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 Γ’€“ 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 CokerΓ’€™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 ]Γ’ˆ’ 0, 1+ [. The neutrosophic components T, I, F represents the truth value, indeterminacy value and falsehood value respectively. Definition 2.1 [7]: Let Δ‘?‘‹Δ‘?‘‹ be a non-empty fixed set. A fuzzy neutrosophic set (FNS for short) Δ‘??Β΄Δ‘??Β΄ is an object having the form Δ‘??Β΄Δ‘??Β΄ = {Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½)Γ’ŒŞ: Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹} where the functions Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ : Δ‘?‘‹Δ‘?‘‹ Γ’†’]Γ’ˆ’ 0, 1+ [, Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ : Δ‘?‘‹Δ‘?‘‹ Γ’†’]Γ’ˆ’ 0, 1+ [ Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ : Δ‘?‘‹Δ‘?‘‹ Γ’†’]Γ’ˆ’ 0, 1+ [, denote the degree of membership function (namely Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½)), the degree of indeterminacy function (namely Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½)) and the degree of non-membership (namely Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½)) respectively of each element Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹ to the set Δ‘??Β΄Δ‘??Β΄ and + 0 Γ’‰Β€ Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½) + Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½) + Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½) Γ’‰Β€ 1+ , for each Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹. Remark 2.2 [7]: Every fuzzy set Δ‘??Β΄Δ‘??Β΄ on a non-empty set Δ‘?‘‹Δ‘?‘‹ is obviously a FNS having the form Δ‘??Β΄Δ‘??Β΄ = {Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), 1 Γ’ˆ’ Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½)Γ’ŒŞ: Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹}

A fuzzy neutrosophic set Δ‘??Β΄Δ‘??Β΄ = {Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½)Γ’ŒŞ: Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹} can be identified to an ordered triple Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ , Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ , Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ Γ’ŒŞ in ]Γ’ˆ’ 0, 1+ [ on Δ‘?‘‹Δ‘?‘‹.

Definition 2.3[1]: Let Δ‘?‘‹Δ‘?‘‹ be a non-empty set and the FNSs Δ‘??Β΄Δ‘??Β΄ and Δ‘??ΔΎΔ‘??ΔΎ be in the form Δ‘??Β΄Δ‘??Β΄ = {Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½)Γ’ŒŞ: Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹ and Δ‘??ΔΎΔ‘??ΔΎ=Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??ΔΎΔ‘??ΔΎΔ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œŽΔ‘?œŽΔ‘??ΔΎΔ‘??ΔΎΔ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œˆΔ‘?œˆΔ‘??ΔΎΔ‘??ΔΎΔ‘?‘Δ½Δ‘?‘Δ½:Δ‘?‘Δ½Δ‘?‘Δ½Γ’ˆˆΔ‘?‘‹Δ‘?‘‹ on Δ‘?‘‹Δ‘?‘‹ and let Δ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘–:Δ‘?‘–Δ‘?‘–Γ’ˆˆΔ‘??˝đ??˝ be an arbitrary family of FNSΓ’€™s in Δ‘?‘‹Δ‘?‘‹, where Δ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘–=Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘–Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘–Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘–Δ‘?‘Δ½Δ‘?‘Δ½:Δ‘?‘Δ½Δ‘?‘Δ½Γ’ˆˆΔ‘?‘‹Δ‘?‘‹e. a) Δ‘??Β΄Δ‘??Β΄ Γ’Š† Δ‘??ΔΎΔ‘??ΔΎ iff Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½) Γ’‰Β€ Δ‘?œ‡Δ‘?œ‡Δ‘??ΔΎΔ‘??ΔΎ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½) Γ’‰Β€ Δ‘?œŽΔ‘?œŽΔ‘??ΔΎΔ‘??ΔΎ (Δ‘?‘Δ½Δ‘?‘Δ½) and Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½) Γ’‰Δ½ Δ‘?œˆΔ‘?œˆΔ‘??ΔΎΔ‘??ΔΎ (Δ‘?‘Δ½Δ‘?‘Δ½) for all Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹. b) Δ‘??Β΄Δ‘??Β΄ = Δ‘??ΔΎΔ‘??ΔΎ iff Δ‘??Β΄Δ‘??Β΄ Γ’Š† Δ‘??ΔΎΔ‘??ΔΎ and Δ‘??ΔΎΔ‘??ΔΎ Γ’Š† Δ‘??Β΄Δ‘??Β΄. c) Δ‘??Β΄Δ‘??´Ě… = {Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), 1 Γ’ˆ’ Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ (Δ‘?‘Δ½Δ‘?‘Δ½)Γ’ŒŞ: Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹} d) Γ’ˆŞ Δ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘– = ďż˝ÒŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Γ’‹ Δ‘?‘–Δ‘?‘–Γ’ˆˆΔ‘??˝đ??˝ Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘– (Δ‘?‘Δ½Δ‘?‘Δ½) , Γ’‹ Δ‘?‘–Δ‘?‘–Γ’ˆˆΔ‘??˝đ??˝ Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘– (Δ‘?‘Δ½Δ‘?‘Δ½) , Γ’‹€Δ‘?‘–Δ‘?‘–Γ’ˆˆΔ‘??˝đ??˝ Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄Δ‘?‘–Δ‘?‘– (Δ‘?‘Δ½Δ‘?‘Δ½)Γ’ŒŞ: Δ‘?‘Δ½Δ‘?‘Δ½ Γ’ˆˆ Δ‘?‘‹Δ‘?‘‹ďż˝

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

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

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