CODEN (USA): IJCRGG, ISSN: 0974-4290, ISSN(Online):2455-9555 Vol.10 No.10, pp 449-458, 2017

Abstract : The purpose of this article is to extend some more topics of topology in neutrosophic topology. Eerily introduce neutrosophic topological space and open sets, closed sets, semi-open and semi closed sets, now in this manuscript we extend up to neutrosophic pre-open and pre-closed (NPO and NPC). Keywords : Neutrosophic set, Neutrosophic Topology, Neutrosophic pre-open set, Neutrosophic pre-closed set.

Neutrosophictechnique is a special technique is based on Neutrosophy. This theory developed by “FlorentinSmarandache“in 1995. In Neutrosophy consider every notion or idea is together with

,and here is the opposite or negation of

, is the field of “neutralities”.Neutrosophic method is derived fromFuzzy logic or in intuitionistic Fuzzy logic. The first book was published on Neutrosophy as a title of book is Neutrosophy probability, set and logic in 1998. , Rehoboth. The word Neutrosophy introduced from Latin “neuter” – neutral, Greek “Sophia”-skill/wisdom. Neutrosophy means sill on neutrals. The main task of this study is to apply neutrosophic method to the general theory of relativity, aiming to discover new hidden effects. Here it’s why we decided to employ neutrosophic method in this field. Neutrosophic method means to find common features to uncommon entities. arecalled neutrosophic components will represent the truth value, indeterminacy value and false hood value respectively referring to neutrosophic methods.

Not

Opposite ofnor

Aversion of

V. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

is different to .

Example 1.1:

, = yellow But =green, red, blue…etc =green, red, blue…etc . A’ = any shade of pink.

1.4Remark: Between an idea

A and its opposite Anti-A there is a continuum power spectrum of neutralities. Neutrosophic theory was studied in neutrosophic metric space, smooth topological spaces, arithmetic operations, rough sets, neutrosophic geometry, and neutrosophic probability….etc. Now, in natural way make longer neutrosophic theory in neutrosophic pre-open and pre-closed set and neutrosophic topological space.

A logic in which each proposition is estimated to have the percentage of truth in subset

the percentage of indeterminacy in subset and percentage of falsity in a subset where are defined above is called neutrosophic logic. Truth values where are standard or non-standard subsets of nonstandard interval

, whereAnd .they are many neutrosophic rules of inference[2].

The degree of membership to , degree of non-membership-to

, such that when one obtain the fuzzy set, and if there is an indeterminacy .Zadeh et. al. fuzzy set theory as a mathematical tool in for dealing with uncertainties where each element had each element had degree of membership acquire form [4-7].The intuitionistic fuzzy set was introduced y Atanassov in as a generalization of fuzzy set where besides the degree of membership and the degree non-membership of each element. The neutrosophic set was introduced by Smarandache and explained neutrosophic set is a generalization of intuitionistic fuzzy set. In Salama,Alblow

introduced the concept of topological space. In introduced the concept of neutrosophic semi-open sets in neutrosophic topological space by p.Ishwarya and dr.k.Bageerathi.Currently, in this manuscript we recall the neutrosophic pre-open sets and neutrosophic per-closed in neutrosophic topological spaces.

We recollect some important basic preliminaries, and in particular, the work of Smarandache in [1], Atanassov in [4, 5] and Salama [8, 9]. Smarandache introduced the neutrosophic components which represent the membership, indeterminacy, and non-membership values respectively, where

is nonstandard unit interval.Definition 2.2: Let

be a non-empty fixed set. Neutrosophic set (NSfor short) is an object having the form where which represent the degreeV. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

ofmember ship function

the degree of indeterminacy and the degree ofnon-member shiprespectively of each element

to the set A [7]. Definition 2.3: The Neutrosophic subsets and in as follows: may be defined as: may be defined asDefinition 2.4: Let beaNSon

, then the complement set of the may be definedas three kinds of complements.

Example 2.1: The complement of is

and the complement of is .are complement each other Definition 2.5: Let be a non empty set and neutrosophic sets

and in the formthen we consider two possible definitions for subsets.

V. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

Proposition 2.1: For any neutrosophicset

, then the following conditions are holds.Definition 2.6: Let

be a non-empty set and are neutrosophic sets thenMay be defined as

We can easily generalize the operations of intersection and union in def

to arbitrary family ofneutrosophic sets. Proposition 2.2:For all are two neutrosophic sets then the following conditions are true.

Definition 2.7: A neutrosophic topology

is a non-empty set is a familyof a neutrosophic sets in

satisfying the following condition.In this case

is called a neutrosophic topological space.Example 2.2:

Let and

Then the family of neutrosophic set in

is neutrosophic topology .Definition 2.8: The elements of

is neutrosophic open sets the complement of neutrosophic open set is called neutrosophic closed set.Definition 2.9: Let

be neutrosophic topological space and be a neutrosophic set in then neutrosophic closure and neutrosophic interior of are defined byV. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

It can also show that is neutrosophic closed set and is neutrosophic open set in

Proposition 2.3: Let be neutrosophic topological space and we have

Proposition 2.4: Let

be neutrosophic topological space and are two neutrosophic set in thenthe following properties are hold.

Definition 2.10: Let

be neutrosophic topological space and be a neutrosophic set in then is called a neutrosophic semi-open if and also neutrosophic semi-closed if .The complement of neutrosophic semi open set is a neutrosophic semi closed set.In this section the concept of neutrosophic pre-open set of

is introduced and also characterizations ofneutrosophic pre-open sets. Definition 3.1: Let

be neutrosophic set of a neutrosophic topology. Then is said to be Neutrosophic preopen[NPO]set of

if there exists a neutrosophic open set NO such that ..

V. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

Conversely suppose that Let

be neutrosophic topological set in .i.e. for some NO But , thus HenceTherefore Hence proved the theorem Theorem3.2:In neutrosophic topology the union of two neutrosophic pre-open sets again a neutrosophic pre-open set. Proof: Let neutrosophic pre-open sets in

By the definition is a neutrosophic open set in

. Theorem3.3:let be an NTS.If is a collection of NPO sets in a NTS X then is NPOset in X. | ||
---|---|---|

Proof: for each | ,we have a neutrosophic open set | such that |

, then |

Hence theorem proved. Remark3.1:The intersection of any two NPO sets need not be a NPO set in X as shown by the following example. Example: let

V. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

Then is NTS Consider

From this example is not NPO set. Theorem 3.4:Every neutrosophic open set in the NTS in X is NPO set in X. Proof: let

be NO set in NTS. Then Clearly is a NPO set in . Theorem 3.5:Letbe neutrosophic pre-open set in the neutrosophic topological space

and supposethen

is neutrosophic pre-open set in . Proof: Let be NO set in neutrosophic topological space . Then alsoHence the theorem proved. Lemma3.1:Let

beanNO set in and a neutrosophic pre-open set in then there exists an NO setin

such thatit follows that

Now, since is open, from the above (theorem 5) lemma, is neutrosophic pre-open set in

. Proposition 3.1: Let and are neutrosophic topological space such that is neutrosophic product related tothen the neutrosophic product of a neutrosophicpre open set of

and a neutrosophic pre open set B of Y is a neutrosophic pre open set of the neutrosophic product topological space .Proof: let and

V. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

Then

Hence | is NPO set in | |||||
---|---|---|---|---|---|---|

4. Neutrosophic Pre-closed sets: | ||||||

Definition 4.1: Let be neutrosophic set of a neutrosophic pre-closed sets of if there exists a | neutrosophic topology spaces set such that | . | Then A is . | said | to | be |

Theorem4.1:A subset A in a NTS | is NCS set if and only if | |||||

Proof: consider | ||||||

Then clearly Therefore is NPC set Conversely, suppose that Let A be NPC set in X Then for some NS closed set | .but |

Hence the theorem proved. Theorem4.2:Let be NTS and

be a neutrosophicsubset of then is a neutrosophic pre-closed sets if and onlyis neutrosophic pre-open set in X. Proof: Let

be a neutrosophic pre-closed set subset of .Clearly Taking complement on both sides

Hence

is neutrosophic pre-open set Conversely suppose that is neutrosophic pre-open set i.e. Taking complement on both sides we get , neutrosophic pre-closed set Hence the theorem proved. Theorem4.3:Let be a neutrosophic topological spaces. Then intersection of two neutrosophic pre- closed set is also a neutrosophic pre-closed set. Proof: let are neutrosophic pre-closed sets onThen , Consider

V. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

Hence is neutrosophic pre-closed set. Remark4.1: The union of any two neutrosophic pre-closed sets need not be a neutrosophicclosed set on

. Theorem 4.4: Let be a collection of neutrosophic pre-closed sets on then isneutrosophic pre-closed sets on

. Proof: we have a neutrosophic set such that for all ThenHence is neutrosophic pre-closed set on

.Theorem 4.5: Every neutrosophic closed set in the neutrosophic topological spaces

is neutrosophic pre-closed set in .

Proof: Let be neutrosophic closed set means and also From that,

, since Hence A is a neutrosophic pre-closed sets. Theorem 4.6:Let

be a neutrosophic closed set in neutrosophic topological spaces and suppose then is neutrosophic pre-closed set onProof: Let be a neutrosophic set in neutrosophic topological spaces

Suppose

There exist a neutrosophic closed set , such that . Then and also Thus,Hence B is neutrosophic pre-closed set on

. Theorem4.7: Let and are neutrosophic topological space such that is neutrosophic product related to then the neutrosophic product is a neutrosophic pre closed set of the neutrosophic product topological space . Where neutrosophic pre closed set of and a neutrosophic pre closed set B of Y.V. Venkateswara Rao et al /International Journal of ChemTech Research, 2017,10(10): 449-458.

Form the above,

Hence is neutrosophic pre-closed set in neutrosophic topological space .

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