International Journal of Mathematical Archive-8(3), 2017, 144-149
Available online through www.ijma.info ISSN 2229 – 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 Coker’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 ]− 0, 1+ [. The neutrosophic components T, I, F represents the truth value, indeterminacy value and falsehood value respectively. Definition 2.1 [7]: Let đ?‘‹đ?‘‹ be a non-empty fixed set. A fuzzy neutrosophic set (FNS for short) đ??´đ??´ is an object having the form đ??´đ??´ = {⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ), đ?œŽđ?œŽđ??´đ??´ (đ?‘Ľđ?‘Ľ), đ?œˆđ?œˆđ??´đ??´ (đ?‘Ľđ?‘Ľ)âŒŞ: đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹} where the functions đ?œŽđ?œŽđ??´đ??´ : đ?‘‹đ?‘‹ →]− 0, 1+ [, đ?œˆđ?œˆđ??´đ??´ : đ?‘‹đ?‘‹ →]− 0, 1+ [ đ?œ‡đ?œ‡đ??´đ??´ : đ?‘‹đ?‘‹ →]− 0, 1+ [, denote the degree of membership function (namely đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ)), the degree of indeterminacy function (namely đ?œŽđ?œŽđ??´đ??´ (đ?‘Ľđ?‘Ľ)) and the degree of non-membership (namely đ?œˆđ?œˆđ??´đ??´ (đ?‘Ľđ?‘Ľ)) respectively of each element đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹ to the set đ??´đ??´ and + 0 ≤ đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ) + đ?œŽđ?œŽđ??´đ??´ (đ?‘Ľđ?‘Ľ) + đ?œˆđ?œˆđ??´đ??´ (đ?‘Ľđ?‘Ľ) ≤ 1+ , for each đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹. Remark 2.2 [7]: Every fuzzy set đ??´đ??´ on a non-empty set đ?‘‹đ?‘‹ is obviously a FNS having the form đ??´đ??´ = {⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ), đ?œŽđ?œŽđ??´đ??´ (đ?‘Ľđ?‘Ľ), 1 − đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ)âŒŞ: đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹}
A fuzzy neutrosophic set đ??´đ??´ = {⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ), đ?œŽđ?œŽđ??´đ??´ (đ?‘Ľđ?‘Ľ), đ?œˆđ?œˆđ??´đ??´ (đ?‘Ľđ?‘Ľ)âŒŞ: đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹} can be identified to an ordered triple ⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´ , đ?œŽđ?œŽđ??´đ??´ , đ?œˆđ?œˆđ??´đ??´ âŒŞ in ]− 0, 1+ [ on đ?‘‹đ?‘‹.
Definition 2.3[1]: Let đ?‘‹đ?‘‹ be a non-empty set and the FNSs đ??´đ??´ and đ??ľđ??ľ be in the form đ??´đ??´ = {⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ), đ?œŽđ?œŽđ??´đ??´ (đ?‘Ľđ?‘Ľ), đ?œˆđ?œˆđ??´đ??´ (đ?‘Ľđ?‘Ľ)âŒŞ: đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹ and đ??ľđ??ľ=đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??ľđ??ľđ?‘Ľđ?‘Ľ, đ?œŽđ?œŽđ??ľđ??ľđ?‘Ľđ?‘Ľ, đ?œˆđ?œˆđ??ľđ??ľđ?‘Ľđ?‘Ľ:đ?‘Ľđ?‘Ľâˆˆđ?‘‹đ?‘‹ on đ?‘‹đ?‘‹ and let đ??´đ??´đ?‘–đ?‘–:đ?‘–đ?‘–∈đ??˝đ??˝ be an arbitrary family of FNS’s in đ?‘‹đ?‘‹, where đ??´đ??´đ?‘–đ?‘–=đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´đ?‘–đ?‘–đ?‘Ľđ?‘Ľ, đ?œŽđ?œŽđ??´đ??´đ?‘–đ?‘–đ?‘Ľđ?‘Ľ, đ?œˆđ?œˆđ??´đ??´đ?‘–đ?‘–đ?‘Ľđ?‘Ľ:đ?‘Ľđ?‘Ľâˆˆđ?‘‹đ?‘‹e. a) đ??´đ??´ ⊆ đ??ľđ??ľ iff đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ) ≤ đ?œ‡đ?œ‡đ??ľđ??ľ (đ?‘Ľđ?‘Ľ), đ?œŽđ?œŽđ??´đ??´ (đ?‘Ľđ?‘Ľ) ≤ đ?œŽđ?œŽđ??ľđ??ľ (đ?‘Ľđ?‘Ľ) and đ?œˆđ?œˆđ??´đ??´ (đ?‘Ľđ?‘Ľ) ≼ đ?œˆđ?œˆđ??ľđ??ľ (đ?‘Ľđ?‘Ľ) for all đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹. b) đ??´đ??´ = đ??ľđ??ľ iff đ??´đ??´ ⊆ đ??ľđ??ľ and đ??ľđ??ľ ⊆ đ??´đ??´. c) đ??´đ??´Ě… = {⌊đ?‘Ľđ?‘Ľ, đ?œˆđ?œˆđ??´đ??´ (đ?‘Ľđ?‘Ľ), 1 − đ?œŽđ?œŽđ??´đ??´ (đ?‘Ľđ?‘Ľ), đ?œ‡đ?œ‡đ??´đ??´ (đ?‘Ľđ?‘Ľ)âŒŞ: đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹} d) âˆŞ đ??´đ??´đ?‘–đ?‘– = ďż˝âŒŠđ?‘Ľđ?‘Ľ, â‹ đ?‘–đ?‘–∈đ??˝đ??˝ đ?œ‡đ?œ‡đ??´đ??´đ?‘–đ?‘– (đ?‘Ľđ?‘Ľ) , â‹ đ?‘–đ?‘–∈đ??˝đ??˝ đ?œŽđ?œŽđ??´đ??´đ?‘–đ?‘– (đ?‘Ľđ?‘Ľ) , â‹€đ?‘–đ?‘–∈đ??˝đ??˝ đ?œˆđ?œˆđ??´đ??´đ?‘–đ?‘– (đ?‘Ľđ?‘Ľ)âŒŞ: đ?‘Ľđ?‘Ľ ∈ đ?‘‹đ?‘‹ďż˝
Corresponding Author: Y. Veereswari* Research Scholar, Govt. Arts College, Coimbatore, (T.N.), India.
International Journal of Mathematical Archive- 8(3), March – 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 = �〈𝑥𝑥, ⋀𝑖𝑖∈𝐽𝐽 𝜇𝜇𝐴𝐴𝑖𝑖 (𝑥𝑥) , ⋀𝑖𝑖∈𝐽𝐽 𝜎𝜎𝐴𝐴𝑖𝑖 (𝑥𝑥) , ⋁𝑖𝑖∈𝐽𝐽 𝜈𝜈𝐴𝐴𝑖𝑖 (𝑥𝑥)〉: 𝑥𝑥 ∈ 𝑋𝑋� 𝐴𝐴 − 𝐵𝐵 = �〈𝑥𝑥, 𝜇𝜇𝐴𝐴 (𝑥𝑥) ∧ 𝜈𝜈𝐵𝐵 (𝑥𝑥), 𝜎𝜎𝐴𝐴 (𝑥𝑥) ∧ �1 − 𝜎𝜎𝐵𝐵 (𝑥𝑥)�, 𝜈𝜈𝐴𝐴 (𝑥𝑥) ∨ 𝜇𝜇𝐵𝐵 (𝑥𝑥)〉: 𝑥𝑥 ∈ 𝑋𝑋� []𝐴𝐴 = {〈𝑥𝑥, 𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜎𝜎𝐴𝐴 (𝑥𝑥), 1 − 𝜇𝜇𝐴𝐴 (𝑥𝑥)〉: 𝑥𝑥 ∈ 𝑋𝑋} <> 𝐴𝐴 = {〈𝑥𝑥, 1 − 𝜈𝜈𝐴𝐴 (𝑥𝑥), 𝜎𝜎𝐴𝐴 (𝑥𝑥), 𝜈𝜈𝐴𝐴 (𝑥𝑥)〉: 𝑥𝑥 ∈ 𝑋𝑋} 0𝑁𝑁 = 〈𝑥𝑥, 0, 0, 1〉 and 1𝑁𝑁 = 〈𝑥𝑥, 1, 1, 0〉
Example 2.4: Let 𝑋𝑋 = {𝑎𝑎, 𝑏𝑏, 𝑐𝑐} and 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 , �,� , , �,� , , �〉, 𝐴𝐴 = 〈𝑥𝑥, � , 0.5 0.2 0.3 0.3 0.2 0.2 0.3 0.4 0.5 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝐵𝐵 = 〈𝑥𝑥, � , , �,� , , �,� , , �〉, 0.5 0.5 0.4 0.4 0.5 0.2 0.2 0.4 0.4 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝐶𝐶 = 〈𝑥𝑥, � , , �,� , , �,� , , �〉, 0.5 0.5 0.5 0.5 0.5 0.3 0.2 0.2 0.2 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝐷𝐷 = 〈𝑥𝑥, � , , �,� , , �,� , , �〉 0.3 0.4 0.4 0.4 0.2 0.2 0.3 0.4 0.4
𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 Here 𝐴𝐴̅ = 〈𝑥𝑥, � , , � , � , , � , � , , �〉 and 𝐴𝐴 ⊆ 𝐶𝐶 because 𝜇𝜇𝐴𝐴 (𝑥𝑥) ≤ 𝜇𝜇𝐵𝐵 (𝑥𝑥), 𝜎𝜎𝐴𝐴 (𝑥𝑥) ≤ 𝜎𝜎𝐵𝐵 (𝑥𝑥) and 𝜈𝜈𝐴𝐴 (𝑥𝑥) ≥ 0.3 0.2 0.2 0.5 0.8 0.7 0.3 0.4 0.5 𝜈𝜈𝐵𝐵 (𝑥𝑥) for all 𝑥𝑥 ∈ 𝑋𝑋. Further, 𝐴𝐴 ∪ 𝐵𝐵 = 〈𝑥𝑥, 𝜇𝜇𝐴𝐴 (𝑥𝑥) ∨ 𝜇𝜇𝐵𝐵 (𝑥𝑥), 𝜎𝜎𝐴𝐴 (𝑥𝑥) ∨ 𝜎𝜎𝐵𝐵 (𝑥𝑥), 𝜈𝜈𝐴𝐴 (𝑥𝑥) ∧ 𝜈𝜈𝐵𝐵 (𝑥𝑥)〉 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 = 〈𝑥𝑥, � , , �,� , , �,� , , �〉 0.5 0.5 0.5 0.5 0.5 0.3 0.2 0.2 0.2
𝐴𝐴 ∩ 𝐵𝐵 = 〈𝑥𝑥, 𝜇𝜇𝐴𝐴 (𝑥𝑥) ∧ 𝜇𝜇𝐵𝐵 (𝑥𝑥), 𝜎𝜎𝐴𝐴 (𝑥𝑥) ∧ 𝜎𝜎𝐵𝐵 (𝑥𝑥), 𝜈𝜈𝐴𝐴 (𝑥𝑥) ∨ 𝜈𝜈𝐵𝐵 (𝑥𝑥)〉 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑎𝑎 𝑏𝑏 𝑐𝑐 = 〈𝑥𝑥, � , , �,� , , �,� , , �〉 0.3 0.4 0.4 0.4 0.2 0.2 0.3 0.4 0.4
Corollary 2.5: Let 𝐴𝐴, 𝐵𝐵, 𝐶𝐶 and 𝐴𝐴𝑖𝑖 be FNS’s in 𝑋𝑋(𝑖𝑖 ∈ 𝐽𝐽). Then a) 𝐴𝐴𝑖𝑖 ⊆ 𝐵𝐵 for each 𝑖𝑖 ∈ 𝐽𝐽 ⟹ ∪ 𝐴𝐴𝑖𝑖 ⊆ 𝐵𝐵 b) 𝐵𝐵 ⊆ 𝐴𝐴𝑖𝑖 for each 𝑖𝑖 ∈ 𝐽𝐽 ⟹ 𝐵𝐵 ⊆ ∩ 𝐴𝐴𝑖𝑖 ������ c) ∪ 𝐴𝐴𝑖𝑖 =∩ 𝐴𝐴�𝑖𝑖 and ������ ∩ 𝐴𝐴𝑖𝑖 =∪ 𝐴𝐴�𝑖𝑖 d) 𝐴𝐴 ⊆ 𝐵𝐵 ⟺ 𝐵𝐵� ⊆ 𝐴𝐴̅ �����̅) = 𝐴𝐴 e) (𝐴𝐴 �� ���� f) 0�� 𝑁𝑁 = 1𝑁𝑁 and 1𝑁𝑁 = 0𝑁𝑁
Now we shall define the image and preimage of FNS’s. Let 𝑋𝑋, 𝑌𝑌 be two non-empty sets and 𝑓𝑓: 𝑋𝑋 ⟶ 𝑌𝑌 a function.
Definition 2.6[4]: a) If 𝐵𝐵 = {〈𝑦𝑦, 𝜇𝜇𝐵𝐵 (𝑦𝑦), 𝜎𝜎𝐵𝐵 (𝑦𝑦), 𝜈𝜈𝐵𝐵 (𝑦𝑦)〉: 𝑦𝑦 ∈ 𝑌𝑌} is an FNS in 𝑌𝑌 then the preimage of 𝐵𝐵 under 𝑓𝑓, denoted by 𝑓𝑓 −1 (𝐵𝐵), is the FNS in 𝑋𝑋 defined by 𝑓𝑓 −1 (𝐵𝐵) = {〈𝑥𝑥, 𝑓𝑓 −1 (𝜇𝜇𝐵𝐵 )(𝑥𝑥), 𝑓𝑓 −1 (𝜎𝜎𝐵𝐵 )(𝑥𝑥), 𝑓𝑓 −1 (𝜈𝜈𝐵𝐵 )(𝑥𝑥)〉: 𝑥𝑥 ∈ 𝑋𝑋} −1 (𝜇𝜇 )(𝑥𝑥) = 𝜇𝜇𝐵𝐵 𝑓𝑓(𝑥𝑥), 𝑓𝑓 −1 (𝜎𝜎𝐵𝐵 )(𝑥𝑥) = 𝜎𝜎𝐵𝐵 𝑓𝑓(𝑥𝑥) and 𝑓𝑓 −1 (𝜈𝜈𝐵𝐵 )(𝑥𝑥) = 𝜈𝜈𝐵𝐵 𝑓𝑓(𝑥𝑥) Where 𝑓𝑓 𝐵𝐵 b) If 𝐴𝐴 = {〈𝑥𝑥, 𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜎𝜎𝐴𝐴 (𝑥𝑥), 𝜈𝜈𝐴𝐴 (𝑥𝑥)〉: 𝑥𝑥 ∈ 𝑋𝑋} is a FNS in 𝑋𝑋, then the image of 𝐴𝐴 under 𝑓𝑓, denoted by 𝑓𝑓(𝐴𝐴), is the FNS in 𝑌𝑌 defined by 𝑓𝑓(𝐴𝐴) = �〈𝑦𝑦, 𝑓𝑓(𝜇𝜇𝐴𝐴 )(𝑦𝑦), 𝑓𝑓(𝜎𝜎𝐴𝐴 )(𝑦𝑦), �1 − 𝑓𝑓(1 − 𝜈𝜈𝐴𝐴 )�(𝑦𝑦)〉: 𝑦𝑦 ∈ 𝑌𝑌� sup 𝜇𝜇𝐴𝐴 (𝑥𝑥) 𝑖𝑖𝑖𝑖 𝑓𝑓 −1 (𝑦𝑦) ≠ ∅ 𝑓𝑓(𝜇𝜇𝐴𝐴 )(𝑦𝑦) = �𝑥𝑥∈𝑓𝑓 −1 (𝑦𝑦) 0 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 sup 𝜎𝜎𝐴𝐴 (𝑥𝑥) 𝑖𝑖𝑖𝑖 𝑓𝑓 −1 (𝑦𝑦) ≠ ∅ 𝑓𝑓(𝜎𝜎𝐴𝐴 )(𝑦𝑦) = �𝑥𝑥∈𝑓𝑓 −1 (𝑦𝑦) 0 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 inf 𝜈𝜈𝐴𝐴 (𝑥𝑥) 𝑖𝑖𝑖𝑖 𝑓𝑓 −1 (𝑦𝑦) ≠ ∅ −1 �1 − 𝑓𝑓(1 − 𝜈𝜈𝐴𝐴 )�(𝑦𝑦) = �𝑥𝑥∈𝑓𝑓 (𝑦𝑦) 1 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 For the sake of simplicity, let us use the symbol 𝑓𝑓_(𝜈𝜈𝐴𝐴 ) for �1 − 𝑓𝑓(1 − 𝜈𝜈𝐴𝐴 )�. Corollary 2.7[4]: Let 𝐴𝐴, 𝐴𝐴𝑖𝑖 (𝑖𝑖 ∈ 𝐽𝐽) be FNS’s in 𝑋𝑋, 𝐵𝐵, 𝐵𝐵𝑗𝑗 (𝑗𝑗 ∈ 𝐾𝐾) be FNS’s in 𝑌𝑌, and 𝑓𝑓: 𝑋𝑋 ⟶ 𝑌𝑌 a function. a) 𝐴𝐴1 ⊆ 𝐴𝐴2 ⟹ 𝑓𝑓(𝐴𝐴1 ) ⊆ 𝑓𝑓(𝐴𝐴2 ), 𝐵𝐵1 ⊆ 𝐵𝐵2 ⟹ 𝑓𝑓 −1 (𝐵𝐵1 ) ⊆ 𝑓𝑓 −1 (𝐵𝐵2 ), b) 𝐴𝐴 ⊆ 𝑓𝑓 −1 (𝑓𝑓(𝐴𝐴)), and if 𝑓𝑓 is injective then 𝐴𝐴 = 𝑓𝑓 −1 (𝑓𝑓(𝐴𝐴)) c) 𝑓𝑓 −1 (𝑓𝑓(𝐵𝐵)) ⊆ 𝐵𝐵, and if 𝑓𝑓 is surjective, then 𝑓𝑓 −1 (𝑓𝑓(𝐵𝐵)) = 𝐵𝐵 d) 𝑓𝑓 −1 �∪ 𝐵𝐵𝑗𝑗 � =∪ 𝑓𝑓 −1 �𝐵𝐵𝑗𝑗 �, 𝑓𝑓 −1 �∩ 𝐵𝐵𝑗𝑗 � =∩ 𝑓𝑓 −1 �𝐵𝐵𝑗𝑗 �, e) 𝑓𝑓(∪ 𝐴𝐴𝑖𝑖 ) =∪ 𝑓𝑓(𝐴𝐴𝑖𝑖 ), 𝑓𝑓(∩ 𝐴𝐴𝑖𝑖 ) ⊆∩ 𝑓𝑓(𝐴𝐴𝑖𝑖 ); and if 𝑓𝑓 is injective, then 𝑓𝑓(∩ 𝐴𝐴𝑖𝑖 ) =∩ 𝑓𝑓(𝐴𝐴𝑖𝑖 ) f) 𝑓𝑓 −1 (1𝑁𝑁 ) = 1𝑁𝑁 , 𝑓𝑓 −1 (0𝑁𝑁 ) = 0𝑁𝑁 g) 𝑓𝑓(0𝑁𝑁 ) = 0𝑁𝑁 , 𝑓𝑓(1𝑁𝑁 ) = 1𝑁𝑁 , if 𝑓𝑓 is surjective © 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 đ?‘“đ?‘“ is surjective, then ������ đ?‘“đ?‘“(đ??´đ??´) ⊆ đ?‘“đ?‘“(đ??´đ??´Ě…). If furthermore, đ?‘“đ?‘“ is injective, then ������ đ?‘“đ?‘“(đ??´đ??´) = đ?‘“đ?‘“(đ??´đ??´Ě…) −1 (đ??ľđ??ľ −1 ��������� ďż˝ ) = đ?‘“đ?‘“ (đ??ľđ??ľ) i) đ?‘“đ?‘“
3. FUZZY NEUTROSOPHIC TOPOLOGICAL SPACES
Definition 3.1: A fuzzy neutrosophic topology (FNT for short) a non-empty set đ?‘‹đ?‘‹ is a family đ?œ?đ?œ? of fuzzy neutrosophic subsets in đ?‘‹đ?‘‹ satisfying the following axioms. (NT1) 0đ?‘ đ?‘ , 1đ?‘ đ?‘ ∈ đ?œ?đ?œ? (NT2) đ??şđ??ş1 ∊ đ??şđ??ş2 ∈ đ?œ?đ?œ? for any đ??şđ??ş1 , đ??şđ??ş2 ∈ đ?œ?đ?œ? (NT3) âˆŞ đ??şđ??şđ?‘–đ?‘– ∈ đ?œ?đ?œ?, ∀{đ??şđ??şđ?‘–đ?‘– : đ?‘–đ?‘– ∈ đ??˝đ??˝} ⊆ đ?œ?đ?œ?
In this case the pair (đ?‘‹đ?‘‹, đ?œ?đ?œ?) is called a Fuzzy neutrosophic topological space (FNTS for short) and any fuzzy neutrosophic set in đ?œ?đ?œ? is known as fuzzy neutrosophic open set (FNOS for short) in đ?‘‹đ?‘‹. The elements of đ?œ?đ?œ? are called open fuzzy neutrosophic sets. The complement of FNOS in the FNTS (đ?‘‹đ?‘‹, đ?œ?đ?œ?) is called fuzzy neutrosophic closed set (FNCS for short).
Example 3.2: Let đ?‘‹đ?‘‹ = {đ?‘Žđ?‘Ž, đ?‘?đ?‘?, đ?‘?đ?‘?} and consider the value of đ??´đ??´, đ??ľđ??ľ, đ??śđ??ś, đ??ˇđ??ˇ are defined in Example 2.4, the family đ?œ?đ?œ? = {0đ?‘ đ?‘ , 1đ?‘ đ?‘ , đ??´đ??´, đ??ľđ??ľ, đ??śđ??ś, đ??ˇđ??ˇ} of FNS in đ?‘‹đ?‘‹. Then (đ?‘‹đ?‘‹, đ?œ?đ?œ?) is FNTS on đ?‘‹đ?‘‹.
Example 3.3: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) be a fuzzy topological space such that đ?œ?đ?œ? is not indiscrete. Suppose now that đ?œ?đ?œ? = {0, 1} âˆŞ {đ??şđ??şđ?‘–đ?‘– : đ?‘–đ?‘– ∈ đ??˝đ??˝}. Then we can construct two FNT’s on đ?‘‹đ?‘‹ as follows: a) đ?œ?đ?œ?1 = {0đ?‘ đ?‘ , 1đ?‘ đ?‘ } âˆŞ {⌊đ?‘Ľđ?‘Ľ, đ??şđ??şđ?‘–đ?‘– , đ?œŽđ?œŽ(đ?‘Ľđ?‘Ľ), 0âŒŞ: đ?‘–đ?‘– ∈ đ??˝đ??˝}, b) đ?œ?đ?œ?2 = {0đ?‘ đ?‘ , 1đ?‘ đ?‘ } âˆŞ {⌊đ?‘Ľđ?‘Ľ, 0, đ?œŽđ?œŽ(đ?‘Ľđ?‘Ľ), đ??şđ??şđ?‘–đ?‘– âŒŞ: đ?‘–đ?‘– ∈ đ??˝đ??˝}
Proposition 3.4: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) be a FNTS on đ?‘‹đ?‘‹. Then we can also construct several FNTS’s on đ?‘‹đ?‘‹ in the following way đ?œ?đ?œ?0,1 = {[ ]đ??şđ??ş: đ??şđ??ş ∈ đ?œ?đ?œ?}, b) đ?œ?đ?œ?0,2 = {⌊ âŒŞđ??şđ??ş: đ??şđ??ş ∈ đ?œ?đ?œ?}
Definition 3.5: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?1 ), (đ?‘‹đ?‘‹, đ?œ?đ?œ?2 ) be two FNTS’s on đ?‘‹đ?‘‹. Then đ?œ?đ?œ?1 is said to be contained in đ?œ?đ?œ?2 (in symbols, đ?œ?đ?œ?1 ⊆ đ?œ?đ?œ?2 ), if đ??şđ??ş ∈ đ?œ?đ?œ?2 for each đ??şđ??ş ∈ đ?œ?đ?œ?1 . In this case, we also say that đ?œ?đ?œ?1 is coarser than đ?œ?đ?œ?2 .
Proposition 3.6: Let {đ?œ?đ?œ?đ?‘–đ?‘– : đ?‘–đ?‘– ∈ đ??˝đ??˝} be a family of FNT’s on đ?‘‹đ?‘‹. Then â‹‚đ?‘–đ?‘–∈đ??˝đ??˝ đ?œ?đ?œ?đ?‘–đ?‘– is FNT on đ?‘‹đ?‘‹. Furthermore, â‹‚đ?‘–đ?‘–∈đ??˝đ??˝ đ?œ?đ?œ?đ?‘–đ?‘– is the coarsest FNT on đ?‘‹đ?‘‹ containing all đ?œ?đ?œ?đ?‘–đ?‘– ’s.
Definition 3.7: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) be a FNTS on đ?‘‹đ?‘‹. a) A family đ?›˝đ?›˝ ⊆ đ?œ?đ?œ? is called a base for (đ?‘‹đ?‘‹, đ?œ?đ?œ?) iff each member of đ?œ?đ?œ? can written as a union of elements of đ?›˝đ?›˝. b) A family đ?›žđ?›ž ⊆ đ?œ?đ?œ? is called a subbase for (đ?‘‹đ?‘‹, đ?œ?đ?œ?) iff the family of finite intersections of elements in đ?›žđ?›ž forms a base for (đ?‘‹đ?‘‹, đ?œ?đ?œ?). In this case the FNT đ?œ?đ?œ? is said to be generated by đ?›žđ?›ž.
Definition 3.8: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) be a FNTS and đ??´đ??´ = ⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´ , đ?œŽđ?œŽđ??´đ??´ , đ?œˆđ?œˆđ??´đ??´ âŒŞ be a FNS in đ?‘‹đ?‘‹. Then the fuzzy interior and fuzzy closure of đ??´đ??´ are defined by đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) =∊ {đ??žđ??ž: đ??žđ??žis a FNCS in đ?‘‹đ?‘‹ and đ??´đ??´ ⊆ đ??žđ??ž}, đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) =âˆŞ {đ??şđ??ş: đ??şđ??şis a FNOS in đ?‘‹đ?‘‹ and đ??şđ??ş ⊆ đ??´đ??´} Now that đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) is a FNCS and đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) is a FNOS in đ?‘‹đ?‘‹. Further, a) đ??´đ??´ is a FNCS in đ?‘‹đ?‘‹ iff đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) = đ??´đ??´ b) đ??´đ??´ is a FNOS in đ?‘‹đ?‘‹ iff đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) = đ??´đ??´
Example 3.9: Consider the FNTS (đ?‘‹đ?‘‹, đ?œ?đ?œ?) is defined in Example 3.2. If đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? , ďż˝,ďż˝ , , ďż˝,ďż˝ , , ďż˝âŒŞ đ??šđ??š = ⌊đ?‘Ľđ?‘Ľ, ďż˝ , 0.4 0.3 0.2 0.2 0.3 0.3 0.4 0.5 0.4 Then, đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) = đ??ˇđ??ˇ and đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) = 1đ?‘ đ?‘ .
đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´). Proposition 3.10: For any FNS đ??´đ??´ in (đ?‘‹đ?‘‹, đ?œ?đ?œ?), we have đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´Ě…) = ��������� đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´), đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´Ě…) = �������
Proposition 3.11: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) be a FNTS and đ??´đ??´, đ??ľđ??ľ be FNS’s in đ?‘‹đ?‘‹. Then the following properties hold: a) đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) ⊆ đ??´đ??´, đ??´đ??´ ⊆ đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) b) đ??´đ??´ ⊆ đ??ľđ??ľ â&#x;š đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) ⊆ đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??ľđ??ľ), đ??´đ??´ ⊆ đ??ľđ??ľ â&#x;š đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) ⊆ đ?‘?đ?‘?đ?‘?đ?‘?(đ??ľđ??ľ) c) đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–ďż˝đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´)ďż˝ = đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´), đ?‘?đ?‘?đ?‘?đ?‘?ďż˝đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´)ďż˝ = đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) d) đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´ ∊ đ??ľđ??ľ) = đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) ∊ đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??ľđ??ľ), đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´ âˆŞ đ??ľđ??ľ) = đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) âˆŞ đ?‘?đ?‘?đ?‘?đ?‘?(đ??ľđ??ľ) e) đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(1đ?‘ đ?‘ ) = 1đ?‘ đ?‘ , đ?‘?đ?‘?đ?‘?đ?‘?(0đ?‘ đ?‘ ) = 0đ?‘ đ?‘ Š 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 (đ?‘‹đ?‘‹, đ?œ?đ?œ?) be a FNTS. If đ??´đ??´ = ⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´ , đ?œŽđ?œŽđ??´đ??´ , đ?œˆđ?œˆđ??´đ??´ âŒŞ be a FNS in đ?‘‹đ?‘‹. Then we have a) đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) ⊆ ⌊đ?‘Ľđ?‘Ľ, đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?œ?đ?œ?1 (đ?œ‡đ?œ‡đ??´đ??´ ), đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?œ?đ?œ?1 (đ?œŽđ?œŽđ??´đ??´ ), đ?‘?đ?‘?đ?‘?đ?‘?đ?œ?đ?œ?2 (đ?œˆđ?œˆđ??´đ??´ )âŒŞ ⊆ đ??´đ??´ b) đ??´đ??´ ⊆ ⌊đ?‘Ľđ?‘Ľ, đ?‘?đ?‘?đ?‘?đ?‘?đ?œ?đ?œ?2 (đ?œ‡đ?œ‡đ??´đ??´ ), đ?‘?đ?‘?đ?‘?đ?‘?đ?œ?đ?œ?2 (đ?œŽđ?œŽđ??´đ??´ ), đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?œ?đ?œ?1 (đ?œˆđ?œˆđ??´đ??´ )âŒŞ ⊆ đ?‘?đ?‘?đ?‘?đ?‘?(đ??´đ??´) where đ?œ?đ?œ?1 and đ?œ?đ?œ?2 are FTS on đ?‘‹đ?‘‹ defined by đ?œ?đ?œ?1 ={đ?œ‡đ?œ‡đ??´đ??´ : đ??şđ??ş ∈ đ?œ?đ?œ?}, đ?œ?đ?œ?2 = {1 − đ?œˆđ?œˆđ??´đ??´ : đ??şđ??ş ∈ đ?œ?đ?œ?} Corollary 3.13: Let đ??´đ??´ = ⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??´đ??´ , đ?œŽđ?œŽđ??´đ??´ , đ?œˆđ?œˆđ??´đ??´ âŒŞ be a FNS in (đ?‘‹đ?‘‹, đ?œ?đ?œ?) a) If đ??´đ??´ is a FNCS, then đ?œ‡đ?œ‡đ??´đ??´ is fuzzy closed in (đ?‘‹đ?‘‹, đ?œ?đ?œ?2 ) and đ?œˆđ?œˆđ??´đ??´ is fuzzy open in (đ?‘‹đ?‘‹, đ?œ?đ?œ?1 ). b) If đ??´đ??´ is a FNOS, then đ?œ‡đ?œ‡đ??´đ??´ is fuzzy open in (đ?‘‹đ?‘‹, đ?œ?đ?œ?1 ) and đ?œˆđ?œˆđ??´đ??´ is fuzzy closed in (đ?‘‹đ?‘‹, đ?œ?đ?œ?2 ).
Example 3.14: Consider the FNTS (đ?‘‹đ?‘‹, đ?œ?đ?œ?) is defined in Example 3.2. If đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ??šđ??š = ⌊đ?‘Ľđ?‘Ľ, ďż˝ , , ďż˝,ďż˝ , , ďż˝,ďż˝ , , ďż˝âŒŞ 0.4 0.5 0.4 0.4 0.3 0.2 0.2 0.3 0.3 Then, đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??´đ??´) = đ??ˇđ??ˇ. Noting that we have đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? , ďż˝,ďż˝ , , ďż˝,ďż˝ , , ďż˝,ďż˝ , , �� đ?œ?đ?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 đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?œ?đ?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 đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? đ?‘Žđ?‘Ž đ?‘?đ?‘? đ?‘?đ?‘? and đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?œ?đ?œ?1 (đ?œ‡đ?œ‡đ??´đ??´ ) = ďż˝ , , ďż˝ , đ?‘?đ?‘?đ?‘?đ?‘?đ?œ?đ?œ?2 (đ?œˆđ?œˆđ??´đ??´ ) = ďż˝ , , ďż˝ 0.3 0.4 0.4
4. FUZZY NEUTROSOPHIC CONTINUITY
0.3 0.4 0.4
Definition 4.1: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) and (đ?‘Œđ?‘Œ, đ?œ‘đ?œ‘) be two FNTSs and let đ?‘“đ?‘“: đ?‘‹đ?‘‹ → đ?‘Œđ?‘Œ be a function. Then đ?‘“đ?‘“ is said to be fuzzy continuous iff the preimage of each FNS in đ?œ‘đ?œ‘ is a FNS in đ?œ?đ?œ?. Definition 4.2: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) and (đ?‘Œđ?‘Œ, đ?œ‘đ?œ‘) be two FNTSs and let đ?‘“đ?‘“: đ?‘‹đ?‘‹ → đ?‘Œđ?‘Œ be a function. Then đ?‘“đ?‘“ is said to be fuzzy open iff the image of each FNS in đ?œ?đ?œ? is a FNS in đ?œ‘đ?œ‘. Example 4.3: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?0 ) and (đ?‘Œđ?‘Œ, đ?œ‘đ?œ‘0 ) be two fuzzy topological spaces. a) If đ?‘“đ?‘“: đ?‘‹đ?‘‹ → đ?‘Œđ?‘Œ is fuzzy continuous in the usual sense, then in this case, đ?‘“đ?‘“ is fuzzy continuous iff the preimage of each FNS in đ?œ‘đ?œ‘0 is a FNS in đ?œ?đ?œ?0 . Consider the FNTs on đ?‘‹đ?‘‹ and đ?‘Œđ?‘Œ respectively, as follows: đ?œ?đ?œ? = {⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??şđ??ş , 0, 1 − đ?œ‡đ?œ‡đ??şđ??ş âŒŞ: đ?œ‡đ?œ‡đ??şđ??ş ∈ đ?œ?đ?œ?0 }, đ?œ‘đ?œ‘ = {⌊đ?‘Śđ?‘Ś, đ?›żđ?›żđ??ťđ??ť , 0, 1 − đ?›żđ?›żđ??ťđ??ť âŒŞ: đ?›żđ?›żđ??ťđ??ť ∈ đ?œ‘đ?œ‘0 } In this case we have for each ⌊đ?‘Śđ?‘Ś, đ?›żđ?›żđ??ťđ??ť , 0, 1 − đ?›żđ?›żđ??ťđ??ť âŒŞ ∈ đ?œ‘đ?œ‘, đ?›żđ?›żđ??ťđ??ť ∈ đ?œ‘đ?œ‘0 , đ?‘“đ?‘“ −1 ⌊đ?‘Śđ?‘Ś, đ?›żđ?›żđ??ťđ??ť , 0, 1 − đ?›żđ?›żđ??ťđ??ť âŒŞ = ⌊đ?‘Ľđ?‘Ľ, đ?‘“đ?‘“ −1 (đ?›żđ?›żđ??ťđ??ť ), đ?‘“đ?‘“ −1 (0), đ?‘“đ?‘“ −1 (1 − đ?›żđ?›żđ??ťđ??ť )âŒŞ = ⌊đ?‘Ľđ?‘Ľ, đ?‘“đ?‘“ −1 (đ?›żđ?›żđ??ťđ??ť ), đ?‘“đ?‘“ −1 (0), 1 − đ?‘“đ?‘“ −1 (đ?›żđ?›żđ??ťđ??ť )âŒŞ ∈ đ?œ?đ?œ? b) Let đ?‘“đ?‘“: đ?‘‹đ?‘‹ → đ?‘Œđ?‘Œ be a fuzzy open in the usual sense, then đ?‘“đ?‘“ is fuzzy openaccording to Definition 4.2. In this case we have, for each ⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??şđ??ş , 0, 1 − đ?œ‡đ?œ‡đ??şđ??ş âŒŞ ∈ đ?œ?đ?œ?, đ?œ‡đ?œ‡đ??şđ??ş ∈ đ?œ?đ?œ?0 , đ?‘“đ?‘“⌊đ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??şđ??ş , 0, 1 − đ?œ‡đ?œ‡đ??şđ??ş âŒŞ = ⌊đ?‘Ľđ?‘Ľ, đ?‘“đ?‘“(đ?œ‡đ?œ‡đ??şđ??ş ), 0, đ?‘“đ?‘“_(1 − đ?œ‡đ?œ‡đ??şđ??ş )âŒŞ = ⌊đ?‘Ľđ?‘Ľ, đ?‘“đ?‘“(đ?œ‡đ?œ‡đ??şđ??ş ), 0, 1 − đ?‘“đ?‘“_(đ?œ‡đ?œ‡đ??şđ??ş )âŒŞ ∈ đ?œ‘đ?œ‘
Proposition 4.4: đ?‘“đ?‘“: (đ?‘‹đ?‘‹, đ?œ?đ?œ?) → (đ?‘Œđ?‘Œ, đ?œ‘đ?œ‘) is fuzzy continuous iff the preimage of each FNCS in đ?œ‘đ?œ‘ is a FNCS in đ?œ?đ?œ?. Proposition 4.5: The following are equivalent to each other a) đ?‘“đ?‘“: (đ?‘‹đ?‘‹, đ?œ?đ?œ?) → (đ?‘Œđ?‘Œ, đ?œ‘đ?œ‘) is fuzzy continuous b) đ?‘“đ?‘“ −1 (đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ??ľđ??ľ)) ⊆ đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–đ?‘–(đ?‘“đ?‘“ −1 (đ??ľđ??ľ)) for each FNS đ??ľđ??ľ in đ?‘Œđ?‘Œ. c) đ?‘?đ?‘?đ?‘?đ?‘?(đ?‘“đ?‘“ −1 (đ??ľđ??ľ)) ⊆ đ?‘“đ?‘“ −1 (đ?‘?đ?‘?đ?‘?đ?‘?(đ??ľđ??ľ))for each FNS đ??ľđ??ľ in đ?‘Œđ?‘Œ.
Example 4.6: Let (đ?‘Œđ?‘Œ, đ?œ‘đ?œ‘) be a FNTS and đ?‘‹đ?‘‹ be a non-empty set and đ?‘“đ?‘“: đ?‘‹đ?‘‹ → đ?‘Œđ?‘Œ a function. In this case đ?œ?đ?œ? = {đ?‘“đ?‘“ −1 (đ??ťđ??ť): đ??ťđ??ť ∈ đ?œ‘đ?œ‘} is a FNT on đ?‘‹đ?‘‹. Indeed, đ?‘Ąđ?‘Ą is the coarsest FNT on đ?‘‹đ?‘‹ which makes the function đ?‘“đ?‘“: đ?‘‹đ?‘‹ → đ?‘Œđ?‘Œ continuous. One may call the FNT đ?‘Ąđ?‘Ą on đ?‘‹đ?‘‹ the initial FNT with respect to đ?‘“đ?‘“. 5. FUZZY NEUTROSOPHIC COMPACTNESS
Definition 5.1: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) be a FNTS. a) If a family đ??şđ??şđ?‘–đ?‘– = ďż˝âŒŠđ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??şđ??şđ?‘–đ?‘– , đ?œŽđ?œŽđ??şđ??şđ?‘–đ?‘– , đ?œˆđ?œˆđ??şđ??şđ?‘–đ?‘– âŒŞ: đ?‘–đ?‘– ∈ đ??˝đ??˝ďż˝ of FNOS is đ?‘‹đ?‘‹ satisfy the condition âˆŞ đ??şđ??şđ?‘–đ?‘– = 1đ?‘ đ?‘ , then it is called a fuzzy open cover of đ?‘‹đ?‘‹. A finite subfamily of fuzzy open cover đ??şđ??şđ?‘–đ?‘– of đ?‘‹đ?‘‹, which is also a fuzzy open cover of đ?‘‹đ?‘‹ is called a finite subcover of đ??şđ??şđ?‘–đ?‘– . b) A family đ??žđ??žđ?‘–đ?‘– = ďż˝âŒŠđ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??žđ??žđ?‘–đ?‘– , đ?œŽđ?œŽđ??žđ??žđ?‘–đ?‘– , đ?œˆđ?œˆđ??žđ??žđ?‘–đ?‘– âŒŞ: đ?‘–đ?‘– ∈ đ??˝đ??˝ďż˝ of FNCSs in đ?‘‹đ?‘‹ satisfies the finite intersection property iff every finite subfamily ďż˝âŒŠđ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??žđ??žđ?‘–đ?‘– , đ?œŽđ?œŽđ??žđ??žđ?‘–đ?‘– , đ?œˆđ?œˆđ??žđ??žđ?‘–đ?‘– âŒŞ: đ?‘–đ?‘– = 1, 2, ‌ , đ?‘›đ?‘›ďż˝ of the family satisfies the condition â‹‚đ?‘›đ?‘›đ?‘–đ?‘–=1ďż˝âŒŠđ?‘Ľđ?‘Ľ, đ?œ‡đ?œ‡đ??žđ??žđ?‘–đ?‘– , đ?œŽđ?œŽđ??žđ??žđ?‘–đ?‘– , đ?œˆđ?œˆđ??žđ??žđ?‘–đ?‘– âŒŞďż˝ ≠0đ?‘ đ?‘ .
Definition 5.2: A FNTS (đ?‘‹đ?‘‹, đ?œ?đ?œ?) is fuzzy compact iff every fuzzy open cover of đ?‘‹đ?‘‹ has a finite subcover. Š 2017, IJMA. All Rights Reserved
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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 = 〈𝑥𝑥, �
1 𝑛𝑛
𝑛𝑛+1
,
2 𝑛𝑛
𝑛𝑛+2
,
3 𝑛𝑛
𝑛𝑛+3
1 2 3 1 2 3 � , � 𝑛𝑛+1 , 𝑛𝑛+2 , 𝑛𝑛+3 � , � 1 , 1 , 1 �〉 𝑛𝑛+2
𝑛𝑛+3
𝑛𝑛+4
𝑛𝑛+2
𝑛𝑛+3
𝑛𝑛+4
and 𝜏𝜏 = {0𝑁𝑁 , 1𝑁𝑁 } ∪ {𝐺𝐺𝑛𝑛 : 𝑛𝑛 ∈ ℕ}. Note that ⋃𝑛𝑛∈ℕ 𝐺𝐺𝑛𝑛 is an open cover for 𝑋𝑋, but this cover has no finite subcover. Consider 1 2 3 1 2 3 1 2 3 , �,� , , �,� , , �〉 𝐺𝐺1 = 〈𝑥𝑥, � , 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 = 〈𝑥𝑥, � , , �,� , , �,� , , �〉 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 = 〈𝑥𝑥, � , , �,� , , �,� , , �〉 0.75 0.6 0.5 0.8 0.8 0.9 0.2 0.2 0.1 and observe that 𝐺𝐺1 ∪ 𝐺𝐺2 ∪ 𝐺𝐺3 = 𝐺𝐺3 . So, for any finite subcollection �𝐺𝐺𝑛𝑛 𝑖𝑖 : 𝑖𝑖 ∈ I, where I is a finite subset of ℕ�, ⋃𝑛𝑛 𝑖𝑖 ∈I 𝐺𝐺𝑛𝑛 𝑖𝑖 = 𝐺𝐺𝑚𝑚 ≠ 1𝑁𝑁 , where 𝑚𝑚 = 𝑚𝑚𝑚𝑚𝑚𝑚{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 � is fuzzy compact.
Proof: Let (𝑋𝑋, 𝜏𝜏) be fuzzy compact and consider fuzzy open cover �[ ]𝐺𝐺𝑗𝑗 : 𝑗𝑗 ∈ 𝐾𝐾� of 𝑋𝑋 in �𝑋𝑋, 𝜏𝜏0,1 �. Since ∪ �[ ]𝐺𝐺𝑗𝑗 � = 1𝑁𝑁 . We obtain ∨ 𝜇𝜇𝐺𝐺𝑗𝑗 = 1,∨ 𝜎𝜎𝐺𝐺𝑗𝑗 = 1 and hence, by 𝜈𝜈𝐺𝐺𝑗𝑗 ≤ 1 − 𝜇𝜇𝐺𝐺𝑗𝑗 ⟹∧ 𝜈𝜈𝐺𝐺𝑗𝑗 ≤ 1 −∨ 𝜇𝜇𝐺𝐺𝑗𝑗 = 0 ⟹∧ 𝜈𝜈𝐺𝐺𝑗𝑗 = 0, we deduce ∪ 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 𝜇𝜇𝐺𝐺𝑖𝑖 = 1, ⋁𝑛𝑛𝑖𝑖=1 𝜎𝜎𝐺𝐺𝑖𝑖 = 1 and ⋀𝑛𝑛𝑖𝑖=1�1 − 𝜇𝜇𝐺𝐺𝑖𝑖 � = 0. That is, �𝑋𝑋, 𝜏𝜏0,1 � is fuzzy compact.
Suppose that �𝑋𝑋, 𝜏𝜏0,1 � is fuzzy compact and consider a fuzzy open cover �𝐺𝐺𝑗𝑗 : 𝑗𝑗 ∈ 𝐾𝐾� of 𝑋𝑋 in (𝑋𝑋, 𝜏𝜏). Since ∪ 𝐺𝐺𝑗𝑗 = 1𝑁𝑁 , we obtain ∨ 𝜇𝜇𝐺𝐺𝑗𝑗 = 1,∨ 𝜎𝜎𝐺𝐺𝑗𝑗 = 1 and 1 −∨ 𝜇𝜇𝐺𝐺𝑗𝑗 = 0. Since �𝑋𝑋, 𝜏𝜏0,1 � is fuzzy compact, there exist 𝐺𝐺1 , 𝐺𝐺2 , 𝐺𝐺3 , … 𝐺𝐺𝑛𝑛 such that ⋃𝑛𝑛𝑖𝑖=1�[ ]𝐺𝐺𝑗𝑗 � = 1𝑁𝑁 , that is, ⋁𝑛𝑛𝑖𝑖=1 𝜇𝜇𝐺𝐺𝑖𝑖 = 1, ⋁𝑛𝑛𝑖𝑖=1 𝜎𝜎𝐺𝐺𝑖𝑖 = 1 and ⋀𝑛𝑛𝑖𝑖=1�1 − 𝜇𝜇𝐺𝐺𝑖𝑖 � = 0. Hence 𝜇𝜇𝐺𝐺𝑖𝑖 ≤ 1 − 𝜈𝜈𝐺𝐺𝑖𝑖 ⟹ 1 = ⋁𝑛𝑛𝑖𝑖=1 𝜇𝜇𝐺𝐺𝑖𝑖 ≤ 1 − ⋀𝑛𝑛𝑖𝑖=1 𝜈𝜈𝐺𝐺𝑖𝑖 = 0. ⟹ ⋀𝑛𝑛𝑖𝑖=1 𝜈𝜈𝐺𝐺𝑖𝑖 = 0. Hence ⋃𝑛𝑛𝑖𝑖=1 𝐺𝐺𝑖𝑖 = 1𝑁𝑁 . Therefore (𝑋𝑋, 𝜏𝜏) is fuzzy compact. Corollary 5.5: Let (𝑋𝑋, 𝜏𝜏) and (𝑌𝑌, 𝜑𝜑) be two FNTSs and let 𝑓𝑓: 𝑋𝑋 → 𝑌𝑌 be a fuzzy continuous surjection. If (𝑋𝑋, 𝜏𝜏) is fuzzy compact, then so is (𝑌𝑌, 𝜑𝜑).
Corollary 5.6: A FNTS (𝑋𝑋, 𝜏𝜏) is fuzzy compact iff every family �〈𝑥𝑥, 𝜇𝜇𝐾𝐾𝑖𝑖 , 𝜎𝜎𝐾𝐾𝑖𝑖 , 𝜈𝜈𝐾𝐾𝑖𝑖 〉: 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 = �〈𝑥𝑥, 𝜇𝜇𝐺𝐺𝑖𝑖 , 𝜎𝜎𝐺𝐺𝑖𝑖 , 𝜈𝜈𝐺𝐺𝑖𝑖 〉: 𝑖𝑖 ∈ 𝐽𝐽� of FNOSs in 𝑋𝑋 satisfies the condition 𝐴𝐴 ⊆∪ 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 = 〈𝑥𝑥, 𝜇𝜇𝐴𝐴 , 𝜎𝜎𝐴𝐴 , 𝜈𝜈𝐴𝐴 〉 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 = 〈𝑥𝑥, 𝜇𝜇𝐴𝐴 , 𝜎𝜎𝐴𝐴 , 𝜈𝜈𝐴𝐴 〉 in a FNTS (𝑋𝑋, 𝜏𝜏) is fuzzy compact iff for each family 𝒢𝒢 = {𝐺𝐺𝑖𝑖 : 𝑖𝑖 ∈ 𝐽𝐽} where 𝐺𝐺𝑖𝑖 = �〈𝑥𝑥, 𝜇𝜇𝐺𝐺𝑖𝑖 , 𝜎𝜎𝐺𝐺𝑖𝑖 , 𝜈𝜈𝐺𝐺𝑖𝑖 〉: 𝑖𝑖 ∈ 𝐽𝐽�, of FNOS in 𝑋𝑋 with properties 𝜇𝜇𝐴𝐴 ≤ ⋁𝑖𝑖∈𝐽𝐽 𝜇𝜇𝐺𝐺𝑖𝑖 , 𝜎𝜎𝐴𝐴 ≤ ⋁𝑖𝑖∈𝐽𝐽 𝜎𝜎𝐺𝐺𝑖𝑖 and 1 − 𝜈𝜈𝐴𝐴 ≤ ⋁𝑖𝑖∈𝐽𝐽 �1 − 𝜈𝜈𝐺𝐺𝑖𝑖 � there exists a finite subfamily {𝐺𝐺𝑖𝑖 : 1, 2, … , 𝑛𝑛} of 𝒢𝒢 such that 𝜇𝜇𝐴𝐴 ≤ ⋁𝑛𝑛𝑖𝑖=1 𝜇𝜇𝐺𝐺𝑖𝑖 , 𝜎𝜎𝐴𝐴 ≤ ⋁𝑛𝑛𝑖𝑖=1 𝜎𝜎𝐺𝐺𝑖𝑖 and 1 − 𝜈𝜈𝐴𝐴 ≤ ⋁𝑛𝑛𝑖𝑖=1�1 − 𝜈𝜈𝐺𝐺𝑖𝑖 �.
Example 5.9: Let 𝑋𝑋 = 𝐼𝐼 and consider the FNSs (𝐺𝐺𝑛𝑛 )𝑛𝑛𝑛𝑛 𝑍𝑍2 , where 𝐺𝐺𝑛𝑛 = 〈𝑥𝑥, 𝜇𝜇𝐺𝐺𝑛𝑛 , 𝜎𝜎𝐺𝐺𝑛𝑛 , 𝜈𝜈𝐺𝐺𝑛𝑛 〉, 𝑛𝑛 = 2, 3, …. and 𝐺𝐺 = 〈𝑥𝑥, 𝜇𝜇𝐺𝐺 , 𝜎𝜎𝐺𝐺 , 𝜈𝜈𝐺𝐺 〉 defined by 0.7, 𝑖𝑖𝑖𝑖 𝑥𝑥 = 0 0.8, 𝑖𝑖𝑖𝑖 𝑥𝑥 = 0 0.1, 𝑖𝑖𝑖𝑖 𝑥𝑥 = 0 ⎧ ⎧ ⎧ ⎪𝑛𝑛𝑛𝑛, 𝑖𝑖𝑖𝑖 0 < 𝑥𝑥 < 1 ⎪(𝑛𝑛 + 1)𝑥𝑥, 𝑖𝑖𝑖𝑖 0 < 𝑥𝑥 < 1 ⎪ 1 − 𝑛𝑛𝑛𝑛, 𝑖𝑖𝑖𝑖 0 < 𝑥𝑥 < 1 𝜇𝜇𝐺𝐺𝑛𝑛 (𝑥𝑥) = 𝜈𝜈𝐺𝐺𝑛𝑛 = 𝑛𝑛 , 𝜎𝜎𝐺𝐺𝑛𝑛 (𝑥𝑥) = 𝑛𝑛 , 𝑛𝑛 ⎨ ⎨ ⎨ 1 1 1 ⎪ 1, 𝑖𝑖𝑖𝑖 < 𝑥𝑥 ≤ 1 ⎪ 1, ⎪ 0, 𝑖𝑖𝑖𝑖 < 𝑥𝑥 ≤ 1 𝑖𝑖𝑖𝑖 < 𝑥𝑥 ≤ 1 ⎩ ⎩ ⎩ 𝑛𝑛 𝑛𝑛 𝑛𝑛 0.7 𝑖𝑖𝑖𝑖 𝑥𝑥 = 0 0.8 𝑖𝑖𝑖𝑖 𝑥𝑥 = 0 and 𝜇𝜇𝐺𝐺 = � , 𝜎𝜎 = � , 1, 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝐺𝐺 1, 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0.1 𝑖𝑖𝑖𝑖 𝑥𝑥 = 0 . 𝜈𝜈𝐺𝐺 = � 1, 𝑜𝑜𝑜𝑜ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 Then 𝜏𝜏 = {0𝑁𝑁 , 1𝑁𝑁 , 𝐺𝐺} ∪ {𝐺𝐺𝑛𝑛 : 𝑛𝑛 ∈ 𝑍𝑍2 } is a FNT on 𝑋𝑋, and consider the FNSs 𝐶𝐶𝛼𝛼 ,𝛽𝛽 ,𝛾𝛾 in (𝑋𝑋, 𝜏𝜏) defined by 𝐶𝐶𝛼𝛼,𝛽𝛽 ,𝛾𝛾 = {〈𝑥𝑥, 𝛼𝛼, 𝛽𝛽, 𝛾𝛾〉: 𝑥𝑥 ∈ 𝑋𝑋}, where 𝛼𝛼, 𝛽𝛽, 𝛾𝛾 ∈ 𝐼𝐼 are arbitrary and 𝛼𝛼 + 𝛽𝛽 + 𝛾𝛾 ≤ 2. Then the FNSs 𝐶𝐶0.75,0.85,0.05 , 𝐶𝐶0.65,0.75,0.15 , 𝐶𝐶0.85,0.850 ,05 are all compact. © 2017, IJMA. All Rights Reserved
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Y. Veereswari* / An Introduction to Fuzzy Neutrosophic Topological Spaces / IJMA- 8(3), March-2017.
Corollary 5.10: Let (đ?‘‹đ?‘‹, đ?œ?đ?œ?) and (đ?‘Œđ?‘Œ, đ?œ‘đ?œ‘) be two FNTSs and let đ?‘“đ?‘“: đ?‘‹đ?‘‹ → đ?‘Œđ?‘Œ be a fuzzy continuous function. If đ??´đ??´ is fuzzy compact in(đ?‘‹đ?‘‹, đ?œ?đ?œ?), then so is đ?‘“đ?‘“(đ??´đ??´) in (đ?‘Œđ?‘Œ, đ?œ‘đ?œ‘). REFERENCES
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