Generalized Pre-semi Homeomorphisms in Intuitionistic Fuzzy Topological Spaces

Generalized Pre-semi Homeomorphisms in Intuitionistic Fuzzy Topological Spaces

1T. Sampoornam, 2Gnanambal Ilango and 3K. M. Arifmohammed

1Associate Professor and Head, Department of Mathematics, Government Arts College, Coimbatore-641018, (T.N.), India.

2Assistant Professor, Department of Mathematics, Government Arts College, Coimbatore-641018, (T.N.), India.

3Department of Mathematics, NGM College, Pollachi-642001, Tamil Nadu, India.

AbstractIn this paper we introduce intuitionistic fuzzy generalized pre-semi homeomorphisms and intuitionistic fuzzy i-generalized pre-semi homeomorphisms. We investigate some of their properties.

KeywordsIntuitionistic fuzzy topology, intuitionistic fuzzy generalized pre-semi closed set, intuitionistic fuzzy generalized pre-semi homeomorphism and intuitionistic fuzzy i-generalized pre-semi homeomorphism.

2010 Mathematical Subject Classification: 54A40, 03F55

1. Introduction

   In 1965, Zadeh [15] introduced fuzzy sets and in 1968, Chang [2] introduced fuzzy topology. After the introduction of fuzzy set and fuzzy topology, several authors were conducted on the generalization of this notion. The notion of intuitionistic fuzzy sets was introduced by Atanassov [1] as a generalization of fuzzy sets. 1997, Coker

   [3] introduced the concept of intuitionistic fuzzy topological spaces. In this paper we introduce intuitionistic fuzzy generalized pre-semi homeomorphisms and intuitionistic fuzzy i-generalized pre-semi homeomorphisms. We investigate some of their properties. We also provide the relationship among various homeomorphisms.

2. Preliminaries

   Definition 2.1: [1] Let X be a non-empty fixed set. An intuitionistic fuzzy set (IFS in short) A in X is an object having the form A = {x, µ (x), (x)/ x X} where the functions

   1. A B if and only if µA(x) µB(x) and A(x) B(x) for all x X.

   2. A = B if and only if A B and B A.

   3. Ac = { x, A(x), µA(x)/ x X }.

   (iv) A B = { x, µA(x) µB(x), A(x) B(x)/ x X }.

   (v) A B = { x, µA(x) µB(x), A(x) B(x)/ x X }.

   For the sake of simplicity, we shall use the notation A = x,

   µA, A instead of A = { x, µA(x), A(x)/ x X }. Also for the sake of simplicity, we shall use the notation A = x, (µA,

   µB), (A, B) instead of A = x, (A/A, B/B), (A/A, B/B). The intuitionistic fuzzy sets 0~ = { x, 0, 1/ x X } and 1~ =

   { x, 1, 0/ x X } are respectively the empty set and the whole set of X.

   Definition 2.3: [3] An intuitionistic fuzzy topology (IFT in short) on X is a family of IFSs in X satisfying the following axioms:

   (i) 0~, 1~ .

   1. G1 G2 , for any G1, G2 .

   2. Gi for any family {Gi / i J} .

   In this case the pair (X, ) is called an intuitionistic fuzzy topological space(IFTS in short) and any IFS in is known as an intuitionistic fuzzy open set(IFOS in short)in X. The complement Ac of an IFOS A in an IFTS (X, ) is called an intuitionistic fuzzy closed set (IFCS in short) in X.

   Definition 2.4: [3] Let (X, ) be an IFTS and A = x, µA, A be an IFS in X. Then

   1. int(A) = { G / G is an IFOS in X and G A }.

   2. cl(A) = { K / K is an IFCS in X and A K }.

   3. cl(Ac) = (int(A))c.

   4. int(Ac) = (cl(A))c.

   A A

   µA: X [0,1] and A : X [0,1] denote the degree of membership (namely µA(x)) and the degree of non- membership (namely A(x)) of each element x X to the set A, respectively, and 0 µA(x) + A(x) 1 for each x X. Denote by IFS(X), the set of all intuitionistic fuzzy sets in X.

   Definition 2.2: [1] Let A and B be IFSs of the form A = {x,

   µA(x), A(x)/ x X} and B = {x, µB(x), B(x)/ x X}. Then

   Definition 2.5: [4] Let A = <x, µA, A> be an IFS in X. Then

   1. pint(A) = {G : G is an IF P OS in X and G A}.

   2. pcl(A) = {K : K is an IF P CS in X and A K}.

   Definition 2.6: [4] An IFS A of an IFTS (X, ) is an

   1. intuitionistic fuzzy semiclosed set (IFSCS in short) if int(cl(A)) A.

   2. intuitionistic fuzzy semiopen set (IFSOS in short) if A

   cl(int(A)).

   Definition 2.7: [8] An IFS A in an IFTS (X, ) is said to be an intuitionistic fuzzy generalized pre-semi closed set (IFGPSCS for short) if pcl(A) U whenever A U and U is an IFSOS in (X, ). An IFS A is said to be an intuitionistic fuzzy generalized pre-semi open set (IFGPSOS for short) in (X, ) if the complement Ac is an IFGPSCS in X.

   Definition 2.8: [4] Let f: (X, ) (Y, ) be a mapping from an IFTS (X, ) into an IFTS (Y, ). Then f is said to be an

   1. intuitionistic fuzzy continuous (IF continuous in short) if f -1(B) IFO(X) for every B .

   2. intuitionistic fuzzy continuous(IF continuous in short) if f -1(B) IFO(X) for every B .

   Definition 2.9: [9] A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy generalized pre-semi continuous (IFGPS

   (IFGSPCM for short) if f (A) is an IFGSPCS in Y for each IFCS A in X.

   Definition 2.17: [6] A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy generalized semipre regular closed mapping (IFGSPRCM for short) if f (A) is an IFGSPRCS in Y for each IFCS A in X.

   Definition 2.18: [10] A mapping f: (X, ) (Y, ) is said to be an intuitionistic fuzzy i-generalized presemi closed mapping (IFiGPSCM for short) if f(A) is an IFGPSCS in Y for every IFGPSCS A in X.

   Definition 2.19: [10] A mapping f: (X, ) (Y, ) is said to be an intuitionistic fuzzy i-generalized presemi open mapping (IFiGPSOM for short) if f(A) is an IFGPSOS in Y for every IFGPSOS A in X.

   Definition 2.20: [8] If every IFGPSCS in (X, ) is an IFPCS in (X, ), then the space can be called as an intuitionistic fuzzy

   continuous for short) mappings if f -1(V) is an IFGPSCS in (X,

   ) for every IFCS V of (Y, ).

   presemi T

   1/2

   (IFPST

   1/2

   space for short) space.

   1/2

   1/2

   Definition 2.21: [8] An IFTS (X, ) is said to be an

   Definition 2.10: [11] A mapping f: (X, ) (Y, ) is called an

   intuitionistic fuzzy presemi T*1/2 space (IFPST* space for

   intuitionistic fuzzy generalized semi-pre continuous (IFGSP continuous for short) mapping if f -1(V) is an IFGSPCS in (X, ) for every IFCS V of (Y, ).

   Definition 2.11: [5] A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy generalized semipre regular continuous (IFGSPR continuous for short) mapping if f -1(V) is an IFGSPRCS in (X, ) for every IFCS V of (Y, ).

   Definition 2.12: [9] A mapping f: (X, ) (Y, ) be an intuitionistic fuzzy generalized pre-semi irresolute (IFGPS irresolute) mapping if f -1(V) is an IFGPSCS in (X, ) for every IFGPSCS V of (Y, ).

   Definition 2.13: [14] A map f: (X, ) (Y, ) is called an

   1. intuitionistic fuzzy closed mapping (IFCM for short) if f(A) is an IFCS in Y for each IFCS A in X.

   2. intuitionistic fuzzy -open mapping (IFOM for short) if f(A) is an IFOS in Y for each IFOS A in X.

   Definition 2.14: [10] A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy generalized pre-semi closed mapping (IFGPSCM for short) if f(A) is an IFGPSCS in Y for each IFCS A in X.

   Definition 2.15: [10] A mapping f: (X, ) (Y, ) is said to be an intuitionistic fuzzy generalized presemi open mapping (IFGPSOM for short) if f(A) is an IFGPSOS in Y for each IFOS in X.

   Definition 2.16 [12] A mapping f: (X, ) (Y, ) is called an intuitionistic fuzzy generalized semipre closed mapping

   short) if every IFGPSCS is an IFCS in (X, ).

   Definition 2.22: [7] Let f be a bijection mapping from an IFTS (X, ) into an IFTS (Y, ). Then f is said to be an

   1. intuitionistic fuzzy homeomorphism (IF homeomorphism in short) if f and f 1 are IF continuous mappings.

   2. intuitionistic fuzzy homeomorphism (IF homeomorphism in short) if f and f 1 are IF continuous mappings.

   Definition 2.23: [6] Let f : (X, ) (Y, ) be a bijective mapping. Then f is said to be an intuitionistic fuzzy generalized semi-pre regular homeomorphism (IFGSPRHM for short) if f is both an IFGSPR continuous mapping and an IFGSPR closed mapping.

   Definition 2.24: [13] Let f : (X, ) (Y, ) be a bijective mapping. Then f is said to be an intuitionistic fuzzy generalized semi-pre homeomorphism (IFGSPHM for short) if f is both an IFGSP continuous mapping and an IFGSP closed mapping.

3. Generalized Pre-Semi Homeomorphisms in Intuitionistic Fuzzy Topological Spaces

   In this paper we have introduced intuitionistic fuzzy generalized pre-semi homeomorphisms and investigated some properties.

   Definition 3.1: Let f : (X, ) (Y, ) be a bijective mapping. Then f is said to be an intuitionistic fuzzy generalized pre- semi homeomorphism (IFGPSHM for short) if f is both an IFGPS continuous mapping and an IFGPS closed mapping.

   For the sake of simplicity, we shall use the notation A= x, (µ,

   µ), (, ) instead of A= x,(a/a, b/b), (a/a, b/b) in all the examples used in this paper. Similarly we shall use the notation B= x, (µ, µ), (, ) instead of B= x,(u/u, v/v), (u/u, v/v) in the following examples.

   Definition 3.2: Let A be an IFS in an IFTS (X, ). Then generalized pre-semi interior of A (gpsint(A) for short) and generalized pre-semi closure of A (gpscl(A) for short) are defined by

   1. gpsint (A) = { G / G is an IFGPSOS in X and G A }.

   2. gpscl (A) = { K / K is an IFGPSCS in X and A K }. Note that for any IFS A in (X, ), we have gpscl(Ac) = (gpsint(A))c and gpsint(Ac) = (gpscl(A))c.

   Theorem 3.3: Every IFHM is an IFGPSHM.

   Proof: Let f : (X, ) (Y, ) be an IFHM. Then f is IF continuous and IF closed. Since every IF continuous function is IFGPS continuous and every IF closed mapping is an IFGPS closed mapping, f is IFGPS continuous and IFGPS closed. Hence f is an IFGPSHM.

   Example 3.6: In Example the bijective mapping f : (X, ) (Y, ) by f(a) = u and f(b) = v is an IFGPSHM but not an IFHM.

   Theorem 3.7: Every IFGPSHM is an IFGSPRHM.

   Proof: Let f : (X, ) (Y, ) be an IFGPSHM. Then f is IFGPS continuous and IFGPS closed. Since every IFGPS continuous function is IFGSPR continuous and every IFGPS closed mapping is an IFGSPR closed mapping, f is IFGSPR continuous and IFGSPR closed. Hence f is an IFGSPRHM.

   Example 3.8: Let X = {a, b}, Y = {u, v} and G1 = x, (0.3, 0.2), (0.7, 0.8), G2 = y, (0.5, 0.6), (0.5, 0.4). Then = {0~,

   G1, 1~} and = {0~, G2, 1~} are IFTs on X and Y respectively. Define a bijective mapping f : (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an IFGSPRHM but not an IFGPSHM.

   Theorem 3.9: Every IFGPSHM is an IFGSPHM.

   Proof: Let f : (X, ) (Y, ) be an IFGPSHM. Then f is IFGPS continuous and IFGPS closed. Since every IFGPS continuous function is IFGSP continuous and every IFGPS closed mapping is an IFGSP closed mapping, f is IFGSP continuous and IFGSP closed. Hence f is an IFGSPHM.

   Example 3.10: In Example the bijective mapping f : (X, ) (Y, ) by f(a) = u and f(b) = v is an IFGSPHM but not an IFGPSHM.

   1/2

   1/2

   Theorem 3.11: Let f: (X, ) (Y, ) be an IFGPSHM, then f is an IFHM if X and Y are IFPST* space.

   1/2

   1/2

   1/2

   1/2

   Proof: Let B be an IFCS in Y. Then f -1(B) is an IFGPSCS in X, by hypothesis. Since X is an IFPST* space, f -1(B) is an IFCS in X. Hence f is an IF continuous mapping. By hypothesis f-1: (Y, ) (X, ) is an IFGPS continuous mapping. Let A be an IFCS in X. Then (f -1)-1(A) = f(A) is an

   Example 3.4: Let X = {a, b}, Y = {u, v} and G1 = x, (0.6,

   IFGPSCS in Y, by hypothesis. Since Y is an IFPST* space,

   0.7), (0.4, 0.2), G2 = y, (0.5, 0.4), (0.5, 0.6). Then = {0~,

   G1, 1~} and = {0~, G2, 1~} are IFTs on X and Y respectively. Define a bijective mapping f : (X, ) (Y, ) by f(a) = u and f(b) = v. Then f is an IFGPSHM but not an IFHM.

   Theorem 3.5: Every IFHM is an IFGPSHM.

   Proof: Let f : (X, ) (Y, ) be an IFHM. Then f is IF continuous and IF closed. Since every IF continuous function is IFGPS continuous and every IF closed mapping is an IFGPS closed mapping, f is IFGPS continuous and IFGPS closed. Hence f is an IFGPSHM.

   f(A) is an IFCS in Y. Hence f-1 is an IF continuous mapping. Therefore the mapping f is an IFHM.

   Theorem 3.12: Let f : (X, ) (Y, ) be a bijective mapping. If f is an IFGPS continuous mapping, then the following statements are equivalent:

   1. f is an IFGPS open mapping.

   2. f is an IFGPSHM.

   3. f is an IFGPS closed mapping.

   Proof: Straightforward.

   Remark 3.13: The composition of two IFGPSHMs need not be an IFGPSHM in general.

   Example 3.14: Let X = {a, b}, Y = {c, d} and Z = {e, f}. Let G1 = x, (0.3, 0.2), (0.7, 0.8), G2 = y, (0.1, 0.9), (0.9, 0.1),

   G3 = z, (0.5, 0.6), (0.5, 0.4). Then = {0~, G1, 1~}, = {0~,

   G2, 1~} and = {0~, G3, 1~} are IFTs on X, Y and Z respectively. Define a mapping f : (X, ) (Y, ) by f(a) = c and f(b) = d and g : (Y, ) (Z, ) by g(c) = e and g(d) = f. Then f and g are IFGPSHMs but their composition g f : (X, ) (Z, ) is not an IFGPSHM.

4. i-Generalized Pre-Semi Homeomorphisms in Intuitionistic Fuzzy Topological Spaces

In this paper we have introduced intuitionistic fuzzy i-generalized pre-semi homeomorphisms and investigated some properties.

Definition 4.1: Let f : (X, ) (Y, ) be a bijective mapping. Then f is said to be an intuitionistic fuzzy i-generalized pre- semi homeomorphism (IFiGPSHM for short) if f is both an IFGPS irresolute mapping and an IFiGPS open mapping.

The family of all IFiGPSHM in X is denoted by IFiGPSHM (X).

Theorem 4.2: Every IFiGPSHM is an IFGPSHM but not conversely.

Proof: Assume that f : (X, ) (Y, ) be an IFiGPSHM. Let A Y be an IFCS. Then A is an IFGPSCS in Y. By hypothesis, f -1(A) is an IFGPSCS in X. Hence f is an IFGPS continuous mapping. Let B X be an IFOS. Then B is an IFGPSOS in X. By hypothesis, f(B) is an IFGPSOS in Y. Hence f is an IFGPS open mapping. Thus f is an IFGPSHM.

Proof: Straightforward.

Theorem 4.6: If f : (X, ) (Y, ) is an IFiGPSHM, then gpscl(f -1(B)) f -1(pcl(B)) for every IFS B in Y.

Proof: Let B Y. Then pcl(B) is an IFGPSCS in Y. Since f is an IFSPG irresolute mapping, f -1(pcl(B)) is an IFGPSCS in

X. This implies gpscl(f -1(pcl(B))) = f -1(pcl(B)). Now gpscl (f -1(B)) gpscl(f -1(pcl(B))) = f -1(pcl(B)).

Theorem 4.7: If f : (X, ) (Y, ) is an IFiGPSHM, where X and Y are IFPST1/2 spaces, then pcl(f -1(B)) = f -1(pcl(B)) for every IFS B in Y.

Proof: Since f is an IFiGPSHM, f is an IFGPS irresolute mapping. Let B Y. Then since pcl(B) is an IFGPSCS in Y, f -1(pcl(B)) is an IFGPSCS in X. Since X is an IFPST1/2 spaces, f -1(pcl(B)) is an IFPCS in X. Now, f -1(B) f -1(pcl(B)). We have pcl(f -1(B)) pcl(f -1(pcl(B))) =

f -1(pcl(B)). This implies pcl(f -1(B)) f -1(pcl(B)) (*). Again since f is an IFiGPSHM, f -1 is IFGPS irresolute mapping. Since pcl(f -1(B)) is an IFGPSCS in X, (f -1) -1(pcl(f -1(B))) =

f(pcl(f -1(B))) is an IFGPSCS in Y. Now B (f -1) -1(f -1(B))

(f -1) -1(pcl(f -1(B))) = f(pcl(f -1(B))). Therefore pcl(B)

pcl(f(pcl(f -1(B)))) = f(pcl(f -1(B))), since Y is an IFPST1/2

spaces. Hence f -1(pcl(B)) f -1(f(cl(f -1(B)))) pcl(f -1(B)).

That is f -1(pcl(B)) pcl(f -1(B)) (**). Thus from (*) and (**) we get pcl(f -1(B)) = f -1(pcl(B)) and hence the proof.

Corollary 4.8: If f : (X, ) (Y, ) is an IFiGPSHM, where X and Y are IFPST1/2 spaces, then pcl(f(B)) = f(pcl(B)) for every IFS B in X.

Example 4.3: Let X = {a, b} and Y = {u, v}. Let G

= x,

Proof: Since f is an IFiGPSHM, f -1 is also an IFiGPSHM.

(0.3, 0.2), (0.7, 0.8) and G2

1

= y, (0.1, 0.9), (0.9, 0.1). Then

Therefore by Theorem 4.7, pcl((f -1) -1(B)) = (f -1) -1(pcl(B))

= {0~, G1, 1~} and = {0~, G2, 1~} are IFTs on X and Y respectively. Define a mapping f : (X, ) (Y, ) by f(a) = u and f(b) = v is an IFGPSHM but not an IFiGSPHM.

Theorem 4.4: The composition of two IFiGPSHMs is an IFiGPSHM.

Proof: Assume that f : (X, ) (Y, ) and g : (Y, ) (Z, ) are any two IFiGPSHMs. Let A Z be an IFGPSCS. Then by hypothesis, g -1(A) is an IFGPSCS in Y. Again by hypothesis, f -1(g -1(A)) is an IFGPSCS in X. Therefore g f is an IFGPS irresolute mapping. Now let B X be an IFGPSOS. Then by hypothesis, f(B) is an IFGPSOS in Y and also g(f(B)) is an IFGPSOS in Z. This implies g f is an IFiGPSHM.

Theorem 4.5: Let f : (X, ) (Y, ) be a bijective mapping. If f is an IFGPS irresolute mapping, then the following statements are equivalent.

1. f is an IFiGPSOM.

2. f is an IFiGPSHM.

3. f is an IFiGPSCM.

for every B X. That is pcl(f(B)) = f(pcl(B)) for every IFS B in X.

Corollary 4.9: If f : (X, ) (Y, ) is an IFiGPSHM, where X and Y are IFPST1/2 spaces, then pint(f(B)) = f(pint(B)) for every IFS B in X.

Proof: For any IFS B X, pint(B) = (pcl(Bc))c. By Corollary 4.8, f(pint(B)) = f(pcl(Bc))c = (f(pcl(Bc)))c = (pcl(f(Bc)))c = pint(f(Bc))c = pint(f(B)).

Corollary 4.10: If f : (X, ) (Y, ) is an IFiGPSHM, where X and Y are IFPST1/2 spaces, then pint(f -1(B)) = f -1(pint(B)) for every IFS B in Y.

Proof: The proof is trivial.

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