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AN INTRODUCTION TO FUZZY NEUTROSOPHIC TOPOLOGICAL SPACES

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International Journal of Mathematical Archive-8(3), 2017, 144-149

Available online through www.ijma.info ISSN 2229 รข€“ 5046 AN INTRODUCTION TO FUZZY NEUTROSOPHIC TOPOLOGICAL SPACES Y. VEERESWARI* Research Scholar, Govt. Arts College, Coimbatore, (T.N.), India. (Received On: 05-02-17; Revised & Accepted On: 23-03-17)

ABSTRACT

In

this paper we introduce fuzzy neutrosophic topological spaces and its some properties. Also we provide fuzzy continuous and fuzzy compactness of fuzzy neutrosophic topological space and its some properties and examples. Keywords: fuzzy neutrosophic set, fuzzy neutrosophic topology, fuzzy neutrosophic topological spaces, fuzzy continuous and fuzzy compactness.

1. INTRODUCTION Fuzzy sets were introduced by Zadeh in 1965. The concepts of intuitionistic fuzzy sets by K. Atanassov several researches were conducted on the generalizations of the notion of intuitionistic fuzzy sets. Florentin Smarandache [5, 6] developed Neutrosophic set &logic ofA Generalization of the Intuitionistic Fuzzy Logic& set respectively. A.A.Salama & S.A.Alblowi [1] introduced and studied Neutrosophic Topological spaces and its continuous function in [2]. In this paper, we define thenotion of fuzzy neutrosophic topological spaces and investigate continuity and compactness by using Cokerรข€™s intuitionistic topological spaces in [4]. We discuss New examples of FNTS. 2. PRELIMINARIES Here we shall present the fundamental definitions. The following one is obviously inspired by Haibin Wang and Florentin Smarandache in [7] and A.A.Salama, S.S.Alblow in [1]. Smarandache introduced the neutrosophic set and neutrosophic components. The sets T, I, F are not necessarily intervals but may be any real sub-unitary subsets of ]รขˆ’ 0, 1+ [. The neutrosophic components T, I, F represents the truth value, indeterminacy value and falsehood value respectively. Definition 2.1 [7]: Let ฤ‘?‘‹ฤ‘?‘‹ be a non-empty fixed set. A fuzzy neutrosophic set (FNS for short) ฤ‘??ยดฤ‘??ยด is an object having the form ฤ‘??ยดฤ‘??ยด = {รขŒล ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)รขŒลž: ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹} where the functions ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด : ฤ‘?‘‹ฤ‘?‘‹ รข†’]รขˆ’ 0, 1+ [, ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด : ฤ‘?‘‹ฤ‘?‘‹ รข†’]รขˆ’ 0, 1+ [ ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด : ฤ‘?‘‹ฤ‘?‘‹ รข†’]รขˆ’ 0, 1+ [, denote the degree of membership function (namely ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)), the degree of indeterminacy function (namely ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)) and the degree of non-membership (namely ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)) respectively of each element ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹ to the set ฤ‘??ยดฤ‘??ยด and + 0 รข‰ยค ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) + ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) + ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) รข‰ยค 1+ , for each ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹. Remark 2.2 [7]: Every fuzzy set ฤ‘??ยดฤ‘??ยด on a non-empty set ฤ‘?‘‹ฤ‘?‘‹ is obviously a FNS having the form ฤ‘??ยดฤ‘??ยด = {รขŒล ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), 1 รขˆ’ ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)รขŒลž: ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹}

A fuzzy neutrosophic set ฤ‘??ยดฤ‘??ยด = {รขŒล ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)รขŒลž: ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹} can be identified to an ordered triple รขŒล ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด , ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด , ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด รขŒลž in ]รขˆ’ 0, 1+ [ on ฤ‘?‘‹ฤ‘?‘‹.

Definition 2.3[1]: Let ฤ‘?‘‹ฤ‘?‘‹ be a non-empty set and the FNSs ฤ‘??ยดฤ‘??ยด and ฤ‘??ฤพฤ‘??ฤพ be in the form ฤ‘??ยดฤ‘??ยด = {รขŒล ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)รขŒลž: ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹ and ฤ‘??ฤพฤ‘??ฤพ=ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ฤพฤ‘??ฤพฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œŽฤ‘?œŽฤ‘??ฤพฤ‘??ฤพฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œˆฤ‘?œˆฤ‘??ฤพฤ‘??ฤพฤ‘?‘ฤฝฤ‘?‘ฤฝ:ฤ‘?‘ฤฝฤ‘?‘ฤฝรขˆˆฤ‘?‘‹ฤ‘?‘‹ on ฤ‘?‘‹ฤ‘?‘‹ and let ฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘–:ฤ‘?‘–ฤ‘?‘–รขˆˆฤ‘??หฤ‘??ห be an arbitrary family of FNSรข€™s in ฤ‘?‘‹ฤ‘?‘‹, where ฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘–=ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘–ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘–ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘–ฤ‘?‘ฤฝฤ‘?‘ฤฝ:ฤ‘?‘ฤฝฤ‘?‘ฤฝรขˆˆฤ‘?‘‹ฤ‘?‘‹e. a) ฤ‘??ยดฤ‘??ยด รขŠ† ฤ‘??ฤพฤ‘??ฤพ iff ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) รข‰ยค ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ฤพฤ‘??ฤพ (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) รข‰ยค ฤ‘?œŽฤ‘?œŽฤ‘??ฤพฤ‘??ฤพ (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) and ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) รข‰ฤฝ ฤ‘?œˆฤ‘?œˆฤ‘??ฤพฤ‘??ฤพ (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) for all ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹. b) ฤ‘??ยดฤ‘??ยด = ฤ‘??ฤพฤ‘??ฤพ iff ฤ‘??ยดฤ‘??ยด รขŠ† ฤ‘??ฤพฤ‘??ฤพ and ฤ‘??ฤพฤ‘??ฤพ รขŠ† ฤ‘??ยดฤ‘??ยด. c) ฤ‘??ยดฤ‘??ยดฤš… = {รขŒล ฤ‘?‘ฤฝฤ‘?‘ฤฝ, ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), 1 รขˆ’ ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ), ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยด (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)รขŒลž: ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹} d) รขˆลž ฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘– = ฤลผหรขŒล ฤ‘?‘ฤฝฤ‘?‘ฤฝ, รข‹ ฤ‘?‘–ฤ‘?‘–รขˆˆฤ‘??หฤ‘??ห ฤ‘?œ‡ฤ‘?œ‡ฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘– (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) , รข‹ ฤ‘?‘–ฤ‘?‘–รขˆˆฤ‘??หฤ‘??ห ฤ‘?œŽฤ‘?œŽฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘– (ฤ‘?‘ฤฝฤ‘?‘ฤฝ) , รข‹€ฤ‘?‘–ฤ‘?‘–รขˆˆฤ‘??หฤ‘??ห ฤ‘?œˆฤ‘?œˆฤ‘??ยดฤ‘??ยดฤ‘?‘–ฤ‘?‘– (ฤ‘?‘ฤฝฤ‘?‘ฤฝ)รขŒลž: ฤ‘?‘ฤฝฤ‘?‘ฤฝ รขˆˆ ฤ‘?‘‹ฤ‘?‘‹ฤลผห

Corresponding Author: Y. Veereswari* Research Scholar, Govt. Arts College, Coimbatore, (T.N.), India.

International Journal of Mathematical Archive- 8(3), March รข€“ 2017

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AN INTRODUCTION TO FUZZY NEUTROSOPHIC TOPOLOGICAL SPACES by Ioan Degฤƒu - Issuu