# Introduction to Non-Standard Neutrosophic Topology

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction to Non-Standard Analysis

^{−}0, 1

^{+}]. Then T (absolute truth) = 1

^{+}= μ (1

^{+}), while T (relative truth) = 1. This is analogously for absolute falsehood vs. relative falsehood, and absolute indeterminacy vs. relative indeterminacy.

_{+}* we denote the set of positive non-zero hyperreal numbers.

^{−}a) or only

^{–}a} was defined as:

^{+}) or only by a

^{+}} was defined as:

#### 1.1. Non-Standard Analysis’s First Extension

^{−}a

^{+}) or only

^{–}a

^{+}} was defined as:

^{−}a) had to be united with a right monad μ (a

^{+}), as such producing a pierced binad: μ (

^{−}a) $\cup $ μ (a

^{+}) =

_{N}μ (

^{−}a

^{+}). Without this pierced binad we would not have been able to define the non-standard neutrosophic operators.

#### 1.2. Non-Standard Analysis’s Second Extension

#### 1.3. The Best Notations for Monads and Binads

^{−},

^{−0},

^{+},

^{+0},

^{−+},

^{−0+}} = {

^{0},

^{−},

^{−0},

^{+},

^{+0},

^{−+},

^{−0+}};

^{0}, or nothing above), or a left monad (

^{−}), or a left monad closed to the right (

^{−0}), or a right monad (

^{+}), or a right monad closed to the left (

^{0+}), or a pierced binad (

^{−+}), or a unpierced binad (

^{−0+}) respectively.

_{N}associated to each symbol, for example: the classical symbol < (less than), becomes <

_{N}(neutrosophically less than, i.e., some indeterminacy is involved, especially with respect to infinitesimals, monads and binads).

_{N}, $\wedge $ and ${\wedge}_{N}$ etc.

#### 1.4. Non-Standard Neutrosophic Inequalities

^{−}a) <

_{N}a <

_{N}(a

^{+})

^{+}) >

_{N}a >

_{N}(−a)

^{−}a) ≤

_{N}(

^{−}a

^{+}) ≤

_{N}(a

^{+})

^{−}a

^{+}) = (

^{−}a) $\cup $ (a

^{+}) and the number a is in between the subsets (on the real number line)

^{−}a = (a − ε, a) and a

^{+}= (a, a + ε), so:

^{−}a

^{+}) >

_{N}b, (

^{−}a

^{+}) >

_{N}(

^{−}b), (

^{−}a

^{+}) >

_{N}(b

^{+}), (

^{−}a

^{+}) >

_{N}(

^{−}b

^{+}), etc.

^{−}a

^{+}),

^{−0}a

^{+}).

#### 1.5. Neutrosophic Infimum and Neutrosophic Supremum

#### 1.5.1. Neutrosophic Infimum

_{N}) be a set, which is neutrosophically partially ordered, and let M be a subset of S.

_{N}(M), is the neutrosophically greatest element in S, which is neutrosophically less than or equal to all elements of M.

#### 1.5.2. Neutrosophic Supremum

_{N}) be a set, which is neutrosophically partially ordered, and let M be a subset of S.

_{N}(M), is the neutrosophically smallest element in S, which is neutrosophically greater than or equal to all elements of M.

_{MB}defined below.

#### 1.5.3. Property

#### 1.6. Non-Standard Real MoBiNad Set

_{MB}is closed under addition, subtraction, multiplication, division [except division by $\stackrel{m}{a}$, with a = 0 and $m\u220a\{{,}^{-}{,}^{-0}{,}^{+}{,}^{0+}{,}^{-+}{,}^{-0+}\}$], and power

#### 1.7. Remark

_{MB}, in the sense that we can compute inf

_{N}and sup

_{N}of any subset of NR

_{MB}.

#### 1.8. Non-Standard Real Open Monad Unit Interval

^{–}a

^{+}, not between a and (

^{−0}a

^{+}), and we need a total order relationship on the set of non-standard real numbers, we remove all binads and keep only the open left monads and open right monads [we also remove the monads closed to one side].

^{−}0, 1

^{+}[

_{M}includes the previously defined ]

^{−}0, 1

^{+}[ as follows:

_{M}means that the interval includes all open monads and infinitesimals between

^{−}0 and 1

^{+}.

## 2. General Monad Neutrosophic Set

^{−},

^{+}[

_{M}, or

_{N}]

^{−}0, 1

^{+}[

_{M}

#### 2.1. Non-Standard Neutrosophic Set

#### 2.2. Non-Standard Fuzzy t-Norm and Fuzzy t-Conorm

_{1}, and T

_{2}, ∊]

^{−}0, 1

^{+}[

_{M}, be nonstandard real numbers (infinitesimals, or open monads), or standard (classical) real numbers, such that at least one of them is a non-standard real number. T

_{1}and T

_{2}are non-standard fuzzy degrees of membership. Then one has:

_{1}/\

_{F}T

_{2}= inf

_{N}{T

_{1}, T

_{2}}

_{1}\/

_{F}T

_{2}= sup

_{N}{T

_{1}, T

_{2}}

#### 2.3. Aggregation Operators on Non-Standard Neutrosophic Set

_{1}, I

_{1}, F

_{1}and T

_{2}, I

_{2}, F

_{2}∊ ]

^{−}0, 1

^{+}[

_{MB}, be nonstandard real numbers (infinitesimals, or monads), or standard (classical) real numbers, such that at least one of them is a non-standard real number.

_{1}, I

_{1}, F

_{1}) ∧

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∧

_{F}T

_{2}, I

_{1}∨

_{F}I

_{2}, F

_{1}∨

_{F}F

_{2}) =

(inf

_{N}(T

_{1}, T

_{2}), sup

_{N}(I

_{1}, I

_{2}), sup

_{N}(F

_{1}, F

_{2}))

_{1}, I

_{1}, F

_{1}) ∨

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}∨

_{F}T

_{2}, I

_{1}∧

_{F}I

_{2}, F

_{1}∧

_{F}F

_{2}) =

(sup

_{N}(T

_{1}, T

_{2}), inf

_{N}(I

_{1}, I

_{2}), inf

_{N}(F

_{1}, F

_{2}))

_{N}or ${\neg}_{N}$ for the neutrosophic complement.

_{1}, I

_{1}, F

_{1}) =

_{N}(T

_{2}, I

_{2}, F

_{2}) iff (T

_{1}, I

_{1}, F

_{1}) ≤

_{N}(T

_{2}, I

_{2}, F

_{2}) and (T

_{2}, I

_{2}, F

_{2}) ≤

_{N}(T

_{1}, I

_{1}, F

_{1}).

**Definition**

**1.**

_{x}, I

_{x}, F

_{x}), such that T

_{x}represents the degree of truth-membership of the element x with respect to set U, I

_{x}represents the degree of indeterminate-membership of the element x with respect to the set U, and F

_{x}represents the degree of false-membership of the element x with respect to the set U; where T

_{x}, I

_{x}, and F

_{x}are non-standard or standard subsets of the neutrosophic real monad set NR

_{M}, but at least one of all of them is non-standard (i.e., contains infinitesimals, or open monads).

_{x}, I

_{x}, F

_{x}are single-values (either real numbers, or infinitesimals, or open monads) belonging to ]

^{−}0, 1

^{+}[

**Definition**

**2.**

**0**

_{N}= (

^{−}0, 1

^{+}, 1

^{+}), is a set${\Phi}_{N}\subset X$whose all elements have the non-standard neutrosophic components equal to (

^{−}0, 1

^{+}, 1

^{+}). The whole set, denoted by 1

_{N}= (1

^{+},

^{−}0,

^{−}0), is a set W

_{N}⊂ X whose all elements have the non-standard neutrosophic components equal to (1

^{+},

^{−}0,

^{−}0).

**Definition**

**3.**

_{1}, I

_{1}, F

_{1}) and B = (T

_{2}, I

_{2}, F

_{2}) be non-standard neutrosophic numbers. Then:

_{N}(T

_{1}, T

_{2}), sup

_{N}(I

_{1}, I

_{2}), sup

_{N}(F

_{1}, F

_{2}))

_{N}(T

_{1}, T

_{2}), inf

_{N}(I

_{1}, I

_{2}), inf

_{N}(F

_{1}, F

_{2}))

_{N}A = (F

_{1}, I

_{1}, T

_{1})

**Definition**

**4.**

- (i)
**0**_{N}and**1**are in τ._{N}- (ii)
- The intersection of the elements of any finite subcollection of τ is in τ.
- (iii)
- The union of the elements of any subcollection of τ is in τ.

**Example**

**1.**

**0**

_{N}and

**1**Then τ is a topology on X. It is called the non-standard neutrosophic trivial topology.

_{N.}**Example**

**2.**

**0**

_{N},

**1**, A}. Then it can be easily shown that τ is a topology on X.

_{N}**Example**

**3.**

**0**

_{N},

**1**, A, B}. Then since A ∩ B = B and A ∪ B = A we deduce that τ is a topology on X.

_{N}**Example**

**4.**

_{n}is a neutrosophic superset of A

_{n−1}for each

**0**

_{N},

**1**, A

_{N}_{n}: n ∊N}. Then since A

_{i}∩

_{N}A

_{j}= A

_{i}and A

_{i}∪

_{N}A

_{j}= A

_{j}for each i less than j, we deduce that τ is a topology on X.

**Example**

**5.**

_{100}be a family of subsets of X, such that each member A

_{m},

_{n},

_{p}of the family has:

**0**

_{N},

**1**, M

_{N}_{100}} is a non-standard neutrosophic topology.

**Proof.**

^{k}:

_{1}, n

_{1}, p

_{1}, m

_{2}, n

_{2}, p

_{2}∊ {1, 2, …, 100}.

_{100}are in M

_{100}.

_{1}, m

_{2}}, min{n

_{1}, n

_{2}}, min{p

_{1}, p

_{2}} ∊ M

_{100}.

**Definition**

**5.**

**Example**

**6.**

**0**

_{N},

**1**, A} and τ’ = {

_{N}**0**

_{N},

**1**, B}.

_{N}**Example**

**7.**

_{100}, let’s define L

_{100}as follows:

**0**

_{N},

**1**, M

_{N}_{100}} is a finer non-standard neutrosophic topology than the non-standard neutrosophic topology τ’ = {

**0**

_{N},

**1**, L

_{N}_{100}}.

**Definition**

**6.**

_{N}(Z) is open in X.

**Example**

**8.**

_{N}(A) = A.

**Proof.**

_{m},

_{n}∊Y one has:

**Theorem**

**1.**

**0**

_{N}and the non-standard neutrosophic whole set

**1**are not necessarily closed, since they are not the non-standard neutrosophic complement of each other.

_{N}**Proof.**

_{N}(

^{−}0, 1

^{+}, 1

^{+}) =

_{N}(1

^{+},1

^{+},

^{−}0) ≠ (1

^{+},

^{−}0,

^{−}0), and reciprocally:

_{N}(1

^{+},

^{−}0,

^{−}0) =

_{N}(

^{−}0,

^{−}0, 1

^{+}) ≠ (

^{−}0, 1

^{+}, 1

^{+}).

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

_{2}, I

_{2}, F

_{2}) and B = (T

_{1}, I

_{1}, F

_{1}). Note that C

_{N}A = (F

_{2}, I

_{2}, T

_{2}) and C

_{N}B = (F

_{1}, I

_{1}, T

_{1}).

_{N}A ∩

_{N}C

_{N}B = (F

_{2}, I

_{1}, T

_{2}).

_{N}(C

_{N}A ∩

_{N}C

_{N}B) = (T

_{2}, I

_{1}, F

_{2})

_{N}A ∩

_{N}C

_{N}B is not a non-standard neutrosophic closed set in X. Also,

_{N}A ∩

_{N}C

_{N}B = (F

_{1}, I

_{2}, T

_{1}).

_{N}(C

_{N}A ∩

_{N}C

_{N}B) = (T

_{1}, I

_{2}, F

_{1})

_{N}A ∩

_{N}C

_{N}B is not a non-standard neutrosophic closed set in X. □

**General Remark**

**1.**

**classes of operators**(not by exact unique operators) respectively, the classical topological space theorems and properties extended (by the transfer principle) to the non-standard neutrosophic topological space may be valid for some non-standard neutrosophic operators, but invalid for other classes of neutrosophic aggregation operators.

_{N}can be defined in 2 ways:

_{1}, I

_{1}, F

_{1}) /\

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}/\

_{F}T

_{2}, I

_{1}/\

_{F}I

_{2}, F

_{1}/\

_{F}F

_{2})

_{1}, I

_{1}, F

_{1}) /\

_{N}(T

_{2}, I

_{2}, F

_{2}) = (T

_{1}/\

_{F}T

_{2}, I

_{1}\/

_{F}I

_{2}, F

_{1}/\

_{F}F

_{2}).

_{F}) and fuzzy t-conorm (\/

_{F}) are also defined in many ways; for example I know at least 3 types of fuzzy t-norms:

_{F}b = min {a, b}

_{F}b = ab

_{F}b = max {a + b − 1, 0}

_{F}b = max {a, b}

_{F}b = a + b − ab

_{F}b = min {a + b, 1}

_{N}).

_{N}).

_{N}and \/

_{N}for some of them the classical topological theorems extended to non-standard neutrosophic topology may be valid, for others invalid.

**Definition**

**7.**

_{N}(A). The Non-standard Neutrosophic Closure of A is the smallest nonstandard neutrosophic closed set in X that neutrosophically includes A.

**Example**

**9.**

**Proof.**

_{1}∩

_{N}A

_{2}= A

_{1}, A

_{1}∩

_{N}A

_{3}= A

_{1}, A

_{2}∩

_{N}A

_{3}= A

_{3}

_{1}∪

_{N}A

_{2}= A

_{2}, A

_{1}∪

_{N}A

_{3}= A

_{3}, A

_{2}∪

_{N}A

_{3}= A

_{2}, A

_{1}∪

_{N}A

_{2}∪

_{N}A

_{3}= A

_{2}.

_{1}, A

_{2}, A

_{3}are open sets since they belong to τ.

_{2}is the non-standard neutrosophic complement of A

_{1}, or C

_{N}(A

_{2}) = A

_{1}, therefore A

_{2}is a non-standard neutrosophic closed set in X.

_{3}is the non-standard neutrosophic complement of A

_{3}(itself), or C

_{N}(A

_{3}) = A

_{3}, therefore A

_{3}is also a non-standard neutrosophic closed set in X.

_{2}and A

_{3}are nonstandard neutrosophic supersets of A

_{1}, since A

_{1}⊂ A

_{2}and A

_{1}⊂ A

_{3}.

_{1}is the intersection of its non-standard neutrosophic closed supersets A

_{2}and A

_{3}, or

_{N}(A

_{1}) =

_{N}A

_{2}∩

_{N}A

_{3}=

_{N}A

_{3}

**Definition**

**8.**

_{N}(A).

**Example**

**10.**

_{N}(A

_{2}).

_{1}and A

_{3}are non-standard neutrosophic open sets in X, with A

_{1}⊂

_{N}A

_{2}and A

_{3}⊂

_{N}A

_{2}

_{N}(A

_{2}) = A

_{1}∪

_{N}A

_{3}= A

_{3}.

**Definition**

**9.**

**Example**

**11.**

_{3}⊂ X, and the non-standard neutrosophic subspace topology

**Definition**

**10.**

^{−1}(A) is a non-standard neutrosophic open set in X.

**Example**

**12.**

**Example**

**13.**

**Definition**

**11.**

_{1}, τ

_{1}) and (X

_{2}, τ

_{2}) be two non-standard neutrosophic topological spaces. Then${\tau}_{1}\times {\tau}_{2}{=}_{N}\{U\times V:U\u220a{\tau}_{1},V\u220a{\tau}_{2}\}$defines a topology on the product

_{1}$\times $ τ

_{2}is called non-standard neutrosophic product topology.

## 3. Development of Neutrosophic Topologies

## 4. Conclusions

_{M}that is formed by real numbers and positive infinitesimals and open monads, together with several concepts related to them, such as: non-standard neutrosophic open/closed sets, non-standard neutrosophic closure and interior of a given set, and non-standard neutrosophic product topology. Several theorems were proven and non-standard neutrosophic examples were presented.

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Al Shumrani, M.A.; Smarandache, F.
Introduction to Non-Standard Neutrosophic Topology. *Symmetry* **2019**, *11*, 706.
https://doi.org/10.3390/sym11050706

**AMA Style**

Al Shumrani MA, Smarandache F.
Introduction to Non-Standard Neutrosophic Topology. *Symmetry*. 2019; 11(5):706.
https://doi.org/10.3390/sym11050706

**Chicago/Turabian Style**

Al Shumrani, Mohammed A., and Florentin Smarandache.
2019. "Introduction to Non-Standard Neutrosophic Topology" *Symmetry* 11, no. 5: 706.
https://doi.org/10.3390/sym11050706