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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Y. Veereswari* / An Introduction to Fuzzy Neutrosophic Topological Spaces / IJMA- 8(3), March-2017.

e) f) g) h) i)

∩ 𝐴𝐴𝑖𝑖 = �〈π‘₯π‘₯, β‹€π‘–π‘–βˆˆπ½π½ πœ‡πœ‡π΄π΄π‘–π‘– (π‘₯π‘₯) , β‹€π‘–π‘–βˆˆπ½π½ πœŽπœŽπ΄π΄π‘–π‘– (π‘₯π‘₯) , β‹π‘–π‘–βˆˆπ½π½ πœˆπœˆπ΄π΄π‘–π‘– (π‘₯π‘₯)βŒͺ: π‘₯π‘₯ ∈ 𝑋𝑋� 𝐴𝐴 βˆ’ 𝐡𝐡 = �〈π‘₯π‘₯, πœ‡πœ‡π΄π΄ (π‘₯π‘₯) ∧ 𝜈𝜈𝐡𝐡 (π‘₯π‘₯), 𝜎𝜎𝐴𝐴 (π‘₯π‘₯) ∧ οΏ½1 βˆ’ 𝜎𝜎𝐡𝐡 (π‘₯π‘₯)οΏ½, 𝜈𝜈𝐴𝐴 (π‘₯π‘₯) ∨ πœ‡πœ‡π΅π΅ (π‘₯π‘₯)βŒͺ: π‘₯π‘₯ ∈ 𝑋𝑋� []𝐴𝐴 = {〈π‘₯π‘₯, πœ‡πœ‡π΄π΄ (π‘₯π‘₯), 𝜎𝜎𝐴𝐴 (π‘₯π‘₯), 1 βˆ’ πœ‡πœ‡π΄π΄ (π‘₯π‘₯)βŒͺ: π‘₯π‘₯ ∈ 𝑋𝑋} <> 𝐴𝐴 = {〈π‘₯π‘₯, 1 βˆ’ 𝜈𝜈𝐴𝐴 (π‘₯π‘₯), 𝜎𝜎𝐴𝐴 (π‘₯π‘₯), 𝜈𝜈𝐴𝐴 (π‘₯π‘₯)βŒͺ: π‘₯π‘₯ ∈ 𝑋𝑋} 0𝑁𝑁 = 〈π‘₯π‘₯, 0, 0, 1βŒͺ and 1𝑁𝑁 = 〈π‘₯π‘₯, 1, 1, 0βŒͺ

Example 2.4: Let 𝑋𝑋 = {π‘Žπ‘Ž, 𝑏𝑏, 𝑐𝑐} and π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ, 𝐴𝐴 = 〈π‘₯π‘₯, οΏ½ , 0.5 0.2 0.3 0.3 0.2 0.2 0.3 0.4 0.5 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 𝐡𝐡 = 〈π‘₯π‘₯, οΏ½ , , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ, 0.5 0.5 0.4 0.4 0.5 0.2 0.2 0.4 0.4 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 𝐢𝐢 = 〈π‘₯π‘₯, οΏ½ , , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ, 0.5 0.5 0.5 0.5 0.5 0.3 0.2 0.2 0.2 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 𝐷𝐷 = 〈π‘₯π‘₯, οΏ½ , , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ 0.3 0.4 0.4 0.4 0.2 0.2 0.3 0.4 0.4

π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 Here 𝐴𝐴̅ = 〈π‘₯π‘₯, οΏ½ , , οΏ½ , οΏ½ , , οΏ½ , οΏ½ , , οΏ½βŒͺ and 𝐴𝐴 βŠ† 𝐢𝐢 because πœ‡πœ‡π΄π΄ (π‘₯π‘₯) ≀ πœ‡πœ‡π΅π΅ (π‘₯π‘₯), 𝜎𝜎𝐴𝐴 (π‘₯π‘₯) ≀ 𝜎𝜎𝐡𝐡 (π‘₯π‘₯) and 𝜈𝜈𝐴𝐴 (π‘₯π‘₯) β‰₯ 0.3 0.2 0.2 0.5 0.8 0.7 0.3 0.4 0.5 𝜈𝜈𝐡𝐡 (π‘₯π‘₯) for all π‘₯π‘₯ ∈ 𝑋𝑋. Further, 𝐴𝐴 βˆͺ 𝐡𝐡 = 〈π‘₯π‘₯, πœ‡πœ‡π΄π΄ (π‘₯π‘₯) ∨ πœ‡πœ‡π΅π΅ (π‘₯π‘₯), 𝜎𝜎𝐴𝐴 (π‘₯π‘₯) ∨ 𝜎𝜎𝐡𝐡 (π‘₯π‘₯), 𝜈𝜈𝐴𝐴 (π‘₯π‘₯) ∧ 𝜈𝜈𝐡𝐡 (π‘₯π‘₯)βŒͺ π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 = 〈π‘₯π‘₯, οΏ½ , , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ 0.5 0.5 0.5 0.5 0.5 0.3 0.2 0.2 0.2

𝐴𝐴 ∩ 𝐡𝐡 = 〈π‘₯π‘₯, πœ‡πœ‡π΄π΄ (π‘₯π‘₯) ∧ πœ‡πœ‡π΅π΅ (π‘₯π‘₯), 𝜎𝜎𝐴𝐴 (π‘₯π‘₯) ∧ 𝜎𝜎𝐡𝐡 (π‘₯π‘₯), 𝜈𝜈𝐴𝐴 (π‘₯π‘₯) ∨ 𝜈𝜈𝐡𝐡 (π‘₯π‘₯)βŒͺ π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 π‘Žπ‘Ž 𝑏𝑏 𝑐𝑐 = 〈π‘₯π‘₯, οΏ½ , , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ 0.3 0.4 0.4 0.4 0.2 0.2 0.3 0.4 0.4

Corollary 2.5: Let 𝐴𝐴, 𝐡𝐡, 𝐢𝐢 and 𝐴𝐴𝑖𝑖 be FNS’s in 𝑋𝑋(𝑖𝑖 ∈ 𝐽𝐽). Then a) 𝐴𝐴𝑖𝑖 βŠ† 𝐡𝐡 for each 𝑖𝑖 ∈ 𝐽𝐽 ⟹ βˆͺ 𝐴𝐴𝑖𝑖 βŠ† 𝐡𝐡 b) 𝐡𝐡 βŠ† 𝐴𝐴𝑖𝑖 for each 𝑖𝑖 ∈ 𝐽𝐽 ⟹ 𝐡𝐡 βŠ† ∩ 𝐴𝐴𝑖𝑖 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ c) βˆͺ 𝐴𝐴𝑖𝑖 =∩ 𝐴𝐴�𝑖𝑖 and οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ∩ 𝐴𝐴𝑖𝑖 =βˆͺ 𝐴𝐴�𝑖𝑖 d) 𝐴𝐴 βŠ† 𝐡𝐡 ⟺ 𝐡𝐡� βŠ† 𝐴𝐴̅ οΏ½οΏ½οΏ½οΏ½οΏ½Μ…) = 𝐴𝐴 e) (𝐴𝐴 οΏ½οΏ½ οΏ½οΏ½οΏ½οΏ½ f) 0οΏ½οΏ½ 𝑁𝑁 = 1𝑁𝑁 and 1𝑁𝑁 = 0𝑁𝑁

Now we shall define the image and preimage of FNS’s. Let 𝑋𝑋, π‘Œπ‘Œ be two non-empty sets and 𝑓𝑓: 𝑋𝑋 ⟢ π‘Œπ‘Œ a function.

Definition 2.6[4]: a) If 𝐡𝐡 = {βŒ©π‘¦π‘¦, πœ‡πœ‡π΅π΅ (𝑦𝑦), 𝜎𝜎𝐡𝐡 (𝑦𝑦), 𝜈𝜈𝐡𝐡 (𝑦𝑦)βŒͺ: 𝑦𝑦 ∈ π‘Œπ‘Œ} is an FNS in π‘Œπ‘Œ then the preimage of 𝐡𝐡 under 𝑓𝑓, denoted by 𝑓𝑓 βˆ’1 (𝐡𝐡), is the FNS in 𝑋𝑋 defined by 𝑓𝑓 βˆ’1 (𝐡𝐡) = {〈π‘₯π‘₯, 𝑓𝑓 βˆ’1 (πœ‡πœ‡π΅π΅ )(π‘₯π‘₯), 𝑓𝑓 βˆ’1 (𝜎𝜎𝐡𝐡 )(π‘₯π‘₯), 𝑓𝑓 βˆ’1 (𝜈𝜈𝐡𝐡 )(π‘₯π‘₯)βŒͺ: π‘₯π‘₯ ∈ 𝑋𝑋} βˆ’1 (πœ‡πœ‡ )(π‘₯π‘₯) = πœ‡πœ‡π΅π΅ 𝑓𝑓(π‘₯π‘₯), 𝑓𝑓 βˆ’1 (𝜎𝜎𝐡𝐡 )(π‘₯π‘₯) = 𝜎𝜎𝐡𝐡 𝑓𝑓(π‘₯π‘₯) and 𝑓𝑓 βˆ’1 (𝜈𝜈𝐡𝐡 )(π‘₯π‘₯) = 𝜈𝜈𝐡𝐡 𝑓𝑓(π‘₯π‘₯) Where 𝑓𝑓 𝐡𝐡 b) If 𝐴𝐴 = {〈π‘₯π‘₯, πœ‡πœ‡π΄π΄ (π‘₯π‘₯), 𝜎𝜎𝐴𝐴 (π‘₯π‘₯), 𝜈𝜈𝐴𝐴 (π‘₯π‘₯)βŒͺ: π‘₯π‘₯ ∈ 𝑋𝑋} is a FNS in 𝑋𝑋, then the image of 𝐴𝐴 under 𝑓𝑓, denoted by 𝑓𝑓(𝐴𝐴), is the FNS in π‘Œπ‘Œ defined by 𝑓𝑓(𝐴𝐴) = οΏ½βŒ©π‘¦π‘¦, 𝑓𝑓(πœ‡πœ‡π΄π΄ )(𝑦𝑦), 𝑓𝑓(𝜎𝜎𝐴𝐴 )(𝑦𝑦), οΏ½1 βˆ’ 𝑓𝑓(1 βˆ’ 𝜈𝜈𝐴𝐴 )οΏ½(𝑦𝑦)βŒͺ: 𝑦𝑦 ∈ π‘Œπ‘ŒοΏ½ sup πœ‡πœ‡π΄π΄ (π‘₯π‘₯) 𝑖𝑖𝑖𝑖 𝑓𝑓 βˆ’1 (𝑦𝑦) β‰  βˆ… 𝑓𝑓(πœ‡πœ‡π΄π΄ )(𝑦𝑦) = οΏ½π‘₯π‘₯βˆˆπ‘“π‘“ βˆ’1 (𝑦𝑦) 0 π‘œπ‘œπ‘œπ‘œβ„Žπ‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’ sup 𝜎𝜎𝐴𝐴 (π‘₯π‘₯) 𝑖𝑖𝑖𝑖 𝑓𝑓 βˆ’1 (𝑦𝑦) β‰  βˆ… 𝑓𝑓(𝜎𝜎𝐴𝐴 )(𝑦𝑦) = οΏ½π‘₯π‘₯βˆˆπ‘“π‘“ βˆ’1 (𝑦𝑦) 0 π‘œπ‘œπ‘œπ‘œβ„Žπ‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’ inf 𝜈𝜈𝐴𝐴 (π‘₯π‘₯) 𝑖𝑖𝑖𝑖 𝑓𝑓 βˆ’1 (𝑦𝑦) β‰  βˆ… βˆ’1 οΏ½1 βˆ’ 𝑓𝑓(1 βˆ’ 𝜈𝜈𝐴𝐴 )οΏ½(𝑦𝑦) = οΏ½π‘₯π‘₯βˆˆπ‘“π‘“ (𝑦𝑦) 1 π‘œπ‘œπ‘œπ‘œβ„Žπ‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’ For the sake of simplicity, let us use the symbol 𝑓𝑓_(𝜈𝜈𝐴𝐴 ) for οΏ½1 βˆ’ 𝑓𝑓(1 βˆ’ 𝜈𝜈𝐴𝐴 )οΏ½. Corollary 2.7[4]: Let 𝐴𝐴, 𝐴𝐴𝑖𝑖 (𝑖𝑖 ∈ 𝐽𝐽) be FNS’s in 𝑋𝑋, 𝐡𝐡, 𝐡𝐡𝑗𝑗 (𝑗𝑗 ∈ 𝐾𝐾) be FNS’s in π‘Œπ‘Œ, and 𝑓𝑓: 𝑋𝑋 ⟢ π‘Œπ‘Œ a function. a) 𝐴𝐴1 βŠ† 𝐴𝐴2 ⟹ 𝑓𝑓(𝐴𝐴1 ) βŠ† 𝑓𝑓(𝐴𝐴2 ), 𝐡𝐡1 βŠ† 𝐡𝐡2 ⟹ 𝑓𝑓 βˆ’1 (𝐡𝐡1 ) βŠ† 𝑓𝑓 βˆ’1 (𝐡𝐡2 ), b) 𝐴𝐴 βŠ† 𝑓𝑓 βˆ’1 (𝑓𝑓(𝐴𝐴)), and if 𝑓𝑓 is injective then 𝐴𝐴 = 𝑓𝑓 βˆ’1 (𝑓𝑓(𝐴𝐴)) c) 𝑓𝑓 βˆ’1 (𝑓𝑓(𝐡𝐡)) βŠ† 𝐡𝐡, and if 𝑓𝑓 is surjective, then 𝑓𝑓 βˆ’1 (𝑓𝑓(𝐡𝐡)) = 𝐡𝐡 d) 𝑓𝑓 βˆ’1 οΏ½βˆͺ 𝐡𝐡𝑗𝑗 οΏ½ =βˆͺ 𝑓𝑓 βˆ’1 �𝐡𝐡𝑗𝑗 οΏ½, 𝑓𝑓 βˆ’1 �∩ 𝐡𝐡𝑗𝑗 οΏ½ =∩ 𝑓𝑓 βˆ’1 �𝐡𝐡𝑗𝑗 οΏ½, e) 𝑓𝑓(βˆͺ 𝐴𝐴𝑖𝑖 ) =βˆͺ 𝑓𝑓(𝐴𝐴𝑖𝑖 ), 𝑓𝑓(∩ 𝐴𝐴𝑖𝑖 ) βŠ†βˆ© 𝑓𝑓(𝐴𝐴𝑖𝑖 ); and if 𝑓𝑓 is injective, then 𝑓𝑓(∩ 𝐴𝐴𝑖𝑖 ) =∩ 𝑓𝑓(𝐴𝐴𝑖𝑖 ) f) 𝑓𝑓 βˆ’1 (1𝑁𝑁 ) = 1𝑁𝑁 , 𝑓𝑓 βˆ’1 (0𝑁𝑁 ) = 0𝑁𝑁 g) 𝑓𝑓(0𝑁𝑁 ) = 0𝑁𝑁 , 𝑓𝑓(1𝑁𝑁 ) = 1𝑁𝑁 , if 𝑓𝑓 is surjective Β© 2017, IJMA. All Rights Reserved

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Y. Veereswari* / An Introduction to Fuzzy Neutrosophic Topological Spaces / IJMA- 8(3), March-2017.

h) If Δ‘?‘“Δ‘?‘“ is surjective, then ������ Δ‘?‘“Δ‘?‘“(Δ‘??Β΄Δ‘??Β΄) Γ’Š† Δ‘?‘“Δ‘?‘“(Δ‘??Β΄Δ‘??´Ě…). If furthermore, Δ‘?‘“Δ‘?‘“ is injective, then ������ Δ‘?‘“Δ‘?‘“(Δ‘??Β΄Δ‘??Β΄) = Δ‘?‘“Δ‘?‘“(Δ‘??Β΄Δ‘??´Ě…) Γ’ˆ’1 (Δ‘??ΔΎΔ‘??ΔΎ Γ’ˆ’1 ��������� ďż˝ ) = Δ‘?‘“Δ‘?‘“ (Δ‘??ΔΎΔ‘??ΔΎ) i) Δ‘?‘“Δ‘?‘“

3. FUZZY NEUTROSOPHIC TOPOLOGICAL SPACES

Definition 3.1: A fuzzy neutrosophic topology (FNT for short) a non-empty set Δ‘?‘‹Δ‘?‘‹ is a family Δ‘?œ?Δ‘?œ? of fuzzy neutrosophic subsets in Δ‘?‘‹Δ‘?‘‹ satisfying the following axioms. (NT1) 0Δ‘?‘ Δ‘?‘ , 1Δ‘?‘ Δ‘?‘ Γ’ˆˆ Δ‘?œ?Δ‘?œ? (NT2) Δ‘??ΕŸΔ‘??ş1 Γ’ˆΕ  Δ‘??ΕŸΔ‘??ş2 Γ’ˆˆ Δ‘?œ?Δ‘?œ? for any Δ‘??ΕŸΔ‘??ş1 , Δ‘??ΕŸΔ‘??ş2 Γ’ˆˆ Δ‘?œ?Δ‘?œ? (NT3) Γ’ˆŞ Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– Γ’ˆˆ Δ‘?œ?Δ‘?œ?, Γ’ˆ€{Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– : Δ‘?‘–Δ‘?‘– Γ’ˆˆ Δ‘??˝đ??˝} Γ’Š† Δ‘?œ?Δ‘?œ?

In this case the pair (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) is called a Fuzzy neutrosophic topological space (FNTS for short) and any fuzzy neutrosophic set in Δ‘?œ?Δ‘?œ? is known as fuzzy neutrosophic open set (FNOS for short) in Δ‘?‘‹Δ‘?‘‹. The elements of Δ‘?œ?Δ‘?œ? are called open fuzzy neutrosophic sets. The complement of FNOS in the FNTS (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) is called fuzzy neutrosophic closed set (FNCS for short).

Example 3.2: Let Δ‘?‘‹Δ‘?‘‹ = {Δ‘?‘ŽΔ‘?‘Ž, Δ‘?‘?Δ‘?‘?, Δ‘?‘?Δ‘?‘?} and consider the value of Δ‘??Β΄Δ‘??Β΄, Δ‘??ΔΎΔ‘??ΔΎ, Δ‘??Ε›Δ‘??Ε›, Δ‘??Λ‡Δ‘??Λ‡ are defined in Example 2.4, the family Δ‘?œ?Δ‘?œ? = {0Δ‘?‘ Δ‘?‘ , 1Δ‘?‘ Δ‘?‘ , Δ‘??Β΄Δ‘??Β΄, Δ‘??ΔΎΔ‘??ΔΎ, Δ‘??Ε›Δ‘??Ε›, Δ‘??Λ‡Δ‘??Λ‡} of FNS in Δ‘?‘‹Δ‘?‘‹. Then (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) is FNTS on Δ‘?‘‹Δ‘?‘‹.

Example 3.3: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) be a fuzzy topological space such that Δ‘?œ?Δ‘?œ? is not indiscrete. Suppose now that Δ‘?œ?Δ‘?œ? = {0, 1} Γ’ˆŞ {Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– : Δ‘?‘–Δ‘?‘– Γ’ˆˆ Δ‘??˝đ??˝}. Then we can construct two FNTΓ’€™s on Δ‘?‘‹Δ‘?‘‹ as follows: a) Δ‘?œ?Δ‘?œ?1 = {0Δ‘?‘ Δ‘?‘ , 1Δ‘?‘ Δ‘?‘ } Γ’ˆŞ {Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– , Δ‘?œŽΔ‘?œŽ(Δ‘?‘Δ½Δ‘?‘Δ½), 0Γ’ŒŞ: Δ‘?‘–Δ‘?‘– Γ’ˆˆ Δ‘??˝đ??˝}, b) Δ‘?œ?Δ‘?œ?2 = {0Δ‘?‘ Δ‘?‘ , 1Δ‘?‘ Δ‘?‘ } Γ’ˆŞ {Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, 0, Δ‘?œŽΔ‘?œŽ(Δ‘?‘Δ½Δ‘?‘Δ½), Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– Γ’ŒŞ: Δ‘?‘–Δ‘?‘– Γ’ˆˆ Δ‘??˝đ??˝}

Proposition 3.4: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) be a FNTS on Δ‘?‘‹Δ‘?‘‹. Then we can also construct several FNTSΓ’€™s on Δ‘?‘‹Δ‘?‘‹ in the following way Δ‘?œ?Δ‘?œ?0,1 = {[ ]Δ‘??ΕŸΔ‘??ş: Δ‘??ΕŸΔ‘??ş Γ’ˆˆ Δ‘?œ?Δ‘?œ?}, b) Δ‘?œ?Δ‘?œ?0,2 = {Γ’ŒΕ  Γ’ŒΕžΔ‘??ΕŸΔ‘??ş: Δ‘??ΕŸΔ‘??ş Γ’ˆˆ Δ‘?œ?Δ‘?œ?}

Definition 3.5: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?1 ), (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?2 ) be two FNTSΓ’€™s on Δ‘?‘‹Δ‘?‘‹. Then Δ‘?œ?Δ‘?œ?1 is said to be contained in Δ‘?œ?Δ‘?œ?2 (in symbols, Δ‘?œ?Δ‘?œ?1 Γ’Š† Δ‘?œ?Δ‘?œ?2 ), if Δ‘??ΕŸΔ‘??ş Γ’ˆˆ Δ‘?œ?Δ‘?œ?2 for each Δ‘??ΕŸΔ‘??ş Γ’ˆˆ Δ‘?œ?Δ‘?œ?1 . In this case, we also say that Δ‘?œ?Δ‘?œ?1 is coarser than Δ‘?œ?Δ‘?œ?2 .

Proposition 3.6: Let {Δ‘?œ?Δ‘?œ?Δ‘?‘–Δ‘?‘– : Δ‘?‘–Δ‘?‘– Γ’ˆˆ Δ‘??˝đ??˝} be a family of FNTΓ’€™s on Δ‘?‘‹Δ‘?‘‹. Then Γ’‹‚Δ‘?‘–Δ‘?‘–Γ’ˆˆΔ‘??˝đ??˝ Δ‘?œ?Δ‘?œ?Δ‘?‘–Δ‘?‘– is FNT on Δ‘?‘‹Δ‘?‘‹. Furthermore, Γ’‹‚Δ‘?‘–Δ‘?‘–Γ’ˆˆΔ‘??˝đ??˝ Δ‘?œ?Δ‘?œ?Δ‘?‘–Δ‘?‘– is the coarsest FNT on Δ‘?‘‹Δ‘?‘‹ containing all Δ‘?œ?Δ‘?œ?Δ‘?‘–Δ‘?‘– Γ’€™s.

Definition 3.7: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) be a FNTS on Δ‘?‘‹Δ‘?‘‹. a) A family Δ‘?›˝đ?›˝ Γ’Š† Δ‘?œ?Δ‘?œ? is called a base for (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) iff each member of Δ‘?œ?Δ‘?œ? can written as a union of elements of Δ‘?›˝đ?›˝. b) A family Δ‘?›ΕΎΔ‘?›ΕΎ Γ’Š† Δ‘?œ?Δ‘?œ? is called a subbase for (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) iff the family of finite intersections of elements in Δ‘?›ΕΎΔ‘?›ΕΎ forms a base for (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?). In this case the FNT Δ‘?œ?Δ‘?œ? is said to be generated by Δ‘?›ΕΎΔ‘?›ΕΎ.

Definition 3.8: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) be a FNTS and Δ‘??Β΄Δ‘??Β΄ = 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 be a FNS in Δ‘?‘‹Δ‘?‘‹. Then the fuzzy interior and fuzzy closure of Δ‘??Β΄Δ‘??Β΄ are defined by Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) =Γ’ˆΕ  {Δ‘??ΕΎΔ‘??ΕΎ: Δ‘??ΕΎΔ‘??ΕΎis a FNCS in Δ‘?‘‹Δ‘?‘‹ and Δ‘??Β΄Δ‘??Β΄ Γ’Š† Δ‘??ΕΎΔ‘??ΕΎ}, Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) =Γ’ˆŞ {Δ‘??ΕŸΔ‘??ş: Δ‘??ΕŸΔ‘??şis a FNOS in Δ‘?‘‹Δ‘?‘‹ and Δ‘??ΕŸΔ‘??ş Γ’Š† Δ‘??Β΄Δ‘??Β΄} Now that Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) is a FNCS and Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) is a FNOS in Δ‘?‘‹Δ‘?‘‹. Further, a) Δ‘??Β΄Δ‘??Β΄ is a FNCS in Δ‘?‘‹Δ‘?‘‹ iff Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) = Δ‘??Β΄Δ‘??Β΄ b) Δ‘??Β΄Δ‘??Β΄ is a FNOS in Δ‘?‘‹Δ‘?‘‹ iff Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) = Δ‘??Β΄Δ‘??Β΄

Example 3.9: Consider the FNTS (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) is defined in Example 3.2. If Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? , ďż˝,ďż˝ , , ďż˝,ďż˝ , , ďż˝ÒŒŞ Δ‘??Ε‘Δ‘??Ε‘ = Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, ďż˝ , 0.4 0.3 0.2 0.2 0.3 0.3 0.4 0.5 0.4 Then, Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) = Δ‘??Λ‡Δ‘??Λ‡ and Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) = 1Δ‘?‘ Δ‘?‘ .

Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄). Proposition 3.10: For any FNS Δ‘??Β΄Δ‘??Β΄ in (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?), we have Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??´Ě…) = ��������� Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄), Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??´Ě…) = �������

Proposition 3.11: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) be a FNTS and Δ‘??Β΄Δ‘??Β΄, Δ‘??ΔΎΔ‘??ΔΎ be FNSΓ’€™s in Δ‘?‘‹Δ‘?‘‹. Then the following properties hold: a) Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) Γ’Š† Δ‘??Β΄Δ‘??Β΄, Δ‘??Β΄Δ‘??Β΄ Γ’Š† Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) b) Δ‘??Β΄Δ‘??Β΄ Γ’Š† Δ‘??ΔΎΔ‘??ΔΎ Γ’&#x;Ε‘ Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) Γ’Š† Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??ΔΎΔ‘??ΔΎ), Δ‘??Β΄Δ‘??Β΄ Γ’Š† Δ‘??ΔΎΔ‘??ΔΎ Γ’&#x;Ε‘ Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) Γ’Š† Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??ΔΎΔ‘??ΔΎ) c) Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–ďż˝đ?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄)ďż˝ = Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄), Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?ďż˝đ?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄)ďż˝ = Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) d) Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄ Γ’ˆΕ  Δ‘??ΔΎΔ‘??ΔΎ) = Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) Γ’ˆΕ  Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??ΔΎΔ‘??ΔΎ), Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄ Γ’ˆŞ Δ‘??ΔΎΔ‘??ΔΎ) = Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) Γ’ˆŞ Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??ΔΎΔ‘??ΔΎ) e) Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(1Δ‘?‘ Δ‘?‘ ) = 1Δ‘?‘ Δ‘?‘ , Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(0Δ‘?‘ Δ‘?‘ ) = 0Δ‘?‘ Δ‘?‘ Γ‚Ε  2017, IJMA. All Rights Reserved

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Y. Veereswari* / An Introduction to Fuzzy Neutrosophic Topological Spaces / IJMA- 8(3), March-2017.

Proposition 3.12: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) be a FNTS. If Δ‘??Β΄Δ‘??Β΄ = 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 be a FNS in Δ‘?‘‹Δ‘?‘‹. Then we have a) Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) Γ’Š† Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?œ?Δ‘?œ?1 (Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ ), Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?œ?Δ‘?œ?1 (Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ ), Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?œ?Δ‘?œ?2 (Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ )Γ’ŒŞ Γ’Š† Δ‘??Β΄Δ‘??Β΄ b) Δ‘??Β΄Δ‘??Β΄ Γ’Š† Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?œ?Δ‘?œ?2 (Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ ), Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?œ?Δ‘?œ?2 (Δ‘?œŽΔ‘?œŽΔ‘??Β΄Δ‘??Β΄ ), Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?œ?Δ‘?œ?1 (Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ )Γ’ŒŞ Γ’Š† Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??Β΄Δ‘??Β΄) where Δ‘?œ?Δ‘?œ?1 and Δ‘?œ?Δ‘?œ?2 are FTS on Δ‘?‘‹Δ‘?‘‹ defined by Δ‘?œ?Δ‘?œ?1 ={Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ : Δ‘??ΕŸΔ‘??ş Γ’ˆˆ Δ‘?œ?Δ‘?œ?}, Δ‘?œ?Δ‘?œ?2 = {1 Γ’ˆ’ Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ : Δ‘??ΕŸΔ‘??ş Γ’ˆˆ Δ‘?œ?Δ‘?œ?} Corollary 3.13: Let Δ‘??Β΄Δ‘??Β΄ = 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 be a FNS in (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) a) If Δ‘??Β΄Δ‘??Β΄ is a FNCS, then Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ is fuzzy closed in (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?2 ) and Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ is fuzzy open in (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?1 ). b) If Δ‘??Β΄Δ‘??Β΄ is a FNOS, then Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ is fuzzy open in (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?1 ) and Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ is fuzzy closed in (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?2 ).

Example 3.14: Consider the FNTS (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) is defined in Example 3.2. If Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘??Ε‘Δ‘??Ε‘ = Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, ďż˝ , , ďż˝,ďż˝ , , ďż˝,ďż˝ , , ďż˝ÒŒŞ 0.4 0.5 0.4 0.4 0.3 0.2 0.2 0.3 0.3 Then, Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??Β΄Δ‘??Β΄) = Δ‘??Λ‡Δ‘??Λ‡. Noting that we have Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? , ďż˝,ďż˝ , , ďż˝,ďż˝ , , ďż˝,ďż˝ , , �� Δ‘?œ?Δ‘?œ?1 = ďż˝0, 1, ďż˝ , 0.3 0.4 0.5 0.5 0.5 0.4 0.5 0.5 0.5 0.3 0.4 0.4 Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?œ?Δ‘?œ?2 = ďż˝0, 1, ďż˝ , , ďż˝,ďż˝ , , ďż˝,ďż˝ , , ďż˝,ďż˝ , , �� 0.7 0.8 0.8 0.8 0.6 0.6 0.8 0.8 0.8 0.7 0.6 0.6 Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? Δ‘?‘ŽΔ‘?‘Ž Δ‘?‘?Δ‘?‘? Δ‘?‘?Δ‘?‘? and Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?œ?Δ‘?œ?1 (Δ‘?œ‡Δ‘?œ‡Δ‘??Β΄Δ‘??Β΄ ) = ďż˝ , , ďż˝ , Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?œ?Δ‘?œ?2 (Δ‘?œˆΔ‘?œˆΔ‘??Β΄Δ‘??Β΄ ) = ďż˝ , , ďż˝ 0.3 0.4 0.4

4. FUZZY NEUTROSOPHIC CONTINUITY

0.3 0.4 0.4

Definition 4.1: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) and (Δ‘?‘ŒΔ‘?‘Œ, Δ‘?œ‘Δ‘?œ‘) be two FNTSs and let Δ‘?‘“Δ‘?‘“: Δ‘?‘‹Δ‘?‘‹ Γ’†’ Δ‘?‘ŒΔ‘?‘Œ be a function. Then Δ‘?‘“Δ‘?‘“ is said to be fuzzy continuous iff the preimage of each FNS in Δ‘?œ‘Δ‘?œ‘ is a FNS in Δ‘?œ?Δ‘?œ?. Definition 4.2: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) and (Δ‘?‘ŒΔ‘?‘Œ, Δ‘?œ‘Δ‘?œ‘) be two FNTSs and let Δ‘?‘“Δ‘?‘“: Δ‘?‘‹Δ‘?‘‹ Γ’†’ Δ‘?‘ŒΔ‘?‘Œ be a function. Then Δ‘?‘“Δ‘?‘“ is said to be fuzzy open iff the image of each FNS in Δ‘?œ?Δ‘?œ? is a FNS in Δ‘?œ‘Δ‘?œ‘. Example 4.3: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?0 ) and (Δ‘?‘ŒΔ‘?‘Œ, Δ‘?œ‘Δ‘?œ‘0 ) be two fuzzy topological spaces. a) If Δ‘?‘“Δ‘?‘“: Δ‘?‘‹Δ‘?‘‹ Γ’†’ Δ‘?‘ŒΔ‘?‘Œ is fuzzy continuous in the usual sense, then in this case, Δ‘?‘“Δ‘?‘“ is fuzzy continuous iff the preimage of each FNS in Δ‘?œ‘Δ‘?œ‘0 is a FNS in Δ‘?œ?Δ‘?œ?0 . Consider the FNTs on Δ‘?‘‹Δ‘?‘‹ and Δ‘?‘ŒΔ‘?‘Œ respectively, as follows: Δ‘?œ?Δ‘?œ? = {Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş , 0, 1 Γ’ˆ’ Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş Γ’ŒŞ: Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş Γ’ˆˆ Δ‘?œ?Δ‘?œ?0 }, Δ‘?œ‘Δ‘?œ‘ = {Γ’ŒΕ Δ‘?‘ΕšΔ‘?‘Ś, Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ , 0, 1 Γ’ˆ’ Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ Γ’ŒŞ: Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ Γ’ˆˆ Δ‘?œ‘Δ‘?œ‘0 } In this case we have for each Γ’ŒΕ Δ‘?‘ΕšΔ‘?‘Ś, Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ , 0, 1 Γ’ˆ’ Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ Γ’ŒŞ Γ’ˆˆ Δ‘?œ‘Δ‘?œ‘, Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ Γ’ˆˆ Δ‘?œ‘Δ‘?œ‘0 , Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 Γ’ŒΕ Δ‘?‘ΕšΔ‘?‘Ś, Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ , 0, 1 Γ’ˆ’ Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ Γ’ŒŞ = Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ ), Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (0), Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (1 Γ’ˆ’ Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ )Γ’ŒŞ = Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ ), Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (0), 1 Γ’ˆ’ Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (Δ‘?›ΕΌΔ‘?›ΕΌΔ‘??Ε₯Δ‘??Ε₯ )Γ’ŒŞ Γ’ˆˆ Δ‘?œ?Δ‘?œ? b) Let Δ‘?‘“Δ‘?‘“: Δ‘?‘‹Δ‘?‘‹ Γ’†’ Δ‘?‘ŒΔ‘?‘Œ be a fuzzy open in the usual sense, then Δ‘?‘“Δ‘?‘“ is fuzzy openaccording to Definition 4.2. In this case we have, for each Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş , 0, 1 Γ’ˆ’ Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş Γ’ŒŞ Γ’ˆˆ Δ‘?œ?Δ‘?œ?, Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş Γ’ˆˆ Δ‘?œ?Δ‘?œ?0 , Δ‘?‘“Δ‘?‘“Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş , 0, 1 Γ’ˆ’ Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş Γ’ŒŞ = Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?‘“Δ‘?‘“(Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş ), 0, Δ‘?‘“Δ‘?‘“_(1 Γ’ˆ’ Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş )Γ’ŒŞ = Γ’ŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?‘“Δ‘?‘“(Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş ), 0, 1 Γ’ˆ’ Δ‘?‘“Δ‘?‘“_(Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ş )Γ’ŒŞ Γ’ˆˆ Δ‘?œ‘Δ‘?œ‘

Proposition 4.4: Δ‘?‘“Δ‘?‘“: (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) Γ’†’ (Δ‘?‘ŒΔ‘?‘Œ, Δ‘?œ‘Δ‘?œ‘) is fuzzy continuous iff the preimage of each FNCS in Δ‘?œ‘Δ‘?œ‘ is a FNCS in Δ‘?œ?Δ‘?œ?. Proposition 4.5: The following are equivalent to each other a) Δ‘?‘“Δ‘?‘“: (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) Γ’†’ (Δ‘?‘ŒΔ‘?‘Œ, Δ‘?œ‘Δ‘?œ‘) is fuzzy continuous b) Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘??ΔΎΔ‘??ΔΎ)) Γ’Š† Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–Δ‘?‘–(Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (Δ‘??ΔΎΔ‘??ΔΎ)) for each FNS Δ‘??ΔΎΔ‘??ΔΎ in Δ‘?‘ŒΔ‘?‘Œ. c) Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (Δ‘??ΔΎΔ‘??ΔΎ)) Γ’Š† Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (Δ‘?‘?Δ‘?‘?Δ‘?‘?Δ‘?‘?(Δ‘??ΔΎΔ‘??ΔΎ))for each FNS Δ‘??ΔΎΔ‘??ΔΎ in Δ‘?‘ŒΔ‘?‘Œ.

Example 4.6: Let (Δ‘?‘ŒΔ‘?‘Œ, Δ‘?œ‘Δ‘?œ‘) be a FNTS and Δ‘?‘‹Δ‘?‘‹ be a non-empty set and Δ‘?‘“Δ‘?‘“: Δ‘?‘‹Δ‘?‘‹ Γ’†’ Δ‘?‘ŒΔ‘?‘Œ a function. In this case Δ‘?œ?Δ‘?œ? = {Δ‘?‘“Δ‘?‘“ Γ’ˆ’1 (Δ‘??Ε₯Δ‘??Ε₯): Δ‘??Ε₯Δ‘??Ε₯ Γ’ˆˆ Δ‘?œ‘Δ‘?œ‘} is a FNT on Δ‘?‘‹Δ‘?‘‹. Indeed, Δ‘?‘Δ„Δ‘?‘Δ„ is the coarsest FNT on Δ‘?‘‹Δ‘?‘‹ which makes the function Δ‘?‘“Δ‘?‘“: Δ‘?‘‹Δ‘?‘‹ Γ’†’ Δ‘?‘ŒΔ‘?‘Œ continuous. One may call the FNT Δ‘?‘Δ„Δ‘?‘Δ„ on Δ‘?‘‹Δ‘?‘‹ the initial FNT with respect to Δ‘?‘“Δ‘?‘“. 5. FUZZY NEUTROSOPHIC COMPACTNESS

Definition 5.1: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) be a FNTS. a) If a family Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– = ďż˝ÒŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– , Δ‘?œŽΔ‘?œŽΔ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– , Δ‘?œˆΔ‘?œˆΔ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– Γ’ŒŞ: Δ‘?‘–Δ‘?‘– Γ’ˆˆ Δ‘??˝đ??˝ďż˝ of FNOS is Δ‘?‘‹Δ‘?‘‹ satisfy the condition Γ’ˆŞ Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– = 1Δ‘?‘ Δ‘?‘ , then it is called a fuzzy open cover of Δ‘?‘‹Δ‘?‘‹. A finite subfamily of fuzzy open cover Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– of Δ‘?‘‹Δ‘?‘‹, which is also a fuzzy open cover of Δ‘?‘‹Δ‘?‘‹ is called a finite subcover of Δ‘??ΕŸΔ‘??ΕŸΔ‘?‘–Δ‘?‘– . b) A family Δ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– = ďż˝ÒŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– , Δ‘?œŽΔ‘?œŽΔ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– , Δ‘?œˆΔ‘?œˆΔ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– Γ’ŒŞ: Δ‘?‘–Δ‘?‘– Γ’ˆˆ Δ‘??˝đ??˝ďż˝ of FNCSs in Δ‘?‘‹Δ‘?‘‹ satisfies the finite intersection property iff every finite subfamily ďż˝ÒŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– , Δ‘?œŽΔ‘?œŽΔ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– , Δ‘?œˆΔ‘?œˆΔ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– Γ’ŒŞ: Δ‘?‘–Δ‘?‘– = 1, 2, Γ’€Ś , Δ‘?‘›Δ‘?‘›ďż˝ of the family satisfies the condition Γ’‹‚Δ‘?‘›Δ‘?‘›Δ‘?‘–Δ‘?‘–=1ďż˝ÒŒΕ Δ‘?‘Δ½Δ‘?‘Δ½, Δ‘?œ‡Δ‘?œ‡Δ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– , Δ‘?œŽΔ‘?œŽΔ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– , Δ‘?œˆΔ‘?œˆΔ‘??ΕΎΔ‘??ΕΎΔ‘?‘–Δ‘?‘– Γ’ŒŞďż˝ Γ’‰ 0Δ‘?‘ Δ‘?‘ .

Definition 5.2: A FNTS (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) is fuzzy compact iff every fuzzy open cover of Δ‘?‘‹Δ‘?‘‹ has a finite subcover. Γ‚Ε  2017, IJMA. All Rights Reserved

147

Y. Veereswari* / An Introduction to Fuzzy Neutrosophic Topological Spaces / IJMA- 8(3), March-2017.

Example 5.3: An FNTS (𝑋𝑋, 𝜏𝜏), where 𝑋𝑋 = {1, 2, 3}, 𝐺𝐺𝑛𝑛 = 〈π‘₯π‘₯, οΏ½

1 𝑛𝑛

𝑛𝑛+1

,

2 𝑛𝑛

𝑛𝑛+2

,

3 𝑛𝑛

𝑛𝑛+3

1 2 3 1 2 3 οΏ½ , οΏ½ 𝑛𝑛+1 , 𝑛𝑛+2 , 𝑛𝑛+3 οΏ½ , οΏ½ 1 , 1 , 1 οΏ½βŒͺ 𝑛𝑛+2

𝑛𝑛+3

𝑛𝑛+4

𝑛𝑛+2

𝑛𝑛+3

𝑛𝑛+4

and 𝜏𝜏 = {0𝑁𝑁 , 1𝑁𝑁 } βˆͺ {𝐺𝐺𝑛𝑛 : 𝑛𝑛 ∈ β„•}. Note that β‹ƒπ‘›π‘›βˆˆβ„• 𝐺𝐺𝑛𝑛 is an open cover for 𝑋𝑋, but this cover has no finite subcover. Consider 1 2 3 1 2 3 1 2 3 , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ 𝐺𝐺1 = 〈π‘₯π‘₯, οΏ½ , 0.5 0.3 0.25 0.7 0.75 0.8 0.3 0.25 0.2 1 2 3 1 2 3 1 2 3 𝐺𝐺2 = 〈π‘₯π‘₯, οΏ½ , , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ 0.7 0.5 0.4 0.75 0.8 0.8 0.25 0.2 0.2 1 2 3 1 2 3 1 2 3 𝐺𝐺3 = 〈π‘₯π‘₯, οΏ½ , , οΏ½,οΏ½ , , οΏ½,οΏ½ , , οΏ½βŒͺ 0.75 0.6 0.5 0.8 0.8 0.9 0.2 0.2 0.1 and observe that 𝐺𝐺1 βˆͺ 𝐺𝐺2 βˆͺ 𝐺𝐺3 = 𝐺𝐺3 . So, for any finite subcollection �𝐺𝐺𝑛𝑛 𝑖𝑖 : 𝑖𝑖 ∈ I, where I is a finite subset of β„•οΏ½, ⋃𝑛𝑛 𝑖𝑖 ∈I 𝐺𝐺𝑛𝑛 𝑖𝑖 = πΊπΊπ‘šπ‘š β‰  1𝑁𝑁 , where π‘šπ‘š = π‘šπ‘šπ‘šπ‘šπ‘šπ‘š{𝑛𝑛𝑖𝑖 : 𝑛𝑛𝑖𝑖 ∈ I}. therefore the FNTS (𝑋𝑋, 𝜏𝜏) is not compact. Proposition 5.4: Let (𝑋𝑋, 𝜏𝜏) be a FNTS on 𝑋𝑋. Then (𝑋𝑋, 𝜏𝜏) is fuzzy compact iff the FNTS �𝑋𝑋, 𝜏𝜏0,1 οΏ½ is fuzzy compact.

Proof: Let (𝑋𝑋, 𝜏𝜏) be fuzzy compact and consider fuzzy open cover οΏ½[ ]𝐺𝐺𝑗𝑗 : 𝑗𝑗 ∈ 𝐾𝐾� of 𝑋𝑋 in �𝑋𝑋, 𝜏𝜏0,1 οΏ½. Since βˆͺ οΏ½[ ]𝐺𝐺𝑗𝑗 οΏ½ = 1𝑁𝑁 . We obtain ∨ πœ‡πœ‡πΊπΊπ‘—π‘— = 1,∨ πœŽπœŽπΊπΊπ‘—π‘— = 1 and hence, by πœˆπœˆπΊπΊπ‘—π‘— ≀ 1 βˆ’ πœ‡πœ‡πΊπΊπ‘—π‘— ⟹∧ πœˆπœˆπΊπΊπ‘—π‘— ≀ 1 βˆ’βˆ¨ πœ‡πœ‡πΊπΊπ‘—π‘— = 0 ⟹∧ πœˆπœˆπΊπΊπ‘—π‘— = 0, we deduce βˆͺ 𝐺𝐺𝑗𝑗 = 1𝑁𝑁 . Since (𝑋𝑋, 𝜏𝜏) is fuzzy compact there exist 𝐺𝐺1 , 𝐺𝐺2 , 𝐺𝐺3 , … 𝐺𝐺𝑛𝑛 such that ⋃𝑛𝑛𝑖𝑖=1 𝐺𝐺𝑖𝑖 = 1𝑁𝑁 from which we obtain ⋁𝑛𝑛𝑖𝑖=1 πœ‡πœ‡πΊπΊπ‘–π‘– = 1, ⋁𝑛𝑛𝑖𝑖=1 πœŽπœŽπΊπΊπ‘–π‘– = 1 and ⋀𝑛𝑛𝑖𝑖=1οΏ½1 βˆ’ πœ‡πœ‡πΊπΊπ‘–π‘– οΏ½ = 0. That is, �𝑋𝑋, 𝜏𝜏0,1 οΏ½ is fuzzy compact.

Suppose that �𝑋𝑋, 𝜏𝜏0,1 οΏ½ is fuzzy compact and consider a fuzzy open cover �𝐺𝐺𝑗𝑗 : 𝑗𝑗 ∈ 𝐾𝐾� of 𝑋𝑋 in (𝑋𝑋, 𝜏𝜏). Since βˆͺ 𝐺𝐺𝑗𝑗 = 1𝑁𝑁 , we obtain ∨ πœ‡πœ‡πΊπΊπ‘—π‘— = 1,∨ πœŽπœŽπΊπΊπ‘—π‘— = 1 and 1 βˆ’βˆ¨ πœ‡πœ‡πΊπΊπ‘—π‘— = 0. Since �𝑋𝑋, 𝜏𝜏0,1 οΏ½ is fuzzy compact, there exist 𝐺𝐺1 , 𝐺𝐺2 , 𝐺𝐺3 , … 𝐺𝐺𝑛𝑛 such that ⋃𝑛𝑛𝑖𝑖=1οΏ½[ ]𝐺𝐺𝑗𝑗 οΏ½ = 1𝑁𝑁 , that is, ⋁𝑛𝑛𝑖𝑖=1 πœ‡πœ‡πΊπΊπ‘–π‘– = 1, ⋁𝑛𝑛𝑖𝑖=1 πœŽπœŽπΊπΊπ‘–π‘– = 1 and ⋀𝑛𝑛𝑖𝑖=1οΏ½1 βˆ’ πœ‡πœ‡πΊπΊπ‘–π‘– οΏ½ = 0. Hence πœ‡πœ‡πΊπΊπ‘–π‘– ≀ 1 βˆ’ πœˆπœˆπΊπΊπ‘–π‘– ⟹ 1 = ⋁𝑛𝑛𝑖𝑖=1 πœ‡πœ‡πΊπΊπ‘–π‘– ≀ 1 βˆ’ ⋀𝑛𝑛𝑖𝑖=1 πœˆπœˆπΊπΊπ‘–π‘– = 0. ⟹ ⋀𝑛𝑛𝑖𝑖=1 πœˆπœˆπΊπΊπ‘–π‘– = 0. Hence ⋃𝑛𝑛𝑖𝑖=1 𝐺𝐺𝑖𝑖 = 1𝑁𝑁 . Therefore (𝑋𝑋, 𝜏𝜏) is fuzzy compact. Corollary 5.5: Let (𝑋𝑋, 𝜏𝜏) and (π‘Œπ‘Œ, πœ‘πœ‘) be two FNTSs and let 𝑓𝑓: 𝑋𝑋 β†’ π‘Œπ‘Œ be a fuzzy continuous surjection. If (𝑋𝑋, 𝜏𝜏) is fuzzy compact, then so is (π‘Œπ‘Œ, πœ‘πœ‘).

Corollary 5.6: A FNTS (𝑋𝑋, 𝜏𝜏) is fuzzy compact iff every family �〈π‘₯π‘₯, πœ‡πœ‡πΎπΎπ‘–π‘– , πœŽπœŽπΎπΎπ‘–π‘– , πœˆπœˆπΎπΎπ‘–π‘– βŒͺ: 𝑖𝑖 ∈ 𝐽𝐽� of FNCSs in 𝑋𝑋 having the FNP has a non-empty intersection.

Definition 5.7: a) Let (𝑋𝑋, 𝜏𝜏) be a FNTS and 𝐴𝐴 a FNS in 𝑋𝑋. If a family 𝐺𝐺𝑖𝑖 = �〈π‘₯π‘₯, πœ‡πœ‡πΊπΊπ‘–π‘– , πœŽπœŽπΊπΊπ‘–π‘– , πœˆπœˆπΊπΊπ‘–π‘– βŒͺ: 𝑖𝑖 ∈ 𝐽𝐽� of FNOSs in 𝑋𝑋 satisfies the condition 𝐴𝐴 βŠ†βˆͺ 𝐺𝐺𝑖𝑖 , then it is called a fuzzy open cover of 𝐴𝐴. A finite subfamily of the fuzzy open cover 𝐺𝐺𝑖𝑖 of 𝐴𝐴, which is also fuzzy open cover of 𝐴𝐴, is called a finite subcover of 𝐺𝐺𝑖𝑖 . b) A FNS 𝐴𝐴 = 〈π‘₯π‘₯, πœ‡πœ‡π΄π΄ , 𝜎𝜎𝐴𝐴 , 𝜈𝜈𝐴𝐴 βŒͺ in a FNTS (𝑋𝑋, 𝜏𝜏) is called fuzzy compact iff every fuzzy open cover of 𝐴𝐴 has a finite subcover. Corollary 5.8: A FNS 𝐴𝐴 = 〈π‘₯π‘₯, πœ‡πœ‡π΄π΄ , 𝜎𝜎𝐴𝐴 , 𝜈𝜈𝐴𝐴 βŒͺ in a FNTS (𝑋𝑋, 𝜏𝜏) is fuzzy compact iff for each family 𝒒𝒒 = {𝐺𝐺𝑖𝑖 : 𝑖𝑖 ∈ 𝐽𝐽} where 𝐺𝐺𝑖𝑖 = �〈π‘₯π‘₯, πœ‡πœ‡πΊπΊπ‘–π‘– , πœŽπœŽπΊπΊπ‘–π‘– , πœˆπœˆπΊπΊπ‘–π‘– βŒͺ: 𝑖𝑖 ∈ 𝐽𝐽�, of FNOS in 𝑋𝑋 with properties πœ‡πœ‡π΄π΄ ≀ β‹π‘–π‘–βˆˆπ½π½ πœ‡πœ‡πΊπΊπ‘–π‘– , 𝜎𝜎𝐴𝐴 ≀ β‹π‘–π‘–βˆˆπ½π½ πœŽπœŽπΊπΊπ‘–π‘– and 1 βˆ’ 𝜈𝜈𝐴𝐴 ≀ β‹π‘–π‘–βˆˆπ½π½ οΏ½1 βˆ’ πœˆπœˆπΊπΊπ‘–π‘– οΏ½ there exists a finite subfamily {𝐺𝐺𝑖𝑖 : 1, 2, … , 𝑛𝑛} of 𝒒𝒒 such that πœ‡πœ‡π΄π΄ ≀ ⋁𝑛𝑛𝑖𝑖=1 πœ‡πœ‡πΊπΊπ‘–π‘– , 𝜎𝜎𝐴𝐴 ≀ ⋁𝑛𝑛𝑖𝑖=1 πœŽπœŽπΊπΊπ‘–π‘– and 1 βˆ’ 𝜈𝜈𝐴𝐴 ≀ ⋁𝑛𝑛𝑖𝑖=1οΏ½1 βˆ’ πœˆπœˆπΊπΊπ‘–π‘– οΏ½.

Example 5.9: Let 𝑋𝑋 = 𝐼𝐼 and consider the FNSs (𝐺𝐺𝑛𝑛 )𝑛𝑛𝑛𝑛 𝑍𝑍2 , where 𝐺𝐺𝑛𝑛 = 〈π‘₯π‘₯, πœ‡πœ‡πΊπΊπ‘›π‘› , πœŽπœŽπΊπΊπ‘›π‘› , πœˆπœˆπΊπΊπ‘›π‘› βŒͺ, 𝑛𝑛 = 2, 3, …. and 𝐺𝐺 = 〈π‘₯π‘₯, πœ‡πœ‡πΊπΊ , 𝜎𝜎𝐺𝐺 , 𝜈𝜈𝐺𝐺 βŒͺ defined by 0.7, 𝑖𝑖𝑖𝑖 π‘₯π‘₯ = 0 0.8, 𝑖𝑖𝑖𝑖 π‘₯π‘₯ = 0 0.1, 𝑖𝑖𝑖𝑖 π‘₯π‘₯ = 0 ⎧ ⎧ ⎧ βŽͺ𝑛𝑛𝑛𝑛, 𝑖𝑖𝑖𝑖 0 < π‘₯π‘₯ < 1 βŽͺ(𝑛𝑛 + 1)π‘₯π‘₯, 𝑖𝑖𝑖𝑖 0 < π‘₯π‘₯ < 1 βŽͺ 1 βˆ’ 𝑛𝑛𝑛𝑛, 𝑖𝑖𝑖𝑖 0 < π‘₯π‘₯ < 1 πœ‡πœ‡πΊπΊπ‘›π‘› (π‘₯π‘₯) = πœˆπœˆπΊπΊπ‘›π‘› = 𝑛𝑛 , πœŽπœŽπΊπΊπ‘›π‘› (π‘₯π‘₯) = 𝑛𝑛 , 𝑛𝑛 ⎨ ⎨ ⎨ 1 1 1 βŽͺ 1, 𝑖𝑖𝑖𝑖 < π‘₯π‘₯ ≀ 1 βŽͺ 1, βŽͺ 0, 𝑖𝑖𝑖𝑖 < π‘₯π‘₯ ≀ 1 𝑖𝑖𝑖𝑖 < π‘₯π‘₯ ≀ 1 ⎩ ⎩ ⎩ 𝑛𝑛 𝑛𝑛 𝑛𝑛 0.7 𝑖𝑖𝑖𝑖 π‘₯π‘₯ = 0 0.8 𝑖𝑖𝑖𝑖 π‘₯π‘₯ = 0 and πœ‡πœ‡πΊπΊ = οΏ½ , 𝜎𝜎 = οΏ½ , 1, π‘œπ‘œπ‘œπ‘œβ„Žπ‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’ 𝐺𝐺 1, π‘œπ‘œπ‘œπ‘œβ„Žπ‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’ 0.1 𝑖𝑖𝑖𝑖 π‘₯π‘₯ = 0 . 𝜈𝜈𝐺𝐺 = οΏ½ 1, π‘œπ‘œπ‘œπ‘œβ„Žπ‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’π‘’ Then 𝜏𝜏 = {0𝑁𝑁 , 1𝑁𝑁 , 𝐺𝐺} βˆͺ {𝐺𝐺𝑛𝑛 : 𝑛𝑛 ∈ 𝑍𝑍2 } is a FNT on 𝑋𝑋, and consider the FNSs 𝐢𝐢𝛼𝛼 ,𝛽𝛽 ,𝛾𝛾 in (𝑋𝑋, 𝜏𝜏) defined by 𝐢𝐢𝛼𝛼,𝛽𝛽 ,𝛾𝛾 = {〈π‘₯π‘₯, 𝛼𝛼, 𝛽𝛽, 𝛾𝛾βŒͺ: π‘₯π‘₯ ∈ 𝑋𝑋}, where 𝛼𝛼, 𝛽𝛽, 𝛾𝛾 ∈ 𝐼𝐼 are arbitrary and 𝛼𝛼 + 𝛽𝛽 + 𝛾𝛾 ≀ 2. Then the FNSs 𝐢𝐢0.75,0.85,0.05 , 𝐢𝐢0.65,0.75,0.15 , 𝐢𝐢0.85,0.850 ,05 are all compact. Β© 2017, IJMA. All Rights Reserved

148

Y. Veereswari* / An Introduction to Fuzzy Neutrosophic Topological Spaces / IJMA- 8(3), March-2017.

Corollary 5.10: Let (Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?) and (Δ‘?‘ŒΔ‘?‘Œ, Δ‘?œ‘Δ‘?œ‘) be two FNTSs and let Δ‘?‘“Δ‘?‘“: Δ‘?‘‹Δ‘?‘‹ Γ’†’ Δ‘?‘ŒΔ‘?‘Œ be a fuzzy continuous function. If Δ‘??Β΄Δ‘??Β΄ is fuzzy compact in(Δ‘?‘‹Δ‘?‘‹, Δ‘?œ?Δ‘?œ?), then so is Δ‘?‘“Δ‘?‘“(Δ‘??Β΄Δ‘??Β΄) in (Δ‘?‘ŒΔ‘?‘Œ, Δ‘?œ‘Δ‘?œ‘). REFERENCES

[1] A.A.Salama, S.A.Alblowi, Neutrosophic Set and Neutrosophic Topological spaces, IOSR Journal of Mathematics, ISSN:2278-5728, Volume 3, Issue 4, SEP-OCT 2012, PP 31-35 [2] A.A.Salama, Florentin Smarandache and Valeri Kromov, Neutrosophic Closed Set and Neutrosophic Continuous Functions, Neutrosophic Sets and Systems, ISBN 9781599733098, Vol 4, 2014 [3] C.L. Chang, Fuzzy Topological Spaces, J. Math. Anal. Appl. 24 (1968)182-1 90. [4] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88(1) (1997), 81-89. [5] Floretin Smarandache, Neutrosophic Logic-A Generalization of the Intuitionistic Fuzzy Logic, Multispace & Multistructure Neutrosophic Transdisciplinarity, ISBN:978-952-67349-2-7, Vol. 4, 2010,396-402 [6] Floretin Smarandache, Neutrosophic Set-A Generalization of the Intuitionistic Fuzzy Logic, MMNT, ISBN:978-952-67349-2-7, Vol 4, 2010, 403-409 [7] Haibin Wang, Florentin Smarandache, Yanqing Zhang, Rajshekhar Sunderraman, Single valued Neutrosophic set, MMNT, ISBN:978-952-67349-2-7, Vol 4, 2010, 410-413 [8] K. Atanassov, intuitionistic fuzzy sets, in V.Sgurev, ed.,Vii ITKRS Session, Sofia(June 1983 central Sci. and Techn. Library, Bulg.Academy of Sciences (1984)). [9] K. Atanassov, intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986)87-96. [10] K. Atanassov, Review and new result on intuitionistic fuzzy sets, preprint IM-MFAIS-1-88, Sofia, 1988. [11] L.A. Zadeh, Fuzzy Sets, Inform and Control 8(1965), 338-353 [12] Reza Saadati, Jin HanPark, On the intuitionistic fuzzy topological space, Chaos, Solitons and Fractals 27(2006)331-344. [13] S.A. Alblowi, A.A. Salama and Mohmed Eisa, New Concepts of Neutrosophic Sets, IJMCAR, Vol. 3, Issue 3, Oct (2013) 95-102. Source of support: Nil, Conflict of interest: None Declared. [Copy right Γ‚Ε  2017. This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.]

Γ‚Ε  2017, IJMA. All Rights Reserved

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