Special Topological Sets and Their Continous Applications

#### **Chapter 2**

## More Functions Associated with Neutrosophic gsα\*- Closed Sets in Neutrosophic Topological Spaces

*P. Anbarasi Rodrigo and S. Maheswari*

#### **Abstract**

The concept of neutrosophic continuous function was very first introduced by A.A. Salama et al. The main aim of this paper is to introduce a new concept of Neutrosophic continuous function namely Strongly Neutrosophic gsα\* - continuous functions, Perfectly Neutrosophic gsα\* - continuous functions and Totally Neutrosophic gsα\* continuous functions in Neutrosophic topological spaces. These concepts are derived from strongly generalized neutrosophic continuous function and perfectly generalized neutrosophic continuous function. Several interesting properties and characterizations are derived and compared with already existing neutrosophic functions.

**Keywords:** Neutrosophic *gsα*\*- closed set, Neutrosophic *gsα*\*- open set, Strongly Neutrosophic *gsα*\*- continuous function, Perfectly Neutrosophic *gsα*\*- continuous function, Totally Neutrosophic *gsα*\*- continuous function

#### **1. Introduction**

The concept of Neutrosophic set theory was introduced by F. Smarandache [1] and it comes from two concept, one is intuitionistic fuzzy sets introduced by K. Atanassov's [2] and the other is fuzzy sets introduced by L.A. Zadeh's [3]. It includes three components, truth, indeterminancy and false membership function. R. Dhavaseelan and S. Jafari [4] has discussed about the concept of strongly generalized neutrosophic continuous function. Further he also introduced the topic of perfectly generalized neutrosophic continuous function. The real life application of neutrosophic topology is applied in Information Systems, Applied Mathematics etc.

In this paper, we introduce some new concepts related to Neutrosophic *gsα*<sup>∗</sup> � continuous function namely Strongly Neutrosophic *gsα*<sup>∗</sup> � continuous function, Perfectly Neutrosophic *gsα*<sup>∗</sup> � continuous function, Totally Neutrosophic *gsα*<sup>∗</sup> � continuous function.

#### **2. Preliminaries**

**Definition 2.1:** [5] Let be a non-empty fixed set. A Neutrosophic set Ҥ on the universe is defined as <sup>Ҥ</sup><sup>¼</sup> <sup>q</sup>, *<sup>t</sup>*Ҥð Þ <sup>q</sup> , *<sup>i</sup>*Ҥð Þ <sup>q</sup> , *<sup>f</sup>* <sup>Ҥ</sup>ð Þ <sup>q</sup> : <sup>q</sup> *<sup>Є</sup>* where *t*Ҥð Þ q , *i*Ҥð Þ q , *f* <sup>Ҥ</sup>ð Þ q represent the degree of membership function *t*Ҥð Þ q , the degree of indeterminacy *i*Ҥð Þ q and the degree of non-membership function *f* <sup>Ҥ</sup>ð Þ q respectively for each element q *Є* to the set Ҥ. Also, *t*Ҥ, *i*Ҥ, *f* <sup>Ҥ</sup> : !��0, 1þ½ and � 0

≤ *t*Ҥð Þþ q *i*Ҥð Þþ q *f* <sup>Ҥ</sup>ð Þ q ≤3þ. Set of all Neutrosophic set over is denoted by Neu().

**Definition 2.2:** [8] Let be a non-empty set. <sup>Ⱥ</sup> <sup>¼</sup> <sup>q</sup>, *<sup>t</sup> <sup>Ⱥ</sup>* ð Þ <sup>q</sup> , *<sup>i</sup> <sup>Ⱥ</sup>* ð Þ <sup>q</sup> , *<sup>f</sup> <sup>Ⱥ</sup>* ð Þ <sup>q</sup> : <sup>q</sup> *<sup>Є</sup>* and *<sup>Ƀ</sup>* <sup>¼</sup> <sup>q</sup>, *<sup>t</sup>Ƀ*ð Þ <sup>q</sup> , *<sup>i</sup>Ƀ*ð Þ <sup>q</sup> , *<sup>f</sup> <sup>Ƀ</sup>*ð Þ <sup>q</sup> : <sup>q</sup> *<sup>Є</sup>* are neutrosophic sets, then

$$\text{i.i. } \mathsf{A} \subseteq \mathsf{B} \text{ if } t\_{\mathsf{A}}(\mathfrak{p}) \le t\_{\mathsf{B}}(\mathfrak{p}), i\_{\mathsf{A}}(\mathfrak{p}) \le i\_{\mathsf{B}}(\mathfrak{p}), f\_{\mathsf{A}}(\mathfrak{p}) \ge f\_{\mathsf{B}}(\mathfrak{p}) \text{ for all } \mathfrak{p} \in \mathbb{P}.$$


$$\text{iv. } \mathsf{A}^c = \{ \langle \ \mathfrak{p}, \left( f\_{\mathsf{A}}(\mathfrak{p}), \mathbf{1} - i\_{\mathsf{A}}(\mathfrak{p}), t\_{\mathsf{A}}(\mathfrak{p}) \right) \rangle : \mathfrak{p} \in \mathbb{P} \}.$$

$$\text{iv. } \mathbf{0}\_{N\_{\text{in}}} = \{ \langle \mathfrak{p}, (\mathbf{0}, \mathbf{0}, \mathbf{1}) \rangle : \mathfrak{p} \in \mathbb{P} \} \text{ and } \mathbf{1}\_{N\_{\text{in}}} = \{ \langle \mathfrak{p}, (\mathbf{1}, \mathbf{1}, \mathbf{0}) \rangle : \mathfrak{p} \in \mathbb{P} \}.$$

**Definition 2.3:** [5] A neutrosophic topology (NeuT) on a non-empty set is a family *τNeu* of neutrosophic sets in satisfying the following axioms,


In this case, the ordered pair , *τNeu* ð Þ or simply is called a neutrosophic topological space (*Neu*TS). The elements of *τNeu* is neutrosophic open set ð Þ *Neu* � *OS* and *τNeu <sup>c</sup>* is neutrosophic closed set ð Þ *Neu* � *CS* .

**Definition 2.4:** [6] A neutrosophic set Ⱥ in a *Neu*TS , *τNeu* ð Þ is called a neutrosophic generalized semi alpha star closed set *Neugs<sup>α</sup>* ð Þ <sup>∗</sup> � *CS* if *Neu<sup>α</sup>* � *int N*ð Þ *eu<sup>α</sup>* � *cl*ð Þ <sup>Ⱥ</sup> <sup>⊆</sup> *Neu* � *int*(G), whenever <sup>Ⱥ</sup> <sup>⊆</sup> <sup>G</sup> and <sup>G</sup> is *Neuα*<sup>∗</sup> � open set.

**Definition 2.5:** [7] A neutrosophic topological space , *τNeu* ð Þ is called a *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space if every *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ is a *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. **Definition 2.6:** A neutrosophic function *f* : , *τNeu* ð Þ! , *σNeu* ð Þ is said to be


*More Functions Associated with Neutrosophic gsα\*- Closed Sets in Neutrosophic… DOI: http://dx.doi.org/10.5772/intechopen.99464*

**Definition 2.7:** [9] Let *τNeu* ¼ 0*Neu* f g , 1*Neu* is a neutrosophic topological space over . Then , *τNeu* ð Þ is called neutrosophic discrete topological space.

**Definition 2.8:** A neutrosophic topological space , *τNeu* ð Þ is called a neutrosophic clopen set ð*Neu*�clopen set) if it is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ.

#### **3. Strongly neutrosophic** *gsα* **<sup>∗</sup> -continuous function**

**Definition 3.1:** A neutrosophic function *f* : , *τNeu* ð Þ! , *σNeu* ð Þ is said to be strongly *Neugsα*<sup>∗</sup> � continuous if the inverse image of every *Neugsα*<sup>∗</sup> � *CS* in , *σNeu* ð Þ is a *Neu* � *CS* in , *τNeu* ð Þ. (ie) *f* �1 ð Þ Ⱥ is a *Neu* � *CS* in , *τNeu* ð Þ for every *Neugsα*<sup>∗</sup> � *CS* <sup>Ⱥ</sup> in , *<sup>σ</sup>Neu* ð Þ.

**Theorem 3.2:** Every strongly *Neugsα*<sup>∗</sup> � continuous is neutrosophic continuous, but not conversely.

#### **Proof:**

Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be any neutrosophic function. Let Ⱥ be any *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since every *Neu* � *CS* is *Neugsα*<sup>∗</sup> � *CS*, then <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 ð Þ Ⱥ is *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *f* is neutrosophic continuous.

**Example 3.3:** Let =f g q and =f gr . *τNeu* ¼ 0*Neu* , 1*Neu* f g , Ⱥ and *σNeu* ¼ 0*Neu* , 1*Neu* f g , Ƀ are *Neu*TS on , *τNeu* ð Þ and , *σNeu* ð Þ respectively. Also Ⱥ¼ f g h i q, 0ð Þ *:*6, 0*:*4, 0*:*4 and Ƀ¼ f g h i r, 0ð Þ *:*4, 0*:*6, 0*:*2 are *Neu*ð Þ and *Neu*ð Þ . Define a map *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ by *<sup>f</sup>*ð Þ¼ <sup>q</sup> <sup>r</sup> <sup>þ</sup> <sup>0</sup>*:*2. Let <sup>Ƀ</sup>*<sup>c</sup>* <sup>¼</sup> f g h i r, 0ð Þ *:*2, 0*:*4, 0*:*4 be a *Neu* � *CS* in , *σNeu* ð Þ. Then *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ¼ f g h i <sup>q</sup>, 0ð Þ *:*4, 0*:*6, 0*:*<sup>6</sup> . Now, *Neu* � *cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>¼</sup> *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ)

*f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is neutrosophic continuous, but *<sup>f</sup>* is not strongly *Neugsα*<sup>∗</sup> � continuous. Let <sup>Ȼ</sup> <sup>¼</sup> f g h i <sup>r</sup>, 0ð Þ *:*1, 0*:*2, 0*:*<sup>8</sup> be a *Neugsα*<sup>∗</sup> � *CS* in , *σNeu* ð Þ. Then *f* �1 ð Þ¼ <sup>Ȼ</sup> f g h i <sup>q</sup>, 0ð Þ *:*3, 0*:*4, 1 . Now *Neu* � *cl f* �<sup>1</sup> ð Þ <sup>Ȼ</sup> <sup>¼</sup> �1 �1

<sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* 6¼ *<sup>f</sup>* ð Þ) Ȼ *f* ð Þ Ȼ is not *Neu* � *CS* in , *τNeu* ð Þ.

**Theorem 3.4:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be strongly *Neugsα*<sup>∗</sup> � continuous iff the inverse image of every *Neugsα*<sup>∗</sup> � *OS* in , *<sup>σ</sup>Neu* ð Þ is *Neu* � *OS* in , *<sup>τ</sup>Neu* ð Þ. **Proof:**

Assume that *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous function. Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *OS* in , *<sup>σ</sup>Neu* ð Þ. Then <sup>Ⱥ</sup> *<sup>c</sup>* is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �<sup>1</sup> <sup>Ⱥ</sup> *<sup>c</sup>* ð Þ is *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ) *<sup>f</sup>* �1 ð Þ <sup>Ⱥ</sup> *<sup>c</sup>* is *Neu* � *CS* in , *τNeu* ð Þ) *f* �1 ð Þ Ⱥ is *Neu* � *OS* in , *τNeu* ð Þ. Conversely, Let Ⱥ be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then <sup>Ⱥ</sup> *<sup>c</sup>* is *Neugsα*<sup>∗</sup> � *OS* in , *<sup>σ</sup>Neu* ð Þ. By hypothesis, *f* �<sup>1</sup> <sup>Ⱥ</sup> *<sup>c</sup>* ð Þ is *Neu* � *OS* in , *<sup>τ</sup>Neu* ð Þ) *<sup>f</sup>* �1 ð Þ <sup>Ⱥ</sup> *<sup>c</sup>* is *Neu* � *OS* in , *τNeu* ð Þ) *f* �1 ð Þ Ⱥ is *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous.

**Theorem 3.5:** Every strongly *Neugsα*<sup>∗</sup> � continuous is *Neugsα*<sup>∗</sup> � continuous, but not conversely.

**Proof:**

Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be any neutrosophic function. Let Ⱥ be any *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *f* �1 ð Þ Ⱥ is a *Neu* � *CS* in , *τNeu* ð Þ) *f* �1 ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � continuous.

**Example 3.6:** Let ¼ f g q and =f gr . *τNeu* ¼ 0*Neu* , 1*Neu* f g , Ⱥ and *σNeu* ¼ 0*Neu* , 1*Neu* f g , Ƀ are *Neu*TS on , *τNeu* ð Þ and , *σNeu* ð Þ respectively. Also Ⱥ ¼ f g h i q, 0ð Þ *:*4, 0*:*5, 0*:*7 and Ƀ ¼ f g h i r, 0ð Þ *:*6, 0*:*8, 0*:*4 are *Neu*ð Þ and *Neu*ð Þ . Define a map *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ by *<sup>f</sup>*ð Þ¼ <sup>q</sup> <sup>r</sup>. Let <sup>Ƀ</sup>*<sup>c</sup>* <sup>¼</sup> f g h i <sup>r</sup>, 0ð Þ *:*4, 0*:*2, 0*:*<sup>6</sup> be a *Neu* � *CS* in , *σNeu* ð Þ. Then *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ¼ f g h i <sup>q</sup>, 0ð Þ *:*4, 0*:*2, 0*:*<sup>6</sup> . *Neuα*<sup>∗</sup> � *OS* <sup>¼</sup> *Neu<sup>α</sup>* � *OS* <sup>¼</sup> <sup>0</sup>*Neu* , 1*Neu* f g , <sup>Ⱥ</sup> and *Neu<sup>α</sup>* � *CS* <sup>¼</sup> <sup>0</sup>*Neu* , 1*Neu* , <sup>Ⱥ</sup> *<sup>c</sup>* f g. *Neuα*–*cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* . Now, *Neu<sup>α</sup>* � *int Neu<sup>α</sup>* � *cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> Ⱥ ⊆ *Neu* � *int* 1*Neu* ð Þ¼ 1*Neu* , whenever *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>⊆</sup> <sup>1</sup>*Neu* ) *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � continuous. But *<sup>f</sup>* is not strongly *Neugsα*<sup>∗</sup> �continuous. Let *<sup>Ȼ</sup>* <sup>¼</sup> f g h i <sup>r</sup>, 0ð Þ *:*3, 0*:*1, 0*:*<sup>7</sup> be a *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then *f* �1 ð Þ¼ *<sup>Ȼ</sup>* f g h i <sup>q</sup>, 0ð Þ *:*3, 0*:*1, 0*:*<sup>7</sup> . Now *Neu* � *cl f* �<sup>1</sup> ð Þ <sup>Ȼ</sup> <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* 6¼ *f* �1 ð Þ) Ȼ *f* �1 ð Þ Ȼ is not *Neu* � *CS* in , *τNeu* ð Þ.

**Theorem 3.7:** Every strongly neutrosophic continuous is strongly *Neugsα*<sup>∗</sup> � continuous, but not conversely.

#### **Proof:**

Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be any neutrosophic function. Let Ⱥ be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is strongly neutrosophic continuous, then *<sup>f</sup>* �1 ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ) *f* �1 ð Þ Ⱥ is *Neu* � *CS* in , *τNeu* ð Þ. Hence, *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous.

**Example 3.8:** Let =f g q and =f gr . *τNeu* ¼ 0*Neu* , 1*Neu* f g , Ⱥ , Ȼ and *σNeu* ¼ 0*Neu* , 1*Neu* f g , Ƀ are *Neu*TS on , *τNeu* ð Þ and , *σNeu* ð Þ respectively. Also Ⱥ¼ f g h i q, 0ð Þ *:*4, 0*:*6, 0*:*2 , Ȼ ¼ fhq, 0ð½ � *:*4, 1 , 0½ *:*6, 1�, 0, 0 ½ � *:*2 Þig and Ƀ ¼ f g h i r, 0ð Þ *:*4, 0*:*6, 0*:*2 are *Neu*ð Þ and *Neu*ð Þ . Define a map *f* : , *τNeu* ð Þ! , *σNeu* ð Þ by *<sup>f</sup>*ð Þ¼ <sup>q</sup> <sup>r</sup>. Let *<sup>Ⱦ</sup>* <sup>¼</sup> f g h i <sup>r</sup>, 0, 0 ð Þ ½ � *:*<sup>2</sup> , 0, 0 ½ � *:*<sup>4</sup> , 0½ � *:*4, 1 be a *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then *f* �1 ð Þ¼ <sup>Ⱦ</sup> f g h i <sup>q</sup>, 0, 0 ð Þ ½ � *:*<sup>2</sup> , 0, 0 ½ � *:*<sup>4</sup> , 0½ � *:*4, 1 . Now *Neu* � *cl f* �<sup>1</sup> ð Þ <sup>Ⱦ</sup> <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>Ȼ</sup>*<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ȼ</sup>*<sup>c</sup>* <sup>¼</sup> *<sup>f</sup>* �1 ð Þ <sup>Ⱦ</sup> . Therefore, *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous. But *<sup>f</sup>* is not strongly neutrosophic continuous. Let Ɇ¼ f g h i r, 0ð Þ *:*4, 0*:*6, 0*:*2 be a neutrosophic set in , *σNeu* ð Þ. Then *f* �1 ð Þ¼ Ɇ f g h i q, 0ð Þ *:*4, 0*:*6, 0*:*2 . Now *Neu* � *int f* �<sup>1</sup> ð Þ <sup>Ɇ</sup> <sup>¼</sup> <sup>0</sup>*Neu* <sup>∪</sup> <sup>Ⱥ</sup> <sup>¼</sup> <sup>Ⱥ</sup> <sup>¼</sup> *<sup>f</sup>* �1 ð Þ) Ɇ *f* �1 ð Þ Ɇ is *Neu* � *OS* in , *τNeu* ð Þ. Also *Neu* � *cl f* �<sup>1</sup> ð Þ <sup>Ɇ</sup> <sup>¼</sup> <sup>1</sup>*Neu* 6¼ *<sup>f</sup>* �1 ð Þ) Ɇ *f* �1 ð Þ Ɇ is not *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *f* �1 ð Þ Ɇ is not both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ.

**Remark 3.9**: Every strongly neutrosophic continuous is *Neugsα*<sup>∗</sup> � continuous, but not conversely. (by Theorem 3.5 & 3.7).

**Theorem 3.10:** Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be neutrosophic function and , *<sup>σ</sup>Neu* ð Þ be *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space. Then the following are equivalent.

1. *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous.

2. *f* is neutrosophic continuous.

#### **Proof:**


**Theorem 3.11:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be *Neugsα*<sup>∗</sup> � continuous. Both , *<sup>τ</sup>Neu* ð Þ and , *<sup>σ</sup>Neu* ð Þ are *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous.

*More Functions Associated with Neutrosophic gsα\*- Closed Sets in Neutrosophic… DOI: http://dx.doi.org/10.5772/intechopen.99464*

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since , *<sup>σ</sup>Neu* ð Þ is *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 ð Þ Ⱥ is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Since , *<sup>τ</sup>Neu* ð Þ is *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then *f* �1 ð Þ Ⱥ is *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous.

**Theorem 3.12:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be strongly *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � irresolute.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *f* �1 ð Þ Ⱥ is *Neu* � *CS* in , *τNeu* ð Þ) *f* �1 ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Hence, *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � irresolute.

**Theorem 3.13:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be *Neugsα*<sup>∗</sup> � irresolute and , *<sup>τ</sup>Neu* ð Þ be *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � irresolute, then *f* �1 ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Since , *<sup>τ</sup>Neu* ð Þ is *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then *f* �1

ð Þ <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous. **Theorem 3.14:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ and *<sup>g</sup>* : , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be strongly *Neugsα*<sup>∗</sup> � continuous, then *gof* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* is strongly *Neugsα*<sup>∗</sup> � continuous.

### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ) *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 *<sup>g</sup>*ð Þ¼ �<sup>1</sup>ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ Ⱥ is *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *gof* is strongly *Neugsα*<sup>∗</sup> � continuous.

**Theorem 3.15:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be strongly *Neugsα*<sup>∗</sup> � continuous and *<sup>g</sup>* : , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be *Neugsα*<sup>∗</sup> �continuous, then *gof* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* is neutrosophic continuous.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neu* � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is *Neugsα*<sup>∗</sup> � continuous, then *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *f* �1 *<sup>g</sup>*ð Þ¼ �<sup>1</sup>ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ Ⱥ is *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *gof* is neutrosophic continuous.

**Theorem 3.16:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be strongly *Neugsα*<sup>∗</sup> � continuous and *<sup>g</sup>* : , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be *Neugsα*<sup>∗</sup> � irresolute, then *gof* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* is strongly *Neugsα*<sup>∗</sup> � continuous.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is *Neugsα*<sup>∗</sup> � irresolute, then *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *f* �1 *<sup>g</sup>*ð Þ¼ �<sup>1</sup>ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ Ⱥ is *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *gof* is strongly *Neugsα*<sup>∗</sup> � continuous.

**Theorem 3.17:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be *Neugsα*<sup>∗</sup> � continuous and *<sup>g</sup>*: , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be strongly *Neugsα*<sup>∗</sup> �continuous, then *gof* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* is *Neugsα*<sup>∗</sup> � irresolute.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � continuous, then *f* �1 *<sup>g</sup>*�<sup>1</sup> ð Þ¼ ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Hence, *gof* is *Neugsα*<sup>∗</sup> � irresolute.

**Theorem 3.18:** Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be neutrosophic continuous and *g* : , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be strongly *Neugsα*<sup>∗</sup> � continuous, then *gof* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* is strongly *Neugsα*<sup>∗</sup> � continuous. **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *<sup>g</sup>*�1ð Þ <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is neutrosophic continuous, then *f* �1 *<sup>g</sup>*�<sup>1</sup> ð Þ¼ ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Hence, *gof* is strongly *Neugsα*<sup>∗</sup> � continuous.

#### **Inter-relationship 3.19:**

#### **4. Perfectly neutrosophic** *gsα* **<sup>∗</sup> -continuous function**

**Definition 4.1:** A neutrosophic function *f* : , *τNeu* ð Þ! , *σNeu* ð Þ is said to be perfectly *Neugsα*<sup>∗</sup> � continuous if the inverse image of every *Neugsα*<sup>∗</sup> � *CS* in , *σNeu* ð Þ is both *Neu* � *OS* and *Neu* � *CS* (ie, *Neu*� clopen set) in , *τNeu* ð Þ.

**Theorem 4.2:** Every perfectly *Neugsα*<sup>∗</sup> � continuous is strongly *Neugsα*<sup>∗</sup> � continuous, but not conversely.

#### **Proof:**

Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be any neutrosophic function. Let Ⱥ be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ) *f* �1 ð Þ Ⱥ is *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous.

**Example 4.3:** Let =f g q and =f gr . *τNeu* ¼ 0*Neu* , 1*Neu* f g , Ⱥ , Ȼ and *σNeu* ¼ 0*Neu* , 1*Neu* f g ,Ƀ are *Neu*TS on , *τNeu* ð Þ and , *σNeu* ð Þ respectively. Also Ⱥ¼ f g h i q, 0ð Þ *:*7, 0*:*8, 0*:*3 , Ȼ ¼ f g h i q, 0ð Þ ½ � *:*7, 1 , 0½ � *:*8, 1 , 0, 0 ½ � *:*3 and Ƀ ¼ f g h i r, 0ð Þ *:*7, 0*:*8, 0*:*3 are *Neu*ð Þ and *Neu*ð Þ . Define a map *f* : , *τNeu* ð Þ! , *σNeu* ð Þ by *<sup>f</sup>*ð Þ¼ <sup>q</sup> <sup>r</sup>. Let <sup>Ⱦ</sup> <sup>¼</sup> f g h i <sup>r</sup>, 0, 0 ð Þ ½ � *:*<sup>3</sup> , 0, 0 ½ � *:*<sup>2</sup> , 0½ � *:*7, 1 be a *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then *f* �1 ð Þ¼ <sup>Ⱦ</sup> f g h i <sup>q</sup>, 0, 0 ð Þ ½ � *:*<sup>3</sup> , 0, 0 ½ � *:*<sup>2</sup> , 0½ � *:*7, 1 . Now *Neu* � *cl f* �<sup>1</sup> ð Þ <sup>Ⱦ</sup> <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>Ȼ</sup>*<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ȼ</sup> *<sup>c</sup>* <sup>¼</sup> *<sup>f</sup>* �1 ð Þ <sup>Ⱦ</sup> . Therefore, *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous. But *<sup>f</sup>* is not perfectly *Neugsα*<sup>∗</sup> � continuous, because *<sup>f</sup>* �1 ð Þ Ⱦ is not both *Neu* � *OS* and *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Since, *Neu* � *int f* �<sup>1</sup> ð Þ <sup>Ⱦ</sup> <sup>¼</sup> <sup>0</sup>*Neu* ¼6 *<sup>f</sup>* �1 ð Þ) Ⱦ *f* �1 ð Þ Ⱦ is not *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *f* �1 ð Þ Ⱦ is not both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ.

**Theorem 4.4:** Every perfectly *Neugsα*<sup>∗</sup> � continuous is perfectly neutrosophic continuous, but not conversely.

*More Functions Associated with Neutrosophic gsα\*- Closed Sets in Neutrosophic… DOI: http://dx.doi.org/10.5772/intechopen.99464*

#### **Proof:**

Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be any neutrosophic function. Let Ⱥ be any *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous, then *f* �1 ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *f* is perfectly neutrosophic continuous.

**Example 4.5:** Let ¼ f g q and = f gr . *τNeu* ¼ 0*Neu* , 1*Neu* f g , Ⱥ , Ȼ, Ɇ and *σNeu* ¼ 0*Neu* , 1*Neu* f g , Ƀ are *Neu*TS on , *τNeu* ð Þ and , *σNeu* ð Þ respectively. Also Ⱥ¼ f g h i q, 0ð Þ *:*4, 0*:*2, 0*:*6 , Ȼ ¼ f g h i q, 0ð Þ *:*6, 0*:*8, 0*:*4 , Ɇ ¼ f g h i q, 0, 0 ð Þ ½ � *:*4 , 0, 0 ½ � *:*2 , 0½ � *:*6, 1 and Ƀ¼ f g h i r, 0ð Þ *:*6, 0*:*8, 0*:*4 are *Neu*ð Þ and *Neu*ð Þ . Define a map *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ by *<sup>f</sup>*ð Þ¼ <sup>q</sup> <sup>r</sup>. Let <sup>Ƀ</sup>*<sup>c</sup>* <sup>¼</sup> f g h i r, 0ð Þ *:*4, 0*:*2, 0*:*6 be a *Neu* � *CS* in , *σNeu* ð Þ. Then *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ¼ f g h i <sup>q</sup>, 0ð Þ *:*4, 0*:*2, 0*:*<sup>6</sup> . Now *Neu* � *cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>Ȼ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>Ɇ</sup>*<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ȼ</sup> *<sup>c</sup>* <sup>¼</sup> *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ. Also, *Neu* � *int f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>Ⱥ</sup> <sup>∪</sup> <sup>Ɇ</sup> <sup>∪</sup> <sup>0</sup>*Neu* <sup>¼</sup> <sup>Ⱥ</sup> <sup>¼</sup> *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ) *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *f* is perfectly neutrosophic continuous. But *<sup>f</sup>* is not perfectly *Neugsα*<sup>∗</sup> � continuous. Let <sup>Ⱦ</sup> <sup>¼</sup> fhr, 0, 0 ð½ � *:*<sup>4</sup> , 0, 0 ½ � *:*<sup>2</sup> , 0½ Þi *:*6, 1� <sup>g</sup> be *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then *<sup>f</sup>* �1 ð Þ¼ Ⱦ fh <sup>q</sup>, 0, 0 ð½ � *:*<sup>4</sup> , 0, 0 ½ � *:*<sup>2</sup> , 0½ Þi *:*6, 1� <sup>g</sup>. Since, *Neu* � *int f* �<sup>1</sup> ð Þ <sup>Ⱦ</sup> <sup>¼</sup> <sup>Ɇ</sup> <sup>∪</sup> <sup>0</sup>*Neu* <sup>¼</sup> <sup>Ɇ</sup> <sup>¼</sup> *<sup>f</sup>* �1 ð Þ Ⱦ ) *f* �1 ð Þ <sup>Ⱦ</sup> is *Neu* � *OS* in , *<sup>τ</sup>Neu* ð Þ. Also, *Neu* � *cl f* �<sup>1</sup> ð Þ <sup>Ⱦ</sup> <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>Ȼ</sup>*<sup>c</sup>* <sup>∩</sup> <sup>Ɇ</sup>*<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ȼ</sup> *<sup>c</sup>* 6¼ *<sup>f</sup>* �1 ð Þ) Ⱦ *f* �1 ð Þ Ⱦ is not *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *f* �1 ð Þ Ⱦ is not both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ.

**Theorem 4.6:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be perfectly *Neugsα*<sup>∗</sup> � continuous iff the inverse image of every *Neugsα*<sup>∗</sup> � *OS* in , *<sup>σ</sup>Neu* ð Þ is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ.

**Proof:**

Assume that *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous function. Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *OS* in , *<sup>σ</sup>Neu* ð Þ. Then <sup>Ⱥ</sup> *<sup>c</sup>* is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous, then *f* �<sup>1</sup> <sup>Ⱥ</sup> *<sup>c</sup>* ð Þ is both *Neu* � *OS* and *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ) *<sup>f</sup>* �1 ð Þ <sup>Ⱥ</sup> *<sup>c</sup>* is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ) *f* �1 ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Conversely, Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then <sup>Ⱥ</sup> *<sup>c</sup>* is *Neugsα*<sup>∗</sup> � *OS* in , *σNeu* ð Þ. By hypothesis, *f* �<sup>1</sup> <sup>Ⱥ</sup> *<sup>c</sup>* ð Þ is both *Neu* � *OS* and *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ) *f* �1 ð Þ <sup>Ⱥ</sup> *<sup>c</sup>* is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ) *f* �1 ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous.

**Theorem 4.7:** Let , *τNeu* ð Þ be a neutrosophic discrete topological space and , *σNeu* ð Þ be any neutrosophic topological space. Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be a neutrosophic function, then the following statements are true.

1. *<sup>f</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous.

2. *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous.

#### **Proof:**


**Theorem 4.8:** Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be perfectly neutrosophic continuous and , *<sup>σ</sup>Neu* ð Þ be *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since , *<sup>σ</sup>Neu* ð Þ is *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then Ⱥ is *Neu* � *CS* in , *σNeu* ð Þ*:*Since *f* is perfectly neutrosophic continuous, then *f* �1 ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous.

**Theorem 4.9:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ and *<sup>g</sup>* : , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be perfectly *Neugsα*<sup>∗</sup> � continuous, then *gof* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* is perfectly *Neugsα*<sup>∗</sup> � continuous.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous, then *<sup>g</sup>*�1ð Þ <sup>Ⱥ</sup> is both *Neu* � *OS* and *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ) *<sup>g</sup>*�1ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 *<sup>g</sup>*�<sup>1</sup> ð Þ¼ ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *gof* is perfectly *Neugsα*<sup>∗</sup> � continuous.

**Theorem 4.10:** Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be neutrosophic continuous and *g* : , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be perfectly *Neugsα*<sup>∗</sup> � continuous, then *gof* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* is strongly *Neugsα*<sup>∗</sup> � continuous.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous, then *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is both *Neu* � *OS* and *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is neutrosophic continuous, then *f* �1 *<sup>g</sup>*ð Þ¼ �<sup>1</sup>ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ Ⱥ is *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *gof* is strongly *Neugsα*<sup>∗</sup> � continuous.

**Theorem 4.11:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be perfectly *Neugsα*<sup>∗</sup> � continuous and *<sup>g</sup>* : , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be strongly *Neugsα*<sup>∗</sup> � continuous, then *gof* : , *<sup>τ</sup>Neu* ð Þ ! , *<sup>γ</sup>Neu* is perfectly *Neugsα*<sup>∗</sup> � continuous.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is strongly *Neugsα*<sup>∗</sup> � continuous, then *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ) *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 *<sup>g</sup>*ð Þ¼ �<sup>1</sup>ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *gof* is perfectly *Neugsα*<sup>∗</sup> � continuous.

### **5. Totally neutrosophic** *gsα* **<sup>∗</sup>** �**continuous function**

**Definition 5.1:** A neutrosophic function *f* : , *τNeu* ð Þ! , *σNeu* ð Þ is said to be totally *Neugsα*<sup>∗</sup> � continuous if the inverse image of every *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ is both *Neugsα*<sup>∗</sup> � *OS* and *Neugsα*<sup>∗</sup> � *CS* (ie, *Neugsα*<sup>∗</sup> � clopen set) in , *<sup>τ</sup>Neu* ð Þ.

**Definition 5.2:** A neutrosophic topological space , *τNeu* ð Þ is called a *Neugsα*<sup>∗</sup> �clopen set *Neugs<sup>α</sup>* <sup>ð</sup> <sup>∗</sup> �clopen set) if it is both *Neugsα*<sup>∗</sup> � *OS* and *Neugsα*<sup>∗</sup> � *CS* in , *τNeu* ð Þ.

**Example 5.3:** Let ¼ f g q and =f gr . *τNeu* ¼ 0*Neu* , 1*Neu* f g , Ⱥ and *σNeu* ¼ 0*Neu* , 1*Neu* f g , Ƀ are *Neu*TS on , *τNeu* ð Þ and , *σNeu* ð Þ respectively. Also Ⱥ¼ f g h i q, 0ð Þ *:*4, 0*:*5, 0*:*7 and Ƀ¼ f g h i r, 0ð Þ *:*2, 0*:*7, 0*:*8 are *Neu*ð Þ and *Neu*ð Þ . Define a map *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ by *<sup>f</sup>*ð Þ¼ <sup>q</sup> <sup>r</sup>. Let <sup>Ƀ</sup>*<sup>c</sup>* <sup>¼</sup> f g h i <sup>r</sup>, 0ð Þ *:*8, 0*:*3, 0*:*<sup>2</sup> be a *Neu* � *CS* in , *σNeu* ð Þ. Then *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ¼ f g h i <sup>q</sup>, 0ð Þ *:*8, 0*:*3, 0*:*<sup>2</sup> . *Neuα*<sup>∗</sup> � *OS* <sup>¼</sup> *Neu<sup>α</sup>* � *OS* <sup>¼</sup> <sup>0</sup>*Neu* , 1*Neu* f g , <sup>Ⱥ</sup> and *Neu<sup>α</sup>* � *CS* <sup>¼</sup> <sup>0</sup>*Neu* , 1*Neu* , <sup>Ⱥ</sup> *<sup>c</sup>* f g. *Neuα*–*cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>1</sup>*Neu :*Now, *Neu<sup>α</sup>* � *int Neu<sup>α</sup>* � *cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>1</sup>*Neu* <sup>⊆</sup> *Neu* � *int* 1*Neu* ð Þ¼ 1*Neu* , whenever *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>⊆</sup> <sup>1</sup>*Neu* ) *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ.

*More Functions Associated with Neutrosophic gsα\*- Closed Sets in Neutrosophic… DOI: http://dx.doi.org/10.5772/intechopen.99464*

Also, *Neuα*–*int f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>0</sup>*Neu :*Now, *Neu<sup>α</sup>* � *cl Neuα*–*int f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> 0*Neu* ⊇ *Neu* � *cl* 0*Neu* ð Þ¼ 0*Neu* , whenever *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>⊇</sup> <sup>0</sup>*Neu* ) *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is *Neugsα*<sup>∗</sup> � *OS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is totally *Neugsα*<sup>∗</sup> � continuous.

**Theorem 5.4:** Every perfectly *Neugsα*<sup>∗</sup> � continuous is totally *Neugsα*<sup>∗</sup> � continuous, but not conversely.

#### **Proof:**

Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be any neutrosophic function. Let Ⱥ be any *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is perfectly *Neugsα*<sup>∗</sup> � continuous, then *f* �1 ð Þ Ⱥ is both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ) *f* �1 ð Þ Ⱥ is both *Neugsα*<sup>∗</sup> � *OS* and *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is totally *Neugsα*<sup>∗</sup> � continuous.

**Example 5.5:** Let ¼ f g q and =f gr . *τNeu* ¼ 0*Neu* , 1*Neu* f g , Ⱥ and *σNeu* ¼ 0*Neu* , 1*Neu* f g , Ƀ are *Neu*TS on , *τNeu* ð Þ and , *σNeu* ð Þ respectively. Also Ⱥ ¼ f g h i q, 0ð Þ *:*2, 0*:*4, 0*:*6 and Ƀ ¼ f g h i r, 0ð Þ *:*6, 0*:*8, 0*:*4 are *Neu*ð Þ and *Neu*ð Þ . Define a map *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ by *<sup>f</sup>*ð Þ¼ <sup>q</sup> <sup>r</sup>. Let <sup>Ƀ</sup>*<sup>c</sup>* <sup>¼</sup> f g h i <sup>r</sup>, 0ð Þ *:*4, 0*:*2, 0*:*<sup>6</sup> be a *Neu* � *CS* in , *σNeu* ð Þ. Then *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ¼ f g h i <sup>q</sup>, 0ð Þ *:*4, 0*:*2, 0*:*<sup>6</sup> . *Neuα*<sup>∗</sup> � *OS* <sup>¼</sup> *Neu<sup>α</sup>* � *OS* <sup>¼</sup> <sup>0</sup>*Neu* , 1*Neu* f g , <sup>Ⱥ</sup> and *Neu<sup>α</sup>* � *CS* <sup>¼</sup> <sup>0</sup>*Neu* , 1*Neu* , <sup>Ⱥ</sup> *<sup>c</sup>* f g. *Neuα*–*cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup> :*Now, *Neu<sup>α</sup>* � *int Neu<sup>α</sup>* � *cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> Ⱥ ∪ 0*Neu* ¼ Ⱥ ⊆ *Neu* � *int* 1*Neu* ð Þ¼ 1*Neu* , whenever *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>⊆</sup> <sup>1</sup>*Neu* ) *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Also, *Neuα*–*int f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>0</sup>*Neu :* Now, *Neu<sup>α</sup>* � *cl Neu<sup>α</sup>* � *int f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>0</sup>*Neu* <sup>⊇</sup> *Neu* � *cl* <sup>0</sup>*Neu* ð Þ¼ <sup>0</sup>*Neu* , whenever *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>⊇</sup> 0*Neu* ) *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is *Neugsα*<sup>∗</sup> � *OS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is totally *Neugsα*<sup>∗</sup> � continuous. But *<sup>f</sup>* is not perfectly *Neugsα*<sup>∗</sup> � continuous. Let <sup>Ⱦ</sup><sup>¼</sup> f g h i <sup>r</sup>, 0ð Þ *:*3, 0*:*1, 0*:*<sup>8</sup> be *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Then *<sup>f</sup>* �1 ð Þ¼ Ⱦ f g h i q, 0ð Þ *:*3, 0*:*1, 0*:*8 . Now, *Neu*–*int f* �<sup>1</sup> ð Þ <sup>Ⱦ</sup> <sup>¼</sup> <sup>0</sup>*Neu* 6¼ *<sup>f</sup>* �1 ð Þ) Ⱦ *f* �1 ð Þ Ⱦ is not *Neu* � *OS* in , *τNeu* ð Þ. Also, *Neu*–*cl f* �<sup>1</sup> ð Þ <sup>Ⱦ</sup> <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* 6¼ *<sup>f</sup>* �1 ð Þ) Ⱦ *f* �1 ð Þ Ⱦ is not *Neu* � *CS* in , *τNeu* ð Þ. Therefore, *f* �1 ð Þ Ⱦ is not both *Neu* � *OS* and *Neu* � *CS* in , *τNeu* ð Þ.

**Theorem 5.6:** Every totally *Neugsα*<sup>∗</sup> � continuous is *Neugsα*<sup>∗</sup> � continuous. **Proof:**

Let *f* : , *τNeu* ð Þ! , *σNeu* ð Þ be any neutrosophic function. Let Ⱥ be any *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is totally *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 ð Þ <sup>Ⱥ</sup> is both *Neugsα*<sup>∗</sup> � *OS* and *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ) *<sup>f</sup>* �1 ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � continuous.

**Example 5.7:** Let ¼ f g q and =f gr . *τNeu* ¼ 0*Neu* , 1*Neu* f g , Ⱥ and *σNeu* ¼ 0*Neu* , 1*Neu* f g , Ƀ are *Neu*TS on , *τNeu* ð Þ and , *σNeu* ð Þ respectively. Also Ⱥ¼ f g h i q, 0ð Þ *:*7, 0*:*6, 0*:*5 and Ƀ ¼ f g h i r, 0ð Þ *:*7, 0*:*8, 0*:*3 are *Neu*ð Þ and *Neu*ð Þ . Define a map *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ by *<sup>f</sup>*ð Þ¼ <sup>q</sup> <sup>r</sup>. Let <sup>Ƀ</sup>*<sup>c</sup>* <sup>¼</sup> f g h i <sup>r</sup>, 0ð Þ *:*3, 0*:*2, 0*:*<sup>7</sup> be a *Neu* � *CS* in , *σNeu* ð Þ. Then *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ¼ f g h i <sup>q</sup>, 0ð Þ *:*3, 0*:*2, 0*:*<sup>7</sup> . *Neuα*<sup>∗</sup> � *OS* <sup>¼</sup> *Neu<sup>α</sup>* � *OS* <sup>¼</sup> <sup>0</sup>*Neu* , 1*Neu* f g , <sup>Ⱥ</sup> , *<sup>D</sup>* and *Neu<sup>α</sup>* � *CS* <sup>¼</sup> <sup>0</sup>*Neu* , 1*Neu* , <sup>Ⱥ</sup> *<sup>c</sup>* f g , *<sup>E</sup>* , where *D* ¼ f g h i q, 0ð Þ ½ � *:*7, 1 , 0½ � *:*6, 1 , 0, 0 ½ � *:*5 , *E* ¼ f g h i q, 0, 0 ð Þ ½ � *:*5 , 0, 0 ½ � *:*4 , 0½ � *:*7, 1 . *Neuα*–*cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>Ⱥ</sup> *<sup>c</sup>* <sup>∩</sup> *<sup>F</sup>* <sup>∩</sup> <sup>1</sup>*Neu* <sup>¼</sup> *<sup>F</sup>*, where <sup>¼</sup> f g h i <sup>q</sup>, 0ð Þ ½ � *:*3, 0*:*<sup>5</sup> , 0½ � *:*2, 0*:*<sup>4</sup> , 0*:*<sup>7</sup> . Now, *Neu<sup>α</sup>* � *int Neu<sup>α</sup>* � *cl f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>0</sup>*Neu* <sup>⊆</sup> *Neu* � *int*ð Þ <sup>Ⱥ</sup> , *Neu* � *int D*ð Þ, *Neu* � *int* 1*Neu* ð Þ¼ Ⱥ , 1*Neu* , whenever *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>⊆</sup> <sup>Ⱥ</sup> , 1*Neu* ) *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � continuous. But *<sup>f</sup>* is not totally *Neugsα*<sup>∗</sup> � continuous, because *<sup>f</sup>* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is not *Neugsα*<sup>∗</sup> � *OS* in , *<sup>τ</sup>Neu* ð Þ. Since *Neu<sup>α</sup>* � *cl Neu<sup>α</sup>* � *int f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ <sup>¼</sup> <sup>0</sup>*Neu*⊉*Neu* � *cl J*ðÞ¼ <sup>Ⱥ</sup> *<sup>c</sup>* , whenever *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ ⊇ *J*, where *J* ¼ f g h i q, 0, 0 ð Þ ½ � *:*3 , 0, 0 ½ � *:*2 , 0½ � *:*7, 1 ) *f* �<sup>1</sup> <sup>Ƀ</sup>*<sup>c</sup>* ð Þ is not *Neugsα*<sup>∗</sup> � *OS* in , *τNeu* ð Þ.

**Inter-relationship 5.8:**

**Theorem 5.9:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ be totally *Neugsα*<sup>∗</sup> � continuous and , *<sup>σ</sup>Neu* ð Þ be *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � irresolute. **Proof:**

Let <sup>Ⱥ</sup> be any *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since , *<sup>σ</sup>Neu* ð Þ is *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is totally *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 ð Þ Ⱥ is both *Neugsα*<sup>∗</sup> � *OS* and *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ) *<sup>f</sup>* �1 ð Þ <sup>Ⱥ</sup> is *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *<sup>f</sup>* is *Neugsα*<sup>∗</sup> � irresolute.

**Theorem 5.10:** Let *<sup>f</sup>* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>σ</sup>Neu* ð Þ and *<sup>g</sup>* : , *<sup>σ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* be totally *Neugsα*<sup>∗</sup> � continuous and , *<sup>σ</sup>Neu* ð Þ be *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then *gof* : , *<sup>τ</sup>Neu* ð Þ! , *<sup>γ</sup>Neu* is totally *Neugsα*<sup>∗</sup> � continuous.

#### **Proof:**

Let <sup>Ⱥ</sup> be any *Neu* � *CS* in , *<sup>γ</sup>Neu* . Since *<sup>g</sup>* is totally *Neugsα*<sup>∗</sup> � continuous, then *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is both *Neugsα*<sup>∗</sup> � *OS* and *Neugsα*<sup>∗</sup> � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since , *<sup>σ</sup>Neu* ð Þ is *Neugsα*<sup>∗</sup> � *<sup>T</sup>*<sup>1</sup>*=*<sup>2</sup> space, then *<sup>g</sup>*�<sup>1</sup>ð Þ <sup>Ⱥ</sup> is *Neu* � *CS* in , *<sup>σ</sup>Neu* ð Þ. Since *<sup>f</sup>* is totally *Neugsα*<sup>∗</sup> � continuous, then *<sup>f</sup>* �1 *<sup>g</sup>*�<sup>1</sup> ð Þ¼ ð Þ <sup>Ⱥ</sup> ð Þ *gof* �<sup>1</sup> ð Þ <sup>Ⱥ</sup> is both *Neugsα*<sup>∗</sup> � *OS* and *Neugsα*<sup>∗</sup> � *CS* in , *<sup>τ</sup>Neu* ð Þ. Therefore, *gof* is totally *Neugsα*<sup>∗</sup> � continuous.

#### **Author details**

P. Anbarasi Rodrigo<sup>1</sup> and S. Maheswari<sup>2</sup> \*

1 Assistant Professor, Department of Mathematics, St. Mary's College (Autonomous), Thoothukudi, Affiliated by Manonmaniam Sundaranar University, Tirunelveli, India

2 Research Scholar, Register number : 20212212092003, St. Mary's College (Autonomous), Thoothukudi, Affiliated by Manonmaniam Sundaranar University, Tirunelveli, India

\*Address all correspondence to: mahma1295@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*More Functions Associated with Neutrosophic gsα\*- Closed Sets in Neutrosophic… DOI: http://dx.doi.org/10.5772/intechopen.99464*

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[6] Anbarasi Rodrigo, P., Maheswari, S., Neutrosophic Generalized Semi Alpha Star Closed Sets in Neutrosophic Topological Spaces, Paper presented in International Conference on Mathematics, Statistics, Computers And Information Sciences, 2021.

[7] Anbarasi Rodrigo, P., Maheswari, S., Functions Related to Neutrosophic *gsα*<sup>∗</sup> Closed Sets in Neutrosophic Topological Spaces, Paper presented in 24th FAI International Conference on Global Trends of Data Analytics in Business Management, Social Sciences, Medical Sciences and Decision Making 24th FAI-ICDBSMD 2021.

[8] Blessie Rebecca, S., Francina Shalini, A., Neutrosophic Generalized Regular Contra Continuity in Neutrosophic Topological Spaces, Research in Advent Technology, Vol.7, No.2, E-ISSN: 2321-9637, pp.761-765, Feb 2019.

[9] Serkan Karatas, Cemil Kuru, Neutrosophic Topology, Neutrosophic Sets & Systems, University of New Mexico, Vol.13, pp.90-95, 2016.

**Chapter 3**

## 4-Dimensional Canards with Brownian Motion

*Shuya Kanagawa and Kiyoyuki Tchizawa*

#### **Abstract**

Generally speaking, it is impossible to analyze slow-fast system with Brownian motion. If it becomes possible to do using a new approach, we can evaluate the rigidity of the original system. What kind of the behavior of such a system we have? Using non-standard analysis, on a"hyper finite time line" by Anderson, the Brownian motions are described by step functions. Then, the original differential equations are described by the difference equations due to using non-standard analysis. When constructing the difference equations, the corresponding measure is extended topologically. Because the interval of the difference is according to the hyper finite time line, the topological space is well defined. In this paper, we propose a two-region economic model with Brownian motions. This concrete example gives us new results.

**Keywords:** canard solution, slow-fast system, nonstandard analysis, Brownian motion, stochastic differential equation

#### **1. Introduction**

Consider a slow-fast system in **R**<sup>4</sup> with a 2-dimensional slow manifold:

$$\begin{cases} \varepsilon \frac{d\mathbf{x}}{dt} = h(\mathbf{x}, \mathbf{y}, \varepsilon) \\\\ \frac{d\mathbf{y}}{dt} = \mathbf{g}(\mathbf{x}, \mathbf{y}, \varepsilon) \end{cases}, \tag{1}$$

where *ε* is infinitesimal and

$$\begin{aligned} \mathbf{g} &= \begin{pmatrix} \mathbf{g}\_1 \\ \mathbf{g}\_2 \end{pmatrix} : \mathbf{R}^d \to \mathbf{R}^2, \ h = \begin{pmatrix} h\_1 \\ h\_2 \end{pmatrix} : \mathbf{R}^d \to \mathbf{R}^2, \\\ \mathbf{x} = \mathbf{x}(t) &= \begin{pmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \end{pmatrix} \in \mathbf{R}^2, \ \mathbf{y} = \mathbf{y}(t) = \begin{pmatrix} \mathbf{y}\_1 \\ \mathbf{y}\_2 \end{pmatrix} \in \mathbf{R}^2. \end{aligned}$$

The above expression form is based on Nelson's [1]. The slow-fast system (1) is applied to many fields, e.g. electronic circuits, neuron systems, etc. In these applications, effectiveness of random noises always exists. We take up this point of view as being one of the main problems.

#### *Advanced Topics of Topology*

Now, let us consider a stochastic differential equation for a slow-fast system with a Brownian motion *B t*ð Þ as the random noises modifying the slow fast system (1): For *t*∈½ � 0, *T* , *T* >0

$$\begin{cases} \varepsilon d\mathbf{x} = h(\mathbf{x}, \mathbf{y}, \varepsilon) dt \\ d\mathbf{y} = \mathbf{g}(\mathbf{x}, \mathbf{y}, \varepsilon) dt + \sigma dB, \end{cases} \tag{2}$$

where *<sup>B</sup>* <sup>¼</sup> *<sup>B</sup>*<sup>1</sup> *B*2 � � <sup>∈</sup> *<sup>R</sup>*<sup>2</sup> is a 2-dimensional standard Brownian motion and *<sup>σ</sup>* <sup>&</sup>gt;0 is a

positive constant which gives a standard deviation for the Brownian motion *B t*ð Þ. Since the Brownian motion *B t*ð Þ is almost surely non-differentiable everywhere, it is difficult to analyze slow-fast system (2).

On the other hand, Anderson [2] showed that the Brownian motion is described by step functions using non-standard analysis on a hyper finite time line by the following definition. (See also [3, 4]).

**Definition 1.** *Let Nt* <sup>¼</sup> *<sup>t</sup>* <sup>Δ</sup>*<sup>t</sup>* , 0≤*t*≤ *T and N* ¼ *NT. Assume that a sequence of i.i.d. random variables* f g Δ*Bk*, *k* ¼ 1, ⋯, *N has the distribution*

$$P\left\{\Delta B\_k = \sqrt{\Delta t}\right\} = P\left\{\Delta B\_k = -\sqrt{\Delta t}\right\} = \frac{1}{2} \tag{3}$$

for each *k* ¼ 1, ⋯, *N.* An extended Wiener process f g *B t*ð Þ, *t* ≥0 is defined by

$$B(t) = \sum\_{k=1}^{Nt} \Delta B\_k, \quad 0 \le t \le T. \tag{4}$$

Rewriting the system (2) via step functions on the hyper finite time line, the following system (5) is obtained.

$$\begin{cases} \varepsilon(\mathbf{x}\_n - \mathbf{x}\_{n-1}) = h(\mathbf{x}\_{n-1}, \mathbf{y}\_{n-1}, \varepsilon)\Delta t\\ \mathbf{y}\_n - \mathbf{y}\_{n-1} = \mathbf{g}(\mathbf{x}\_{n-1}, \mathbf{y}\_{n-1}, \varepsilon)\Delta t + \sigma \Delta B\_n, \end{cases} \tag{5}$$

for *<sup>n</sup>* <sup>¼</sup> 1, 2, <sup>⋯</sup> , *<sup>N</sup>*, where <sup>Δ</sup>*Bn* <sup>¼</sup> *B n*ð Þ� <sup>Δ</sup>*<sup>t</sup> B n* ð Þ ð Þ � <sup>1</sup> <sup>Δ</sup>*<sup>t</sup>* , <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> *<sup>T</sup> <sup>N</sup>* and *N* is a hyper finite natural number.

Since the system (5) is equivalent to the system (2), taking B(t) in Definition 1, we prove the existence of the solution for the system (2) in Section 3.

#### **2. Slow-fast system in R<sup>4</sup> with co-dimension 2**

We assume that the system (1) sastisfies the following conditions (A1) � (A5): (A1) *h* is of class **C**<sup>1</sup> and *g* is of class **C**<sup>2</sup> .

(A2) The slow manifold *<sup>S</sup>* <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>4</sup>j*h x*ð Þ¼ , *<sup>y</sup>*, 0 <sup>0</sup> � � is a two-dimensional differential manifold and intersects the set *<sup>V</sup>* <sup>¼</sup> ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>4</sup> det *<sup>∂</sup><sup>h</sup> <sup>∂</sup><sup>x</sup>* ð Þ *<sup>x</sup>*, *<sup>y</sup>*, 0 � � <sup>¼</sup> <sup>0</sup> � � � � transversely.

Then, the pli set *PL* ¼ f g ð Þ *x*, *y* ∈ *S* ∩*V* is a one-dimensional differentiable manifold. (A3) Either the value of *g*<sup>1</sup> or that of *g*<sup>2</sup> is nonzero at any point of *PL*.

Note that the pli set *PL* devides the slow manifolds *S*nPL into three parts depending on the signs of the two eigenvalues of *<sup>∂</sup><sup>h</sup> <sup>∂</sup><sup>x</sup>* ð Þ *x*, *y*, 0 .

First consider the following reduced system which is obtained from (1) with *ε* ¼ 0:

*4-Dimensional Canards with Brownian Motion DOI: http://dx.doi.org/10.5772/intechopen.102151*

$$\begin{cases} \mathbf{0} = h(\mathbf{x}, \mathbf{y}, \mathbf{0})\\ \frac{d\mathbf{y}}{dt} = \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{0}). \end{cases} \tag{6}$$

By differentiating *h x*ð Þ , *y*, 0 with respect to *t*, we have

$$\frac{\partial h}{\partial \mathbf{x}}(\mathbf{x}, \mathbf{y}, \mathbf{0}) \frac{d\mathbf{x}}{dt} + \frac{\partial h}{\partial \mathbf{y}}(\mathbf{x}, \mathbf{y}, \mathbf{0}) \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{0}) = \mathbf{0}.\tag{7}$$

Then (6) becomes the following:

$$\begin{cases} \frac{dx}{dt} = -\left[\frac{\partial h}{\partial \mathbf{x}}(\mathbf{x}, \mathbf{y}, \mathbf{0})\right]^{-1} \frac{\partial h}{\partial \mathbf{y}}(\mathbf{x}, \mathbf{y}, \mathbf{0}) \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{0})\\\\ \frac{dy}{dt} = \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{0}) \end{cases},\tag{8}$$

where ð Þ *x*, *y* ∈*S*n*PL*. To avoid degeneracy in (8), we consider the following system:

$$\begin{cases} \frac{d\mathbf{x}}{dt} = \left\{-\text{det}\left[\frac{\partial h}{\partial \mathbf{x}}(\mathbf{x}, \mathbf{y}, \mathbf{0})\right]^{-1}\right\} \left[\frac{\partial h}{\partial \mathbf{x}}(\mathbf{x}, \mathbf{y}, \mathbf{0})\right]^{-1} \frac{\partial h}{\partial \mathbf{y}}(\mathbf{x}, \mathbf{y}, \mathbf{0}) \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{0}) \\\\ \frac{d\mathbf{y}}{dt} = \left\{\det\left[\frac{\partial h}{\partial \mathbf{x}}(\mathbf{x}, \mathbf{y}, \mathbf{0})\right]^{-1}\right\} \mathbf{g}(\mathbf{x}, \mathbf{y}, \mathbf{0}) \end{cases} . \tag{9}$$

The phase portrait of the system (9) is the same as that of (8) except the region where det *<sup>∂</sup><sup>h</sup> <sup>∂</sup><sup>x</sup>* ð Þ *<sup>x</sup>*, *<sup>y</sup>*, 0 � � <sup>¼</sup> 0, but only the orientation of the orbit is different.

**Definition 2.** A singular point of (9), which is on *PL*, is called a pseudo singular point of (1).

(A4) r*ank <sup>∂</sup><sup>h</sup> <sup>∂</sup><sup>x</sup>* ð Þ *<sup>x</sup>*, *<sup>y</sup>*, 0 � � <sup>¼</sup> 2 for any ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> *<sup>S</sup>*nPL.

From (A4), the implicit function theorem guarantees the existence of a unique function *y* ¼ *ξ*ð Þ *x* such that *h x*ð Þ¼ , *ξ*ð Þ *x* , 0 0. By using *y* ¼ *ξ*ð Þ *x* , we obtain the following system:

$$\frac{d\mathbf{x}}{dt} = \left\{-\det\left[\frac{\partial h}{\partial \mathbf{x}}(\mathbf{x}, \boldsymbol{\xi}(\mathbf{x}), \mathbf{0})\right]^{-1}\right\} \left[\frac{\partial h}{\partial \mathbf{x}}(\mathbf{x}, \boldsymbol{\xi}(\mathbf{x}), \mathbf{0})\right]^{-1} \frac{\partial h}{\partial \mathbf{y}}(\mathbf{x}, \boldsymbol{\xi}(\mathbf{x}), \mathbf{0}) \mathbf{g}(\mathbf{x}, \boldsymbol{\xi}(\mathbf{x}), \mathbf{0}).\tag{10}$$

(A5) All singular points of (10) are non-degenerate, that is, the linearization of (10) at a singular point has two nonzero eigenvalues.

**Definition 3**. Let *λ*1, *λ*<sup>2</sup> be two eigenvalues of the linearization of (10) at a pseudo singular point. The pseudo singular point with real eigenvalues is called a pseudo singular saddle point if *λ*<sup>1</sup> < 0<*λ*<sup>2</sup> and a pseudo singular node point if *λ*<sup>1</sup> <*λ*<sup>2</sup> <0 or *λ*<sup>1</sup> > *λ*<sup>2</sup> > 0*.*

The following theorem is established (see, e.g. [5]).

**Theorem 1.** Let *x*0, *y*<sup>0</sup> � � be a pseudo singular point. If trace *<sup>∂</sup><sup>h</sup> <sup>∂</sup><sup>x</sup> <sup>x</sup>*0, *<sup>y</sup>*0, 0 � � � � <sup>&</sup>lt; 0, then there exists a solution which first follows the attractive part and the repulsive part after crossing PL near the pseudo singular point.

**Remark 1.** The condition trace *<sup>∂</sup><sup>h</sup> <sup>∂</sup><sup>x</sup> <sup>x</sup>*0, *<sup>y</sup>*0, 0 � � � � <sup>&</sup>lt;0 implies that one of eigenvalues of *<sup>∂</sup><sup>h</sup> <sup>∂</sup><sup>x</sup> <sup>x</sup>*0, *<sup>y</sup>*0, 0 � � � � is equal to zero and the other one is negative. Notice that the system has two kinds of vector fields: one is 2-dimensional slow and the other is 2-dimensional fast one. The condition provides the state of the fast vector field.

**Remark 2.** The singular solution in Theorem 1 is called a canard in **R**<sup>4</sup> with 2-dimensional slow manifold. As a result, it causes a delayed jumping. The study of canards requires still more precise topological analysis on the slow vector field.

In the next section, we show that a canard exists for the system (2) in which the orbit of the canard of the system (1) is moved to another one by a Brownian motion *B t*ð Þ.

#### **3. Canards with Brownian motion**

Let us prove the following theorem. **Theorem 2.** *In the system (3), if there exists k*f g*<sup>n</sup> such that*

$$|\varkappa\_n - \varkappa\_{n-1}| \le \varepsilon k\_n, \quad n = 1, 2, \dots, N \tag{11}$$

and

$$\sup\_{1 \le n \le N} k\_n \le K \tag{12}$$

for some *K* hyper finite, then there exists a solution of (5) which is called canard in the sense of Remark 2.

*Proof.* From the condition (11), we have

$$|\boldsymbol{\kappa}\_{n} - \boldsymbol{\kappa}\_{n-1}| = \left| \frac{1}{\varepsilon} h(\boldsymbol{\kappa}\_{n-1}, \boldsymbol{y}\_{n-1}, \varepsilon) \Delta t \right| \le ek\_{n}. \tag{13}$$

*<sup>ε</sup>* is an arbitrary constant, therefore putting *<sup>ε</sup>* <sup>¼</sup> <sup>1</sup> *<sup>N</sup>* we have from (13)

$$\left| h(\mathbf{x}\_{n-1}, \mathbf{y}\_{n-1}, \varepsilon) \right| \le \frac{\varepsilon^2 k\_n}{\Delta t} \le \frac{k\_n}{N}, \tag{14}$$

for each *n*≥1.

From Definition 2, the following is satisfied for the pseudo-singular point *x*0, *y*<sup>0</sup> � � of (1);

$$\begin{cases} \left\{-\text{det}\left[\frac{\partial h}{\partial x}(\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{0})\right]^{-1}\right\} \Big|\_{\frac{\partial h}{\partial x}}^{\frac{\partial h}{\partial x}}(\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{0})\Big|^{-1} \frac{\partial h}{\partial \mathbf{y}}(\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{0})\mathbf{g}\left(\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{0}\right) = \mathbf{0} \\\\ h(\mathbf{x}\_{0},\mathbf{y}\_{0},\mathbf{0}) = \mathbf{0} \end{cases} . \tag{15}$$

Assume that *<sup>σ</sup>*<sup>2</sup> in the Brownian motion *B t*ð Þ is sufficiently small. Let *<sup>x</sup>ξ*, *<sup>y</sup><sup>ξ</sup>* � � be a pseudo-singular point of (2) or (5). Note that (2) is equivalent to (5) in the sense of nonstandard analysis. Then there exists a positive number *ξ* such that

$$\left\{-\text{det}\left[\frac{\partial\hbar}{\partial\mathbf{x}}\left(\mathbf{x}\_{\xi},\mathbf{y}\_{\xi},\mathbf{0}\right)\right]^{-1}\right\} \left[\frac{\partial\hbar}{\partial\mathbf{x}}\left(\mathbf{x}\_{\xi},\mathbf{y}\_{\xi},\mathbf{0}\right)\right]^{-1} \frac{\partial\hbar}{\partial\mathbf{y}}\left(\mathbf{x}\_{\xi},\mathbf{y}\_{\xi},\mathbf{0}\right) \left(\mathbf{g}\left(\mathbf{x}\_{\xi},\mathbf{y}\_{\xi},\mathbf{0}\right)\Delta t + \sigma\Delta B\_{\xi}\right) \approx \mathbf{0},\tag{16}$$

where <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> *<sup>T</sup> N* . In this situation, as *σ* ≈0

$$\mathbf{x}\left(\mathbf{x}\_{\xi},\mathbf{y}\_{\xi}\right) \approx \left(\mathbf{x}\_{0},\mathbf{y}\_{0}\right). \tag{17}$$

*4-Dimensional Canards with Brownian Motion DOI: http://dx.doi.org/10.5772/intechopen.102151*

Therefore, the eigenvalues of the linearized system (2) at the point *xξ*, *y<sup>ξ</sup>* � � keeps the almost same as the eigenvalues of the system (1) at the point *x*0, *y*<sup>0</sup> � �.

On the other hand, there exists a canard of (1) from Theorem 1. Since *kn <sup>N</sup>* is small enough, the solution of (5) also first follows the attractive part and the repulsive part follows after crossing *PL* near the pseudo singular point like as the canard of (1).

#### **4. Concrete models**

#### **4.1 Two-region business cycle model**

As a concrete model, we consider a two-region business cycle model between two nations A and B including a Brownian motion *B t*ðÞ¼ ð Þ *B*1ð Þ*t* , *B*2ð Þ*t* as followings. See [6] for more details of the two-region business cycle model.

$$\begin{cases} \varepsilon \frac{d\mathbf{x}\_1}{dt} = -\frac{\mathbf{1} - a + m\_1}{\theta} \mathbf{y}\_1 + \frac{m\_2}{\theta} \mathbf{y}\_2 - \left(\frac{\varepsilon}{\theta} + \mathbf{1} - a\right) (\mathbf{x}\_1 - a) + \frac{\mathbf{1} - n\_1}{\theta} \boldsymbol{\varrho}\_1 (\mathbf{x}\_1 - a) + \frac{n\_2}{\theta} \boldsymbol{\varrho}\_2 (\mathbf{x}\_2 - a) \\\ \varepsilon \frac{d\mathbf{x}\_2}{dt} = \frac{m\_1}{\theta} \mathbf{y}\_1 - \frac{\mathbf{1} - a + m\_2}{\theta} \mathbf{y}\_2 + \frac{\mathbf{1} - n\_1}{\theta} \boldsymbol{\varrho}\_1 (\mathbf{x}\_1 - a) - \left(\frac{\varepsilon}{\theta} + \mathbf{1} - a\right) (\mathbf{x}\_2 - a) + \frac{\mathbf{1} - n\_2}{\theta} \boldsymbol{\varrho}\_2 (\mathbf{x}\_2 - a) \\\ d\mathbf{y}\_1 = (\mathbf{x}\_1 - a) dt + \sigma dB\_1(t) \\\ d\mathbf{y}\_2 = (\mathbf{x}\_2 - a) dt + \sigma dB\_2(t) \end{cases} \tag{18}$$

for 0≤ *t*≤*T*, where *x*1ð Þ*t* and *x*2ð Þ*t* are exports of A and B, *m*1ð Þ*t* and *m*2ð Þ*t* are imports of A and B, *<sup>y</sup>*1ð Þ�*<sup>t</sup> <sup>q</sup>* <sup>1</sup>�*<sup>α</sup>* and *<sup>y</sup>*2ðÞ�*<sup>t</sup> <sup>q</sup>* <sup>1</sup>�*<sup>α</sup>* are national income identities of A and B for some constants *q* and *α*, respectively. (See [6] for more details.)

Now, let us introduce a difference equations for the system (18). Then, the relations <sup>Δ</sup>*<sup>t</sup>* <sup>¼</sup> *<sup>T</sup> <sup>N</sup>* and *tk* <sup>¼</sup> *<sup>k</sup> <sup>T</sup> <sup>N</sup>* , *k* ¼ 0, 1, ⋯ , *N* are satisfied, where *N* is a hyper finite. Put

*ε* Δ*t* f g *<sup>x</sup>*1ð Þ� *tk <sup>x</sup>*1ð Þ *tk*�<sup>1</sup> ¼ � <sup>1</sup> � *<sup>α</sup>* <sup>þ</sup> *<sup>m</sup>*<sup>1</sup> *<sup>θ</sup> <sup>y</sup>*1ð Þþ *tk*�<sup>1</sup> *m*<sup>2</sup> *<sup>θ</sup> <sup>y</sup>*2ð Þ *tk*�<sup>1</sup> � *ε <sup>θ</sup>* <sup>þ</sup> <sup>1</sup> � *<sup>α</sup>* � �ð Þþ *<sup>x</sup>*1ð Þ� *tk*�<sup>1</sup> *<sup>a</sup>* 1 � *n*<sup>1</sup> *<sup>θ</sup> <sup>φ</sup>*1ð Þþ *<sup>x</sup>*1ð Þ� *tk*�<sup>1</sup> *<sup>a</sup> n*2 *<sup>θ</sup> <sup>φ</sup>*2ð Þ *<sup>x</sup>*2ð Þ� *tk*�<sup>1</sup> *<sup>a</sup> ε* Δ*t* f g *<sup>x</sup>*2ð Þ� *tk <sup>x</sup>*2ð Þ *tk*�<sup>1</sup> <sup>¼</sup> *<sup>m</sup>*<sup>1</sup> *<sup>θ</sup> <sup>y</sup>*1ð Þ� *tk*�<sup>1</sup> 1 � *α* þ *m*<sup>2</sup> *<sup>θ</sup> <sup>y</sup>*2ð Þ *tk*�<sup>1</sup> þ 1 � *n*<sup>1</sup> *<sup>θ</sup> <sup>φ</sup>*1ð Þ� *<sup>x</sup>*1ð Þ� *tk*�<sup>1</sup> *<sup>a</sup> <sup>ε</sup> <sup>θ</sup>* <sup>þ</sup> <sup>1</sup> � *<sup>α</sup>* � �ð Þþ *<sup>x</sup>*2ð Þ� *tk*�<sup>1</sup> *<sup>a</sup>* 1 � *n*<sup>2</sup> *<sup>θ</sup> <sup>φ</sup>*2ð Þ *<sup>x</sup>*2ð Þ� *tk*�<sup>1</sup> *<sup>a</sup> y*1ð Þ� *tk y*1ð Þ¼ *tk*�<sup>1</sup> ð Þ *x*1ð Þ� *tk*�<sup>1</sup> *a* Δ*t* þ *σ*f g *B*1ð Þ� *tk B*1ð Þ *tk*�<sup>1</sup> *y*2ð Þ� *tk y*2ð Þ¼ *tk*�<sup>1</sup> ð Þ *x*2ð Þ� *tk*�<sup>1</sup> *a* Δ*t* þ *σ*f g *B*2ð Þ� *tk B*2ð Þ *tk*�<sup>1</sup> 8 >>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>: (19)

*:*

Furthermore put

$$
\varphi\_1(\mathbf{x} - a) = \varphi\_2(\mathbf{x} - a) = (1 - a) \left( \theta \mathbf{x} + \mathbf{x}^2 - \frac{\mathbf{x}^3}{3} \right). \tag{20}
$$

#### **4.2 Simulation results**

In this section, let us provide computer simulations for the two-region business cycle model using the above Eqs. (19) and (20. In (19), we assume that two Brownian motions *B*1ð Þ*t* and *B*2ð Þ*t* are mutually independent and note that

$$\mathbf{B}\_{1}(\mathbf{t}\_{k}) - \mathbf{B}\_{1}(\mathbf{t}\_{k-1}) \sim N(\mathbf{0}, \Delta t \sigma\_{1}^{2}), \quad \mathbf{B}\_{2}(\mathbf{t}\_{k}) - \mathbf{B}\_{2}(\mathbf{t}\_{k-1}) \sim N(\mathbf{0}, \Delta t \sigma\_{2}^{2}), \tag{21}$$

for each 1≤*k*≤ *<sup>T</sup>* Δ*t* .

**Figure 1**, except for the axes, shows the pli set *PL* ¼ f g ð Þ *x*, *y* ∈ *S* ∩*V* with the pseudo singular point 1, ð Þ �1 of (1) defined by (9).

Putting some parameters in (19), we have the following results for some orbits of f g ð Þ *x*1ð Þ*t* , *x*2ð Þ*t* , 0≤*t*≤*T* satisfying the Eq. (1) or (2).

**Figure 2** shows an orbit of f g ð Þ *x*1ð Þ*t* , *x*2ð Þ*t* , 0≤ *t*≤*T* ¼ 0*:*8 satisfying the Eq. (5) with *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0 and starting from 0ð Þ *:*8, �0*:*8 near the pseudo singular point 1, ð Þ �1 . In **Figures 2**–**4**, *ε* ¼ 0*:*01. From **Figure 2** the speed of the orbit ð Þ *x*1ð Þ*t* , *x*2ð Þ*t* for 0≤ *t*≤0*:*2 is not only very fast, but also the orbit jumps near the pseudo singular point 1, ð Þ �1 . The orbit turns at the point ð Þ �2, 2 and returns on the line. Δt = 0.001 in **Figures 2**–**6**.

**Figure 3** shows an orbit of f g ð Þ *x*1ð Þ*t* , *x*2ð Þ*t* , 0≤*t*≤*T* ¼ 0*:*8 satisfying the Eq. (5) with *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0*:*4 and starting from 0ð Þ *:*8, �0*:*8 near the the pseudo singular point ð Þ 1, �1 . From **Figure 3** we observe that the orbit moves on the line from 0ð Þ *:*8, �0*:*8 and separates from the line at *t* ¼ 0*:*2 by the Brownian motion *B t*ð Þ*.*

**Figure 4** shows an orbit of f g ð Þ *x*1ð Þ*t* , *x*2ð Þ*t* , 0≤*t*≤ *T* ¼ 4*:*64 satisfying the Eq. (5) with *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0 and starting from 0ð Þ *:*8, �0*:*8 near the pseudo singular point 1, ð Þ �1 . The orbit separates from the line at *t* ¼ 4*:*61*.*

**Figure 5** shows an orbit of f g ð Þ *x*1ð Þ*t* , *x*2ð Þ*t* , 0≤*t*≤ *T* ¼ 2*:*75 satisfying the Eq. (5) with *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0 and starting from 0ð Þ *:*8, �0*:*8 near the pseudo singular point ð Þ 1, �1 , where *ε* ¼ 0*:*004. The orbit with *ε* ¼ 0*:*004 separates from the line at

**Figure 1.** *Pli set PL.*

*4-Dimensional Canards with Brownian Motion DOI: http://dx.doi.org/10.5772/intechopen.102151*

**Figure 2.**

Δt ¼ 0*:*001, 0 ≤*t* ≤0*:*8, *a* ¼ 0*:*6, *ε* ¼ 0*:*01, *θ* ¼ 0*:*5, *m*1*m*<sup>1</sup> ¼ 0*:*1, *m*<sup>2</sup> ¼ 0*:*2, *n*<sup>1</sup> ¼ 0*:*2, *n*<sup>2</sup> ¼ 0*:*2, *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0*.*

**Figure 3.** Δt ¼ 0*:*001, 0 ≤*t* ≤0*:*8, *a* ¼ 0*:*6, *ε* ¼ 0*:*01, *θ* ¼ 0*:*5, *m*<sup>1</sup> ¼ 0*:*1, *m*<sup>2</sup> ¼ 0*:*2, *n*<sup>1</sup> ¼ 0*:*4, *n*<sup>2</sup> ¼ 0*:*4, *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0*:*4*.*

*t* ¼ 2*:*61. On the other hand the orbit with *ε* ¼ 0*:*01 separates from the line at *t* ¼ 4*:*61 in **Figure 4**. Therefore we see that the orbit changes according to *ε*. **Figure 6** shows an orbit of f g ð Þ *x*1ð Þ*t* , *x*2ð Þ*t* , 0≤ *t*≤*T* ¼ 0*:*8 satisfying the Eq. (5) with *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0*:*4 and starting from 0ð Þ *:*8, �0*:*8 near the pseudo singular point 1, ð Þ �1 , where *ε* ¼ 0*:*004. The orbit with *ε* ¼ 0*:*004 separates from

**Figure 4.**

Δt ¼ 0*:*001, 0 ≤*t* ≤4*:*64, *a* ¼ 0*:*6, *ε* ¼ 0*:*01, *θ* ¼ 0*:*5, *m*<sup>1</sup> ¼ 0*:*1, *m*<sup>2</sup> ¼ 0*:*2, *n*<sup>1</sup> ¼ 0*:*2, *n*<sup>2</sup> ¼ 0*:*2, *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0*.*

**Figure 5.** Δt ¼ 0*:*001, 0 ≤*t* ≤2*:*75, *a* ¼ 0*:*6, *ε* ¼ 0*:*004, *θ* ¼ 0*:*5, *m*<sup>1</sup> ¼ 0*:*1, *m*<sup>2</sup> ¼ 0*:*2, *n*<sup>1</sup> ¼ 0*:*2, *n*<sup>2</sup> ¼ 0*:*2, *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0*.*

the line at *t* ¼ 0*:*1. On the other hand, in **Figure 3**, the orbit with *ε* ¼ 0*:*01 separates from the line at *t* ¼ 0*:*2. Then, the orbit changes according to *ε* also in the non-random case.

*4-Dimensional Canards with Brownian Motion DOI: http://dx.doi.org/10.5772/intechopen.102151*

**Figure 6.**

Δt ¼ 0*:*001, 0 ≤*t* ≤0*:*8, *a* ¼ 0*:*6, *ε* ¼ 0*:*004, *θ* ¼ 0*:*5, *m*<sup>1</sup> ¼ 0*:*1, *m*<sup>2</sup> ¼ 0*:*2, *n*<sup>1</sup> ¼ 0*:*2, *n*<sup>2</sup> ¼ 0*:*2, *σ*<sup>1</sup> ¼ *σ*<sup>2</sup> ¼ 0*:*4*.*

#### **5. Conclusion**

Brownian motions are described by non-differentiable functions almost surely. In order to overcome the difficulty in the system (2) we consider the system (5) using nonstandard analysis. The system (5) makes us possible to analyze the canard with Brownian motions. As the difference equations is determined by according to the hyper finite time line, the measure is extended effectively to do this analysis. In **Figures 1**–**6** obtained by the simulations, we observe the effects of Brownian motions which change the orbit of ð Þ *x*1ð Þ*t* , *x*2ð Þ*t* .

#### **Acknowledgements**

The authors would like to express the reviewer's comments which are useful to explain the structure of canards. The first author is supported in part by Grant-in-Aid Scientific Research (C), No.18 K03431, Ministry of Education, Science and Culture, Japan.

#### **The 2020 AMS classification**

ordinary differential Eqs., dynamical systems and ergodic theory, difference and functional equations.

*Advanced Topics of Topology*

### **Author details**

Shuya Kanagawa<sup>1</sup> \* and Kiyoyuki Tchizawa2


\*Address all correspondence to: kanagawa1954@icloud.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*4-Dimensional Canards with Brownian Motion DOI: http://dx.doi.org/10.5772/intechopen.102151*

#### **References**

[1] Nelson E. Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society. 1977;**83**:1165-1198

[2] Anderson RM. A non-standard representation for brownian motion and ito integration. Israel Journal of Mathematics. 1976;**25**(2):1546

[3] Kanagawa S, Tchizawa K. Proof of Ito's Formula for Ito's Process in Nonstandard Analysis. Applied Mathematics. 2019;**10**(2):561-567

[4] Kanagawa S, Tchizawa K. Extended wiener process in nonstandard analysis. Applied Mathematics. 2020;**11**(2): 247254

[5] Tchizawa K. Four-dimensional canards and their center manifold. Extended Abstracts Spring 2018. Trends in Mathematics. 2019;**11**:193-199. Springer Nature Switzerland AG

[6] Miki H, Nishino H, Tchizawa K. On the possible occurrence of duck solutions in domestic and two-region business cycle models. Nonlinear Studies. 2012;**18**:39-55

### Section 3
