Weakly π Generalized Homeomorphism in Intuitionistic Fuzzy Topology

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American Journal of Engineering Research (AJER) 2016 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-5, Issue-10, pp-277-282 www.ajer.org Research Paper Open Access

Weakly π Generalized Homeomorphism in Intuitionistic Fuzzy Topology 1

K.Kadambavanam and 2K.Vaithiyalingam

1. 2.

Associate Professor, Department of Mathematics, Sri Vasavi College, Erode-638 316, Tamilnadu, India. Assistant Professor, Department of Mathematics, Kongu Engineering College, Perundurai, Erode-638052, Tamilnadu, India.

ABSTRACT: In this paper the concepts of weakly π generalized homeomorphism on intuitionistic fuzzy topological space is introduced with numerical examples. Some of their properties are investigated. Keywords: Intuitionistic fuzzy (IF) topology, IF weakly π generalized closed set and open set, IF completely weakly π generalized continuous mappings and closed mapping, IF weakly π generalized homeomorphism, IFwπ�1 2 space and IFwπg�� space. AMS CLASSIFICATION CODE (2000): 54A40, 03F55, 03E72.

I. INTRODUCTION In 1965, the concept of Fuzzy set (FS) theory and its applications are first proposed by Zadeh [20]. It provides a framework to encounter uncertainty, vagueness and partial truth. It is represented by introducing degree of membership for each member of the universe of discourse to a subset of it. In 1968, the concept of fuzzy topology has been introduced by Chang [3]. They are referred as a concept and its context. The concept of FS has generalized into intuitionistic fuzzy (IF) by Atanassov [1] in 1986. After that many research articles have been published in the study of examining and exploring, how far the basic concepts and theorems, defined in crisp sets and in fuzzy sets remain true in IF sets. In 1997, Coker [4] has initiated the concept of generalization of fuzzy topology into IF topology. In his article, the apprehension of semi closed, ι closed, semi pre-closed, weakly closed are introduced. Further its properties are derived. In this paper, the concept of IF weakly π generalized homeomorphism in IF topological space is introduced and suitable examples are given. Some of its characteristics are obtained. The inter-relationships among the various existed classes, which form IF homeomorphisms are established. Numerical illustrations are also given to substantiate the derived results. This paper is organized into four sections. In the first section, historical development of the concepts is briefed. The basic definitions and results, needed for this work are listed in the second section. Section three discusses the IF weakly π generalized homeomorphism and suitable examples are given. Section four contains the Conclusion remarks.

II. PRELIMINARIES Definition 2.1: [1] Let đ?‘‹ be a non-empty crisp set. An intuitionistic fuzzy (IF) set đ??´, in đ?‘‹ is defined as an object of the form đ??´ = { < đ?‘Ľ, đ?œ‡đ??´ (đ?‘Ľ), đ?œˆđ??´ (đ?‘Ľ) > | đ?‘Ľ đ?œ– đ?‘‹ }, where the functions đ?œ‡đ??´ (đ?‘Ľ) âˆś đ?‘‹ → [0, 1] and đ?œˆđ??´ (đ?‘Ľ) âˆś đ?‘‹ → [0, 1] denote respectively the degree of membership (briefly đ?œ‡đ??´ ) and the degree of non-membership (briefly đ?œˆđ??´ ) of each element đ?‘Ľ đ?œ– đ?‘‹ to the set đ??´, for each đ?‘Ľ đ?œ– đ?‘‹ 0 ≤ đ?œ‡đ??´ đ?‘Ľ + đ?œˆđ??´ đ?‘Ľ ≤ 1 . The collection of all IF sub-sets in đ?‘‹, is denoted by IFS(X). Definition 2.2: [1] Let đ??´ and đ??ľ be two different IFSs defined by, đ??´ = { < đ?‘Ľ, đ?œ‡đ??´ (đ?‘Ľ), đ?œˆđ??´ (đ?‘Ľ) > | đ?‘Ľ đ?œ– đ?‘‹ }, đ??ľ = { < đ?‘Ľ, đ?œ‡đ??ľ (đ?‘Ľ), đ?œˆđ??ľ (đ?‘Ľ) > | đ?‘Ľ đ?œ– đ?‘‹ }. The brief notation of đ??´ = { < đ?‘Ľ, đ?œ‡đ??´ (đ?‘Ľ), đ?œˆđ??´ (đ?‘Ľ) > | đ?‘Ľ đ?œ– đ?‘‹ } and đ??ľ = { < đ?‘Ľ, đ?œ‡đ??ľ (đ?‘Ľ), đ?œˆđ??ľ (đ?‘Ľ) > | đ?‘Ľ đ?œ– đ?‘‹ } are đ??´ = < đ?‘Ľ, Âľđ??´ , đ?œˆđ??´ > and đ??ľ = < đ?‘Ľ, đ?œ‡đ??ľ , đ?œˆđ??ľ > respectively.

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The operations ∧ and ∨ are defined on đ?œ‡đ??´ , đ?œ‡đ??ľ , đ?œˆđ??´ , and đ?œˆđ??ľ as follows: i) đ?œ‡đ??´ ∨ đ?œ‡đ??ľ = max { đ?œ‡đ??´ , đ?œ‡đ??ľ }, ii) đ?œ‡đ??´ ∧ đ?œ‡đ??ľ = min { đ?œ‡đ??´ , đ?œ‡đ??ľ }, iii) đ?œˆđ??´ ∨ đ?œˆđ??ľ = max { đ?œˆđ??´ , đ?œˆđ??ľ }, and iv) đ?œˆđ??´ ∧ đ?œˆđ??ľ = min {đ?œˆđ??´ , đ?œˆđ??ľ }. Then, i) đ??´ ⊆ đ??ľ, if and only if, đ?œ‡đ??´ ≤ đ?œ‡đ??ľ and đ?œˆđ??´ ≼ đ?œˆđ??ľ for all đ?‘Ľ đ?œ– đ?‘‹ , similarly đ??´ ⊇ đ??ľ can be defined. ii) đ??´ = đ??ľ, if and only if, both đ??´ ⊆ đ??ľ and đ??ľ ⊆ đ??´ are valid. iii) đ??´ đ??ś = { < đ?‘Ľ, đ?œ‡đ??´â€˛ , đ?œˆđ??´â€˛ > | đ?‘Ľ đ?œ– đ?‘‹ }, where đ?œ‡đ??´â€˛ = đ?œˆđ??´ and đ?œˆđ??´â€˛ = đ?œ‡đ??´ . iv) đ??´ ∊ đ??ľ = { < đ?‘Ľ, đ?œ‡đ??´ ∧ đ?œ‡đ??ľ , đ?œˆđ??´ ∨ đ?œˆđ??ľ > | đ?‘Ľ đ?œ– đ?‘‹ }, and v) đ??´ âˆŞ đ??ľ = { < đ?‘Ľ, đ?œ‡đ??´ ∨ đ?œ‡đ??ľ , đ?œˆđ??´ ∧ đ?œˆđ??ľ > | đ?‘Ľ đ?œ– đ?‘‹ } . The intuitionistic fuzzy sets 0âˆź and 1âˆź are defined respectively as, 0âˆź = { < đ?‘Ľ, 0, 1 > | đ?‘Ľ đ?œ– đ?‘‹ } and 1âˆź = { < đ?‘Ľ, 1, 0 > | đ?‘Ľ đ?œ– đ?‘‹ }. The sets 0âˆź and 1âˆź are known as the empty IF set and the whole IF set of đ?‘‹ respectively. Definition 2.3: [4] An intuitionistic fuzzy topology (IFT) is a family đ?œ? of IFS defined on đ?‘‹, satisfying the following axioms : i) 0âˆź , 1âˆź đ?œ– đ?œ? , ii) đ??ş1 ∊ đ??ş2 đ?œ– đ?œ? , whenever đ??ş1 , đ??ş2 đ?œ– đ?œ?, iii) âˆŞ đ??şđ?‘– đ?œ– đ?œ? for any arbitrary family đ??şđ?‘– đ?‘– đ?œ– đ??˝ } ⊆ đ?œ?. Then the pair (đ?‘‹, đ?œ?)is called an intuitionistic fuzzy topological space (IFTS) and any IFS in Ď„ are known as an intuitionistic fuzzy open set (IFOS) in đ?‘‹. If, đ??´ is an IFOS, in an IFTS đ?‘‹, đ?œ? , then its complement đ??´ đ??ś is called an intuitionistic fuzzy closed set (IFCS) in đ?‘‹. Definition 2.4: [4] Let đ?‘‹, đ?œ? be an IFTS and đ??´ = < đ?‘Ľ, đ?œ‡đ??´ , đ?œˆđ??´ > be an IFS in đ?‘‹. Then the intuitionistic fuzzy closure and an intuitionistic fuzzy interior are defined by , đ?‘?đ?‘™(đ??´) = ∊ { đ??ş | đ??ş đ?‘–đ?‘ đ?‘Žđ?‘› đ??źđ??šđ??śđ?‘† đ?‘–đ?‘› đ?‘‹ đ?‘Žđ?‘›đ?‘‘ đ??´ ⊆ đ??ş}, and đ?‘–đ?‘›đ?‘Ą (đ??´) = âˆŞ { đ??ž | đ??ž đ?‘–đ?‘ đ?‘Žđ?‘› đ??źđ??šđ?‘‚đ?‘† đ?‘–đ?‘› đ?‘‹ đ?‘Žđ?‘›đ?‘‘ đ??ž ⊆ đ??´}. Note that for any IFS, đ??´ in đ?‘‹, đ?‘?đ?‘™ đ??´đ??ś = đ?‘–đ?‘›đ?‘Ą đ??´

đ?‘?

and đ?‘–đ?‘›đ?‘Ą (đ??´đ?‘? ) =

đ?‘?đ?‘™ đ??´

đ?‘?

.

Definition 2.5: [4] An IFS, đ??´ = < đ?‘Ľ, đ?œ‡đ??´ , đ?œˆđ??´ > in an IFTS( đ?‘‹, đ?œ?) is said to be an, i) intuitionistic fuzzy closed set (IFCS) in đ?‘‹ ⇔ đ?‘?đ?‘™(đ??´) = đ??´, and ii) intuitionistic fuzzy open set (IFOS) in đ?‘‹ ⇔ đ?‘–đ?‘›đ?‘Ą đ??´ = đ??´. Definition 2.6: [14] A subset đ??´ of a space (đ?‘‹, đ?œ?) is called, i) regular open, if, đ??´ = đ?‘–đ?‘›đ?‘Ą đ?‘?đ?‘™ đ??´ , and ii) Ď€ open, if, đ??´ is the union of regular open sets, symbolically đ??´ is an IFĎ€OS in X. Definition 2.7: [5] An IFS đ??´ = < đ?‘Ľ, đ?œ‡đ??´ , đ?œˆđ??´ > in an IFTS( đ?‘‹, đ?œ?) is said to be an, i) intuitionistic fuzzy semi-closed set (IFSCS) if đ?‘–đ?‘›đ?‘Ą đ?‘?đ?‘™ đ??´ ⊆ đ??´, and ii) intuitionistic fuzzy semi-open set (IFSOS) if đ??´ ⊆ đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ??´ . Definition 2.8: [18] Let đ??´ be an IFS of an IFTS (đ?‘‹, đ?œ?). Then the semi-closure of đ??´ (simply đ?‘ đ?‘?đ?‘™(đ??´)) and semi- interior of đ??´ (simply đ?‘ đ?‘–đ?‘›đ?‘Ą(đ??´) ) are defined as, đ?‘– ) đ?‘ đ?‘?đ?‘™(đ??´) = ∊ { đ??ş | đ??ş đ?‘–đ?‘ đ?‘Žđ?‘› đ??źđ??šđ?‘†đ??śđ?‘† đ?‘–đ?‘› đ?‘‹ đ?‘Žđ?‘›đ?‘‘ đ??´ ⊆ đ??ş}, đ?‘–đ?‘– ) đ?‘ đ?‘–đ?‘›đ?‘Ą đ??´ = âˆŞ { đ??ž | đ??ž đ?‘–đ?‘ đ?‘Žđ?‘› đ??źđ??šđ?‘†đ?‘‚đ?‘† đ?‘–đ?‘› đ?‘‹ đ?‘Žđ?‘›đ?‘‘ đ??ž ⊆ đ??´}. Result 2.1: [16] Let đ??´ be an IFS in đ?‘‹, đ?œ? , then i) đ?‘ đ?‘?đ?‘™ đ??´

= đ??´ âˆŞ đ?‘–đ?‘›đ?‘Ą đ?‘?đ?‘™ đ??´ , and

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ii) đ?‘ đ?‘–đ?‘›đ?‘Ą(đ??´) = đ??´ ∊ đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ??´ . Definition 2.9: [5] An IFS đ??´ = < đ?‘Ľ, đ?œ‡đ??´ , đ?œˆđ??´ > in an IFTS( đ?‘‹, đ?œ?) is said to be an,

i) intuitionistic fuzzy Îą closed set (IFÎąCS), if, đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ?‘?đ?‘™ đ??´

⊆ đ??´, and

ii) intuitionistic fuzzy Îą open set (IFÎąOS), if, đ??´ ⊆ đ?‘–đ?‘›đ?‘Ą đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ??´

.

Definition 2.10: [11] Let đ??´ = < đ?‘Ľ, đ?œ‡đ??´ , đ?œˆđ??´ > be an IFS of an IFTS đ?‘‹, đ?œ? . Then, the Îą closure of đ??´ (đ?›ź đ?‘?đ?‘™(đ??´)) and Îą interior of đ??´ đ?›ź đ?‘–đ?‘›đ?‘Ą đ??´ are defined as, đ?›ź đ?‘?đ?‘™(đ??´) đ?›ź đ?‘–đ?‘›đ?‘Ą đ??´

= ∊ { đ??ş | đ??ş đ?‘–đ?‘ đ?‘Žđ?‘› đ??źđ??šđ?›źđ??śđ?‘† đ?‘–đ?‘› đ?‘‹ đ?‘Žđ?‘›đ?‘‘ đ??´ ⊆ đ??ş}, and = âˆŞ { đ??ž | đ??ž đ?‘–đ?‘ đ?‘Žđ?‘› đ??źđ??šđ?›źđ?‘‚đ?‘† đ?‘–đ?‘› đ?‘‹ đ?‘Žđ?‘›đ?‘‘ đ??ž ⊆ đ??´}.

Result 2.2: [12] Let đ??´ be an IFS in đ?‘‹, đ?œ? , then, i) đ?›ź đ?‘?đ?‘™ đ??´ = đ??´ âˆŞ đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ?‘?đ?‘™ đ??´ ii) đ?›ź đ?‘–đ?‘›đ?‘Ą(đ??´) = đ??´ ∊ đ?‘–đ?‘›đ?‘Ą đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ??´

, and .

Definition 2.11: An IFS, đ??´ = < đ?‘Ľ, đ?œ‡đ??´ , đ?œˆđ??´ > in an IFTS đ?‘‹, đ?œ? is said to be an i) intuitionistic fuzzy pre-closed set [5] (IFPCS) if, đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ??´ ⊆ đ??´ , ii) intuitionistic fuzzy regular closed set [5] (IFRCS) if, đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ??´ = đ??´ , iii)intuitionistic fuzzy generalized closed set [17] (IFGCS) if,đ?‘?đ?‘™ đ??´ ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ, đ?‘ˆ is an IFOS in đ?‘‹, iv)intuitionistic fuzzy generalized semi closed set [13] (IFGSCS) if, đ?‘ đ?‘?đ?‘™ đ??´ ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ, đ?‘ˆ is an IFOS in đ?‘‹, v)intuitionistic fuzzy Îą generalized closed set [12] (IFÎąGCS) if, đ?›ź đ?‘?đ?‘™ đ??´ ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ,đ?‘ˆ is an IFOS in đ?‘‹. Definition 2.12: [7] An IFS, đ??´ is said to be an intuitionistic fuzzy weakly Ď€ generalized closed set (IFWĎ€GCS) in đ?‘‹, đ?œ? if, đ?‘?đ?‘™ đ?‘–đ?‘›đ?‘Ą đ??´ ⊆ đ?‘ˆ whenever đ??´ ⊆ đ?‘ˆ and đ?‘ˆ is an IFĎ€OS in đ?‘‹. The family of all IFWĎ€GCS of an IFTS đ?‘‹, đ?œ? is denoted by IFWĎ€GCS(X). Result 2.3: Every IFCS, IFÎąCS, IFGCS, IFRCS, IFPCS, IFÎąGCS are IFWĎ€GCS [7] but the converse need not be true. Definition 2.13: [7] An IFS, đ??´ is said to be an intuitionistic fuzzy weakly Ď€ generalized open set (IFWĎ€GOS) in đ?‘‹, đ?œ? if, the complement đ??´đ?‘? is an IFWĎ€GOS in đ?‘‹. The family of all IFWĎ€GOS of an IFTS đ?‘‹, đ?œ? is denoted by IFWĎ€GOS (đ?‘‹). Definition 2.14: [5] Let đ?‘“ be a mapping defined on an IFTS đ?‘‹, đ?œ? into IFTS đ?‘Œ, đ?œŽ . Then đ?‘“ is said to be intuitionistic fuzzy continuous (IF cts) if, đ?‘“ −1 đ??ľ ∈ IFOS (đ?‘‹) for every đ??ľ ∈ đ?œŽ . Definition 2.15: Let đ?‘“ be a mapping from an IFTS đ?‘‹, đ?œ? into IFTS đ?‘Œ, đ?œŽ . Then đ?‘“ is said to be i) intuitionistic fuzzy weakly Ď€ generalized continuous mapping [8] (IFWĎ€G cts) if, đ?‘“ −1 (đ??ľ) is an IFWĎ€GCS in (đ?‘‹, đ?œ?) for every IFCS, đ??ľ of đ?‘Œ, đ?œŽ , ii) intuitionistic fuzzy semi continuous mapping [19] (IFS cts) if, đ?‘“ −1 (đ??ľ) ∈ IFSO (đ?‘‹), for every đ??ľ ∈ đ?œŽ, iii) intuitionistic fuzzy Îą continuous mapping [19] (IfÎą cts) if, đ?‘“ −1 (đ??ľ) ∈ IFÎąO (đ?‘‹), for every đ??ľ ∈ đ?œŽ, iv) intuitionistic fuzzy pre-continuous mapping [19] (IFP cts) if, đ?‘“ −1 (đ??ľ) ∈ IFPO (đ?‘‹), for every đ??ľ ∈ đ?œŽ, v) intuitionistic fuzzy completely continuous mapping [6] if, đ?‘“ −1 (đ??ľ) ∈ IFRO (đ?‘‹), for every đ??ľ ∈ đ?œŽ, vi) intuitionistic fuzzy generalized continuous mapping [13] (IFG cts) if, đ?‘“ −1 (đ??ľ) ∈ IFGO (đ?‘‹), for every đ??ľ ∈ đ?œŽ,

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vii) intuitionistic fuzzy generalized semi continuous mapping [12] (IFGS cts) if, đ?‘“ −1 (đ??ľ) ∈ IFGSO (đ?‘‹), for every đ??ľ ∈ đ?œŽ, and viii) intuitionistic fuzzy Îą generalized continuous mapping [12](IFÎąG cts) if, đ?‘“ −1 (đ??ľ) ∈ IFÎąGO (đ?‘‹), for every đ??ľ ∈ đ?œŽ. Definition 2.16: [9] Let đ?‘‹, đ?œ? and đ?‘Œ, đ?œŽ be two IFTS. A mapping đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ is called an intuitionistic fuzzy weakly Ď€ generalized closed mapping (IFWĎ€GCM) if, đ?‘“(đ??´) is an IFWĎ€GCS in đ?‘Œ, for every IFCS, đ??´ in đ?‘‹. In other words, every IFCS in đ?‘‹ are mapped into IFWĎ€GCS in đ?‘Œ. Definition 2.17: [10] Let đ?‘‹, đ?œ? and đ?‘Œ, đ?œŽ be two IFTS. A mappingđ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ is called an intuitionistic fuzzy completely weakly Ď€ generalized continuous mapping (IF completely WĎ€G cts) if, đ?‘“ −1 (đ??ľ) is an IFRCS in (đ?‘‹, đ?œ?) for every IFWĎ€GCS, đ??ľ of (đ?‘Œ, đ?œ?). Definition 2.18: [7] An IFTS đ?‘‹, đ?œ? is called an intuitionistic fuzzy wĎ€đ?‘‡1 in đ?‘‹ is an IFCS in đ?‘‹.

2

(IF wπ�1 2 ) space if, every IFWπGCS

Definition 2.19: [7] An IFTS đ?‘‹, đ?œ? is called an intuitionistic fuzzy wĎ€gđ?‘‡đ?‘ž (IF wĎ€gđ?‘‡đ?‘ž ) space, ( 0 < q < 1 ) if, every IFWĎ€GCS in đ?‘‹ is an IFPCS in đ?‘‹.

III. INTUITIONISTIC FUZZY WEAKLY Ď€ GENERALIZEDHOMEOMORPHISM The main objective of this section is to study the weakly Ď€ generalized homeomorphism defined on a topological spaces, in this connection some of its properties are obtained. Definition 3.1: Let đ?‘‹, đ?œ? and đ?‘Œ, đ?œŽ be two IFTS. A bijection mapping đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ is called an intuitionistic fuzzy weakly Ď€ generalized (IFWĎ€G) homeomorphism if, both the functions, đ?‘“ and đ?‘“ −1 are IFWĎ€G continuous mappings. Example 3.1: Let đ?‘‹ = đ?‘Ž, đ?‘?, đ?‘? , đ?‘Œ = {đ?‘˘, đ?‘Ł, đ?‘¤}, đ??ş1 = < đ?‘Ľ, 0.2, 0.3,0.4 , 0.8, 0.6,0.6 > for đ?‘Ľ ∈ đ?‘‹ and đ??ş2 = < đ?‘Ś, 0.8, 0.6,0.6 , 0.2, 0.3,0.4 > for đ?‘Ś ∈ đ?‘Œ. Then đ?œ? ={ 0âˆź , đ??ş1 , 1âˆź } and đ?œŽ = { 0âˆź , đ??ş2 , 1âˆź } are IFTs on đ?‘‹ and đ?‘Œ respectively. Define a bijection mapping đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ by đ?‘“(đ?‘Ž) = đ?‘˘, đ?‘“ đ?‘? = đ?‘Ł and đ?‘“(đ?‘?) = đ?‘¤. Then, đ?‘“ and đ?‘“ −1 are IFWĎ€G continuous mappings. Therefore đ?‘“ is an IFWĎ€G homeomorphism. Proposition 3.1: Every IF homeomorphism is an IFWĎ€G homeomorphism. Proof: Let đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ be an IF homeomorphism. Then đ?‘“ and đ?‘“ −1 are IF continuous mappings. This implies đ?‘“ and đ?‘“ −1 are IFWĎ€G continuous mappings. Therefore đ?‘“ is an IFWĎ€G homeomorphism. Example 3.2: (Converse of Proposition 3.1 need not be true) Let đ?‘‹ = đ?‘Ž, đ?‘?, đ?‘? , đ?‘Œ = {đ?‘˘, đ?‘Ł, đ?‘¤}, đ??ş1 = < đ?‘Ľ, 0.3, 0.4,0.5 , 0.7, 0.6,0.5 > for đ?‘Ľ ∈ đ?‘‹ and đ??ş2 = < đ?‘Ś, 0.7, 0.7,0.6 , 0.3, 0.3,0.4 > for đ?‘Ś ∈ đ?‘Œ. Then đ?œ? ={ 0âˆź , đ??ş1 , 1âˆź } and đ?œŽ = { 0âˆź , đ??ş2 , 1âˆź } are IFTs on đ?‘‹ and đ?‘Œ respectively. Consider a bijective mapping đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ defined by đ?‘“(đ?‘Ž) = đ?‘˘, đ?‘“(đ?‘?) = đ?‘Ł and đ?‘“(đ?‘?) = đ?‘¤ . Then, đ?‘“ is an IFWĎ€G homeomorphism but not an IF homeomorphism, since đ?‘“ and đ?‘“ −1 are not IF continuous mappings. Proposition 3.2: Every IFÎą homeomorphism is an IFWĎ€G homeomorphism. Proof: Let đ?‘‹, đ?œ? and đ?‘Œ, đ?œŽ be two IFTS. Suppose, đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ is an IFÎą homeomorphism. Then both đ?‘“ and đ?‘“ −1 are IFÎą continuous mappings. This implies đ?‘“ and đ?‘“ −1 are IFWĎ€G continuous mappings. Therefore đ?‘“ is an IFWĎ€G homeomorphism. Example 3.3: (Converse of Proposition 3.2 need not be true) Let đ?‘‹ = đ?‘Ž, đ?‘?, đ?‘? , đ?‘Œ = {đ?‘˘, đ?‘Ł, đ?‘¤}, đ??ş1 = < đ?‘Ľ, 0.4, 0.3,0.2 , 0.6, 0.7,0.8 > for đ?‘Ľ ∈ đ?‘‹, and đ??ş2 = < đ?‘Ś, 0.7, 0.8,0.8 , 0.3, 0.2,0.2 > for đ?‘Ś ∈ đ?‘Œ. Then đ?œ? ={ 0âˆź , đ??ş1 , 1âˆź } and đ?œŽ = { 0âˆź , đ??ş2 , 1âˆź } are IFTs on đ?‘‹ and đ?‘Œ respectively. Define a bijective mapping đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ defined by đ?‘“(đ?‘Ž) = đ?‘˘, đ?‘“(đ?‘?) = đ?‘Ł and đ?‘“(đ?‘?) = đ?‘¤ . Then, đ?‘“ is an IFWĎ€G homeomorphism but not an IFÎą homeomorphism, because đ?‘“ and đ?‘“ −1 are not IFÎą continuous mappings. Proposition 3.3: Every IFG homeomorphism is an IFWĎ€G homeomorphism. Proof: Let đ?‘‹, đ?œ? and đ?‘Œ, đ?œŽ be two IFTS and đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ be an IFG homeomorphism. Then đ?‘“ and đ?‘“ −1 are IFG continuous mappings. This implies đ?‘“ and đ?‘“ −1 are IFWĎ€G continuous mappings. So, đ?‘“ is an IFWĎ€G homeomorphism.

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Example 3.4: (Converse of Proposition 3.3 need not be true) Let đ?‘‹ = đ?‘Ž, đ?‘?, đ?‘? , đ?‘Œ = {đ?‘˘, đ?‘Ł, đ?‘¤} đ??ş1 = < đ?‘Ľ, 0.2, 0.3,0.4 , 0.8, 0.7,0.6 > for đ?‘Ľ ∈ đ?‘‹, and đ??ş2 = < đ?‘Ś, 0.8, 0.8,0.7 , 0.2, 0.2,0.2 > for ∈ đ?‘Œ . Then đ?œ? ={ 0âˆź , đ??ş1 , 1âˆź } and đ?œŽ = { 0âˆź , đ??ş2 , 1âˆź } are IFTs on đ?‘‹ and đ?‘Œ respectively. Consider a bijective mapping đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ defined by đ?‘“(đ?‘Ž) = đ?‘˘, đ?‘“(đ?‘?) = đ?‘Ł and đ?‘“(đ?‘?) = đ?‘¤ . Then, đ?‘“ is an IFWĎ€G homeomorphism but not an IFG homeomorphism, since both đ?‘“ and đ?‘“ −1 are not IFG continuous mappings. Proposition 3.4: Every IFÎąG homeomorphism is an IFWĎ€G homeomorphism. Proof: Let đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ be an IFÎąG homeomorphism. Then đ?‘“ and đ?‘“ −1 are IFÎąG continuous mappings. This implies that both đ?‘“ and đ?‘“ −1 are IFWĎ€G continuous mappings. Therefore đ?‘“ is an IFWĎ€G homeomorphism. Example 3.5: Let đ?‘‹ = đ?‘Ž, đ?‘?, đ?‘? , đ?‘Œ = {đ?‘˘, đ?‘Ł, đ?‘¤} and đ??ş1 = < đ?‘Ľ, 0.5, 0.6,0.6 , 0.5, 0.4,0.4 > , đ??ş2 = < đ?‘Ś, 0.6, 0.5,0.5 , 0.3, 0.5,0.5 >. Then đ?œ? ={ 0âˆź , đ??ş1 , 1âˆź } and đ?œŽ = { 0âˆź , đ??ş2 , 1âˆź } are IFTs on đ?‘‹ and đ?‘Œ respectively. Consider a bijective mapping đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ , given by đ?‘“(đ?‘Ž) = đ?‘˘, đ?‘“(đ?‘?) = đ?‘Ł and đ?‘“(đ?‘?) = đ?‘¤ . Then, đ?‘“ is an IFWĎ€G homeomorphism but not an IFÎąG homeomorphism, since đ?‘“ and đ?‘“ −1 are not IFÎąG continuous mappings. Proposition 3.5: Let đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ be a bijective mapping from an IFTS đ?‘‹, đ?œ? onto an IFTS đ?‘Œ, đ?œŽ . Then the following statements are equivalent: a) đ?‘“ is an IFWĎ€G open mapping, b) đ?‘“ is an IFWĎ€G closed mapping, c) đ?‘“ −1 : đ?‘Œ, đ?œŽ → đ?‘‹, đ?œ? is an IFWĎ€G continuous mapping. Proof: (i) (a) ⇒ (b): Let đ??´ be an IFCS in đ?‘‹, then đ??´đ?‘? is an IFOS in đ?‘‹. By Definition, đ?‘“ đ??´đ?‘? = (đ?‘“ đ??´ )đ?‘? is an IFWĎ€G open set in đ?‘Œ. Therefore đ?‘“(đ??´) is an IFWĎ€G closed set in đ?‘Œ . Thus đ?‘“ is an IFWĎ€G closed mapping. (ii) (b) ⇒(c): Let đ??´ be an IFCS in đ?‘‹. Since đ?‘“ is an IFWĎ€G closed mapping, đ?‘“ đ??´ = đ?‘“ −1 closed set in đ?‘Œ. So, đ?‘“ −1 is an IFWĎ€G continuous mapping. (iii)(c) ⇒(a): Let đ??´ be an IF open set in đ?‘‹. By Definition đ?‘“ −1 Therefore đ?‘“ is an IFWĎ€G open mapping.

−1

−1

đ??´ is an IFWĎ€G

đ??´ = đ?‘“ đ??´ is an IFWĎ€G open set in đ?‘Œ.

Hence from (i) to (iii), all the statements (a) to (c) in the proposition (3.5) are equivalent. Proposition 3.6: Let đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ be a bijective mapping from an IFTS đ?‘‹, đ?œ? onto an IFTS đ?‘Œ, đ?œŽ . If đ?‘“ is an IFWĎ€G continuous mapping , then the following statements are equivalent: a) đ?‘“ is an IFWĎ€G closed mapping, b) đ?‘“ is an IFWĎ€G open mapping, c) đ?‘“ is an IFWĎ€G homeomorphism. Note(3.1):The proof of the above proposition (3.6) is similar to the proof of the proposition(3.5). Remark 3.1: The composition of two IFWĎ€G homeomorphism need not be an IFWĎ€G homeomorphism in general. It is illustrated by means of the following example(3.6). Example 3.6: Let đ?‘‹ = đ?‘Ž, đ?‘?, đ?‘? , đ?‘Œ = {đ?‘˘, đ?‘Ł, đ?‘¤} and đ?‘? = đ?‘?, đ?‘ž, đ?‘&#x; . Let đ??ş1 = < đ?‘Ľ, 0.7, 0.5, 0.5 , 0.3, 0.5,0.5 > for đ?‘Ľ ∈ đ?‘‹, đ??ş2 = < đ?‘Ś, 0.7, 0.8,0.8 , 0.3, 0.2,0.2 > for đ?‘Ś ∈ đ?‘Œ and đ??ş3 = < đ?‘§, 0.8, 0.8,0.7 , 0.2, 0.2,0.3 > for z ∈ đ?‘? . Then, đ?œ? ={ 0âˆź , đ??ş1 , 1âˆź } and đ?œŽ = { 0âˆź , đ??ş2 , 1âˆź } and ÂŁ = {0âˆź , đ??ş3 , 1âˆź } are IFTs on đ?‘‹, đ?‘Œ and đ?‘? respectively. Define a bijection mapping đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ by đ?‘“(đ?‘Ž) = đ?‘˘, đ?‘“(đ?‘?) = đ?‘Ł and đ?‘“(đ?‘?) = đ?‘¤ and đ?‘”: đ?‘Œ, đ?œŽ → đ?‘?, ÂŁ by đ?‘”(đ?‘˘) = đ?‘?, đ?‘”(đ?‘Ł) = đ?‘ž and đ?‘” đ?‘¤ = đ?‘&#x;. Then, both the functions đ?‘“ and đ?‘“ −1 are IFWĎ€G continuous mappings. Also đ?‘” and đ?‘”−1 are IFWĎ€G continuous mappings. Therefore đ?‘“ and đ?‘” are IFWĎ€G homeomorphism. But the composition đ?‘” ∘ đ?‘“: (đ?‘‹, đ?œ?) → đ?‘?, ÂŁ is not an IFWĎ€G homeomorphism, since đ?‘” ∘ đ?‘“ is not an IFWĎ€G continuous mapping.

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Proposition 3.7: Let đ?‘“: (đ?‘‹, đ?œ?) → (đ?‘Œ, đ?œŽ) be an IFWĎ€G homeomorphism from an IFTS (X,Ď„) into an IFTS đ?‘Œ, đ?œŽ . Then đ?‘“ is an IF homeomorphism if đ?‘‹, đ?œ? and (đ?‘Œ, đ?œŽ) are IFwĎ€T1 2 spaces. Proof : Let đ??ľ be an IFCS in đ?‘Œ. By definition, đ?‘“ −1 (đ??ľ) is an IFWĎ€GCS in đ?‘‹. Since đ?‘‹, đ?œ? is an IF wĎ€T1 2 space, đ?‘“ −1 (đ??ľ) is an IFCS in đ?‘‹. So, đ?‘“ is an IF continuous mapping. Also by definition, đ?‘“ −1 : đ?‘Œ, đ?œŽ → đ?‘‹, đ?œ? is an IFWĎ€G continuous mapping. Let đ??´ be an IFCS in đ?‘‹. Then (đ?‘“ −1 )−1 đ??´ = đ?‘“(đ??´) is an IFWĎ€GCS in đ?‘Œ, by definition. Since đ?‘Œ, đ?œŽ is an IF wĎ€T1 2 space, đ?‘“(đ??´) is an IFCS in đ?‘Œ. Therefore đ?‘“ −1 is an IF continuous mapping. Thus đ?‘“ is an IF homeomorphism. Proposition 3.8: Let đ?‘‹, đ?œ? and đ?‘Œ, đ?œŽ be two IFTS . Suppose, đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ and đ?‘”: đ?‘Œ, đ?œŽ → đ?‘?, ÂŁ be two IFWĎ€G homeomorphisms and đ?‘Œ, đ?œŽ is an IFwĎ€đ?‘‡1 2 space. Then đ?‘” ∘ đ?‘“ is an IFWĎ€G homeomorphism. Proof: Let đ??´ be an IFCS in đ?‘?. Since đ?‘”: đ?‘Œ, đ?œŽ → đ?‘?, ÂŁ is an IFWĎ€G continuous mapping, đ?‘”−1 (đ??´) is an IFWĎ€GCS in đ?‘Œ. Then đ?‘”−1 (đ??´) is an IFCS in đ?‘Œ as đ?‘Œ, đ?œŽ is an IFwĎ€đ?‘‡1 2 space. Also since đ?‘“: đ?‘‹, đ?œ? → đ?‘Œ, đ?œŽ is an IFWĎ€G continuous mapping, đ?‘“ −1 (đ?‘”−1 đ??´ ) = (đ?‘” ∘ đ?‘“ )−1 (đ??´) is an IFWĎ€GCS in X. Therefore đ?‘” ∘ đ?‘“ is an IFWĎ€G continuous mapping.

IV. CONCLUSION In this paper, a particular type of intuitionistic fuzzy homeomorphism, namely intuitionistic fuzzy weakly π generalized homeomorphism is defined. The relationships among the various existed classes, which form IF homeomorphisms are established. By means of suitable numerical examples, it is established that the converse of the propositions describing the properties, need not be true.

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