpi_2$ there is an $\mathcal{T}_{\mathrm{I} d}$ open set $\varpi_1$ such that $\varpi_1 \in\left\{\varpi_1\right\}, \varpi_2 \notin\left\{\varpi_1\right\}$.

$\varpi_2, \varpi_3 \in \Lambda, \varpi_2 \neq \varpi_3$, there is an $\mathcal{T}_{\text {Id }}$-open set $\left\{\varpi_1, \varpi_3\right\}$ such that $\varpi_3 \in\left\{\varpi_1, \varpi_3\right\}, \varpi_2 \notin\left\{\varpi_1, \varpi_3\right\}$

$\varpi_3, \varpi_4 \in \Lambda, \varpi_3 \neq \varpi_4$, there is $\mathcal{T}_{\text {Id }}$-open set $\left\{\varpi_4\right\}$ such that $\varpi_4 \in\left\{\varpi_4\right\}, \varpi_3 \notin\left\{\varpi_4\right\}$.

$\varpi_4, \varpi_5 \in \Lambda, \varpi_4 \neq \varpi_5$, there is $\mathcal{T}_{\text {I } d}$-open sets $\left\{\varpi_4\right\}$ such that $\varpi_4 \in\left\{\varpi_4\right\}, \varpi_5 \notin\left\{\varpi_4\right\}$.

$\varpi_1, \varpi_3 \in \Lambda, \varpi_1 \neq \varpi_3$, there is $\mathcal{T}_{\text {Id }}$-open set $\left\{\varpi_1\right\}$ such that $\varpi_1 \in\left\{\varpi_1\right\}, \varpi_3 \notin\{\varpi\}$.

$\varpi_1, \varpi_4 \in \Lambda, \varpi_1 \neq \varpi_4$, there is $\mathcal{T}_{\text {I } d}$-open set $\left\{\varpi_1\right\}$ such that $\varpi_1 \in\{\varpi\}, \varpi_4 \notin\left\{\varpi_1\right\}$

$\varpi_1, \varpi_5 \in \Lambda, \varpi_1 \neq \varpi_5$, there is $\mathcal{T}_{\text {Id }}$-open set $\left\{\varpi_1\right\}$ such that $\varpi_1 \in\left\{\varpi_1\right\}, \varpi_5 \notin\left\{\varpi_1\right\}$

$\varpi_2, \varpi_4 \in \Lambda, \varpi_2 \neq \varpi_4$, there is $\mathcal{T}_{\text {I } d}$-open set $\left\{\varpi_1, \varpi_2\right\}$ such that $\varpi_2 \in\left\{\varpi_1, \varpi_2\right\}, \varpi_4 \notin\left\{\varpi_1, \varpi_2\right\}$.

$\varpi_2, \varpi_5 \in \Lambda, \varpi_2 \neq \varpi_5$, there is $\mathcal{T}_{\text {I } d}$-open set $\left\{\varpi_1, \varpi_2\right\}$ such that $\varpi_2 \in\left\{\varpi_1, \varpi_2\right\}, \varpi_5 \notin\left\{\varpi_1, \varpi_2\right\}$.

$\varpi_3, \varpi_5 \in \Lambda, \varpi_3 \neq \varpi_5$, there is $\mathcal{T}_{\text {I } d}$-open set $\left\{\varpi_1, \varpi_3\right\}$ such that $\varpi_3 \in\left\{\varpi_1, \varpi_3\right\}, \varpi_5 \notin\left\{\varpi_1, \varpi_3\right\}$.

Then it is clear, the independent topology $\mathcal{T}_{\mathrm{I} d}$ is represents a $\mathrm{T}_0$-space.

The sub-base of incident topology is the following set:

$S_{\text {Ic }}=\left\{\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_1, \varpi_3\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_2, \varpi_5\right\},\left\{\varpi_3, \varpi_4\right\},\right.\left.\left\{\varpi_4, \varpi_5\right\}\right\}$

By taking finitely intersection the base $\beta_{\text {Ic }}$ obtained,

$\left\{\emptyset,\left\{\varpi_1\right\},\left\{\varpi_3\right\},\left\{\varpi_2\right\},\left\{\varpi_4\right\},\left\{\varpi_5\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_1, \varpi_3\right\}\right.$,

$\left.\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_2, \varpi_5\right\},\left\{\varpi_3, \varpi_4\right\}\left\{\varpi_4, \varpi_5\right\}\right\}$

Then via taking all unions;

$\mathcal{T}_{\mathrm{I} c}=\left\{\Lambda, \emptyset,\left\{\varpi_1\right\},\left\{\varpi_3\right\},\left\{\varpi_2\right\},\left\{\varpi_4\right\},\left\{\varpi_5\right\},\left\{\varpi_1, \varpi_2\right\}\right.$,

$\left\{\varpi_1, \varpi_3\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_2, \varpi_5\right\},\left\{\varpi_3, \varpi_4\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_5\right\}$,

$\left\{\varpi_1, \varpi_4\right\},\left\{\varpi_1, \varpi_5\right\},\left\{\varpi_2, \varpi_4\right\},\left\{\varpi_3, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_4\right\}$,

$\left\{\varpi_1, \varpi_2, \varpi_5\right\},\left\{\varpi_1, \varpi_3, \varpi_4\right\},\left\{\varpi_1, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\}$,

$\left\{\varpi_1, \varpi_3, \varpi_5\right\},\left\{\varpi_3, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_5\right\},\left\{\varpi_2, \varpi_3, \varpi_5\right\}$

$\left\{\varpi_2, \varpi_3, \varpi_4\right\},\left\{\varpi_2, \varpi_4, \varpi_5\right\},\left\{\varpi_2, \varpi_3, \varpi_4, \varpi_5\right\}$,

$\left.\left\{\varpi_1, \varpi_2, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_4\right\},,\left\{\varpi_1, \varpi_3, \varpi_4, \varpi_5\right\}\right\}$

Now,

$\varpi_1, \varpi_2 \in \Lambda, \varpi_1 \neq \varpi_2$, there is an $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_1\right\}$ such that $\varpi_1 \in\left\{\varpi_1\right\}, \varpi_2 \notin\left\{\varpi_1\right\}$.

$\varpi_2, \varpi_3 \in \Lambda, \varpi_2 \neq \varpi_3$, there is an $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_1\right\}$ such that $\varpi_3 \in\left\{\varpi_1\right\}, \varpi_2 \notin\left\{\varpi_1\right\}$.

$\varpi_3, \varpi_4 \in \Lambda, \varpi_3 \neq \varpi_4$, there is $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_4\right\}$ such that $\varpi_4 \in\left\{\varpi_4\right\}, \varpi_3 \notin\left\{\varpi_4\right\}$.

$\varpi_4, \varpi_5 \in \Lambda, \varpi_4 \neq \varpi_5$, there is $\mathcal{T}_{\text {Ic }}$-open sets $\left\{\varpi_4\right\}$ such that $\varpi_4 \in\left\{\varpi_4\right\}, \varpi_5 \notin\left\{\varpi_4\right\}$.

$\varpi_1, \varpi_3 \in \Lambda, \varpi_1 \neq \varpi_3$, there is $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_1\right\}$ such that $\varpi_1 \in\left\{\varpi_1\right\}, \varpi_3 \notin\{\varpi\}$.

$\varpi_1, \varpi_4 \in \Lambda, \varpi_1 \neq \varpi_4$, there is $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_1\right\}$ such that $\varpi_1 \in\{\varpi\}, \varpi_4 \notin\left\{\varpi_1\right\}$.

$\varpi_1, \varpi_5 \in \Lambda, \varpi_1 \neq \varpi_5$, there is $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_1\right\}$ such that $\varpi_1 \in\left\{\varpi_1\right\}, \varpi_5 \notin\left\{\varpi_1\right\}$.

$\varpi_2, \varpi_4 \in \Lambda, \varpi_2 \neq \varpi_4$, there is $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_2\right\}$ such that $\varpi_2 \in\left\{\varpi_2\right\}, \varpi_4 \notin\left\{\varpi_2\right\}$.

$\varpi_2, \varpi_5 \in \Lambda, \varpi_2 \neq \varpi_5$, there is $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_2\right\}$ such that $\varpi_2 \in\left\{\varpi_2\right\}, \varpi_5 \notin\left\{\varpi_2\right\}$.

$\varpi_3, \varpi_5 \in \Lambda, \varpi_3 \neq \varpi_5$, there is $\mathcal{T}_{\text {Ic }}$-open set $\left\{\varpi_3\right\}$ such that $\varpi_3 \in\left\{\varpi_3\right\}, \varpi_5 \notin\left\{\varpi_3\right\}$.

Clearly, the incidence topology $\mathcal{T}_{\text {Ic }}$ represents a $\mathrm{T}_0$-space.

Then the II-induced topology $\mathcal{T}_{\text {II }}=\mathcal{T}_{\text {Id }} \cap \mathcal{T}_{\text {Ic }}: \mathcal{T}_{\text {II }}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_4\right\},\left\{\varpi_5\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_1, \varpi_3\right\},\left\{\varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_4\right\}\right.$, $\left\{\varpi_1, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_4\right\},\left\{\varpi_1, \varpi_2, \varpi_5\right\},\left\{\varpi_1, \varpi_3, \varpi_4\right\}$, $\left\{\varpi_1, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_4\right\}$, $\left.\left\{\varpi_1, \varpi_2, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_3, \varpi_4, \varpi_5\right\},\left\{\varpi_1, \varpi_2, \varpi_3, \varpi_5\right\}\right\}$

It is clear that for every distinct vertex in II-induced topology, there exist II -bi-open set such that one of them contain one vertex but not the other.

Then $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ represents a II-$T_0$-space.

In this example we conclude that the independent $\mathcal{T}_{\text {Id }}$ is the most dominant and the most powerful because it controls the the intersection of the other topology.

Definition 6.1.3. Let $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ be II -induced topological space from bi-topological space $\left(\Lambda, \mathcal{T}_{\text {Id }}, \mathcal{T}_{\text {Ic }}\right)$ over $\Lambda$, then $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ is said to be II-$\mathrm{T}_1$-space if for each pair of distinct vertices $\varpi_1, \varpi_2 \in \Lambda$, there is II-bi-open sets $W_1, W_2$ s. t. '$\varpi_1 \in$ $W_1, \varpi_2 \notin W_1$' and '$\varpi_1 \notin W_2, \varpi_2 \in W_2$'.

Definition 6.1.4. Let $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ be II-induced topological space from bi-topological space $\left(\Lambda, \mathcal{T}_{\text {Id }}, \mathcal{T}_{\text {Ic }}\right)$ over $\Lambda$, then $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ is said to be II-$\mathrm{T}_2$-space if for each pair of distinct vertices $\varpi_1, \varpi_2 \in \Lambda$, there is II-bi-open sets $W_1, W_2$ s. t. '$\varpi_1 \in$ $W_1, \varpi_2 \in W_2$ and $W_1 \cap W_2=\emptyset$'.

Remark 6.1.5. Only if the two topologies are discrete, the $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ will be II-$\mathrm{T}_1$ and II-$\mathrm{T}_2$ as shows the example below.

6.2 II-regular, II-normal and II-T_i: (i=3,4) spaces

In this section, we define $\mathrm{II}-\mathrm{T}_3$ and $\mathrm{II}-\mathrm{T}_4$ spaces using ordinary points and characterize II-regular and II-normal spaces on II-induced topological space ($\Lambda, \mathcal{T}_{\text {II }}$) from bi-topological space $\left(\Lambda, \mathcal{T}_{\text {Id }}, \mathcal{T}_{\text {Ic }}\right)$ over the vertices set $\Lambda$.

Definition 6.2.1. Let $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ be II-induced topological space from bi-topological space $\left(\Lambda, \mathcal{T}_{\text {Id }}, \mathcal{T}_{\text {Ic }}\right)$ over $\Lambda$, then $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ is said to be II-regular-space if for each II-bi-closed set $F$ in $\Lambda$, and each $\varpi \notin F$, there is an II-bi-open sets $W_1, W_2$ s. t. '$\varpi \in W_1, F \subseteq W_2$ and $W_1 \cap W_2=\emptyset$'.

Definition 6.2.2. Let $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ be II-induced topological space from bi-topological space $\left(\Lambda, \mathcal{T}_{\text {Id }}, \mathcal{T}_{\text {Ic }}\right)$ over $\Lambda$, then $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ is said to be II-$\mathrm{T}_3$-space iff it is an II-regular and II-$\mathrm{T}_{1}$-space.

Definition 6.2.3. Let $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ be II-induced topological space from bi-topological space $\left(\Lambda, \mathcal{T}_{\text {Id }}, \mathcal{T}_{\text {Ic }}\right)$ over $\Lambda$, then $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ is said to be II-normal space if for all pair of II-biclosed sets $\mathrm{F}_1$ and $\mathrm{F}_2$ in $\Lambda$ s. t. $F_1 \cap F_2=\emptyset$, there is an II-biopen sets $W_1, W_2$ s. t. '$F_1 \subseteq W_1, F_2 \subseteq W_2$ and $W_1 \cap W_2=\emptyset$'.

Definition 6.2.4. Let $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ be II-induced topological space from bi-topological space $\left(\Lambda, \mathcal{T}_{\text {Id }}, \mathcal{T}_{\text {Ic }}\right)$ over $\Lambda$, then $\left(\Lambda, \mathcal{T}_{\text {II }}\right)$ is said to be II-$\mathrm{T}_4$-space iff it is an II-normal and II-$\mathrm{T}_1$-space.

Example 6.2.5. $\mathfrak{S}=(\Lambda, \mathfrak{L})$ is a cycle $\mathbb{C}_3$ graph as in Figure 6, such that: $\Lambda=\left\{\varpi_1, \varpi_2, \varpi_3\right\}, \mathfrak{L}=\left\{\vartheta_1, \vartheta_2, \vartheta_3\right\}$.

The sub-base of independent topology is the set: $S_{\mathrm{I} d}=$ $\left\{\left\{\varpi_1\right\},\left\{\varpi_2\right\},\left\{\varpi_3\right\}\right\}$ taking the finitely intersection the base $\beta_{I d}$ obtained, $\left\{\emptyset,\left\{\varpi_1\right\},\left\{\varpi_2\right\},\left\{\varpi_3\right\}\right\}$.

6.png

Figure 6. A cycle $\mathbb{C}_3$ graph

Then via taking the unions: independent topology is $\mathcal{T}_{\mathrm{I} d}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_2\right\},\left\{\varpi_3\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3\right\}\right\}$.

It is clear that $\mathcal{T}_{\mathrm{I} d}$ is discrete.

The sub-base of incident topology is the set $S_{\text {Ic }}=$ $\left\{\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3\right\}\right\}$.

Taking the finitely intersection the base $\beta_{\text {Ic }}$ obtained $\left\{\emptyset,\left\{\varpi_1\right\},\left\{\varpi_2\right\},\left\{\varpi_3\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3\right\}\right\}$.

And, incident topology is $\mathcal{T}_{\text {I } c}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_2\right\},\left\{\varpi_3\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3\right\}\right\}$.

It is clear that $\mathcal{T}_{\text {Ic }}$ is discrete.

Since $\mathcal{T}_{\mathrm{II}}=\mathcal{T}_{\mathrm{I} d} \cap \mathcal{T}_{\mathrm{I} c}$, then II-induced topology $\mathcal{T}_{\mathrm{II}}=$ $\left\{\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_2\right\},\left\{\varpi_3\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3\right\}\right\}$, then $\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_2\right\},\left\{\varpi_3\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3\right\}$ are II-biopen sets, and $\emptyset, \Lambda,\left\{\varpi_1\right\},\left\{\varpi_2\right\},\left\{\varpi_3\right\},\left\{\varpi_1, \varpi_2\right\},\left\{\varpi_2, \varpi_3\right\},\left\{\varpi_1, \varpi_3\right\}$ are II-biclosed sets.

Therefore $\mathcal{T}_{\mathrm{I} d}$ and $\mathcal{T}_{\mathrm{I} c}$ on $\Lambda$ give us a discrete bi-topological space $\left(\Lambda, \mathcal{T}_{\mathrm{I} d}, \mathcal{T}_{\mathrm{I} c}\right)$.

Clear that $\mathcal{T}_{\mathrm{I} d}$ and $\mathcal{T}_{\mathrm{I} c}$ are discrete, and they induce $\mathcal{T}_{\mathrm{II}}$ topology, which is also a discrete topology, thus $\mathcal{T}_{\text {II }}$ investigates the II-$\mathrm{T}_2$-space and II-$\mathrm{T}_2$-space, also it investigates the II-regular-space and II-normal-space. And sequentially it is II-$\mathrm{T}_3$-space and II-$\mathrm{T}_4$-space.

Therefore $\mathcal{T}_{\text {II }}$ of the cycle graph investigates the II-$\mathrm{T}_3$ and II-$\mathrm{T}_4$ axioms.