 Open Access
 Total Downloads : 574
 Authors : T. Sampoornam, Gnanambal Ilango, K. M. Arifmohammed
 Paper ID : IJERTV2IS90456
 Volume & Issue : Volume 02, Issue 09 (September 2013)
 Published (First Online): 19092013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Generalized Presemi Homeomorphisms in Intuitionistic Fuzzy Topological Spaces
1T. Sampoornam, 2Gnanambal Ilango and 3K. M. Arifmohammed
1Associate Professor and Head, Department of Mathematics, Government Arts College, Coimbatore641018, (T.N.), India.
2Assistant Professor, Department of Mathematics, Government Arts College, Coimbatore641018, (T.N.), India.
3Department of Mathematics, NGM College, Pollachi642001, Tamil Nadu, India.
AbstractIn this paper we introduce intuitionistic fuzzy generalized presemi homeomorphisms and intuitionistic fuzzy igeneralized presemi homeomorphisms. We investigate some of their properties.
KeywordsIntuitionistic fuzzy topology, intuitionistic fuzzy generalized presemi closed set, intuitionistic fuzzy generalized presemi homeomorphism and intuitionistic fuzzy igeneralized presemi homeomorphism.
2010 Mathematical Subject Classification: 54A40, 03F55

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 presemi homeomorphisms and intuitionistic fuzzy igeneralized presemi homeomorphisms. We investigate some of their properties. We also provide the relationship among various homeomorphisms. 
Preliminaries
Definition 2.1: [1] Let X be a nonempty fixed set. An intuitionistic fuzzy set (IFS in short) A in X is an object having the form A = {x, Âµ (x), (x)/ x X} where the functions

A B if and only if ÂµA(x) ÂµB(x) and A(x) B(x) for all x X.

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

Ac = { x, A(x), ÂµA(x)/ x X }.
(iv) A B = { x, ÂµA(x) ÂµB(x), A(x) B(x)/ x X }.
(v) A B = { x, ÂµA(x) ÂµB(x), A(x) B(x)/ x X }.
For the sake of simplicity, we shall use the notation A = x,
ÂµA, A instead of A = { x, ÂµA(x), A(x)/ x X }. Also for the sake of simplicity, we shall use the notation A = x, (ÂµA,
ÂµB), (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~ .

G1 G2 , for any G1, G2 .

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, ÂµA, A be an IFS in X. Then

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

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

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

int(Ac) = (cl(A))c.
A A
ÂµA: X [0,1] and A : X [0,1] denote the degree of membership (namely ÂµA(x)) and the degree of non membership (namely A(x)) of each element x X to the set A, respectively, and 0 ÂµA(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,
ÂµA(x), A(x)/ x X} and B = {x, ÂµB(x), B(x)/ x X}. Then
Definition 2.5: [4] Let A = <x, ÂµA, A> be an IFS in X. Then

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

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

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

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 presemi 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 presemi 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

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

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 presemi 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 igeneralized 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 igeneralized 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 semipre 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 presemi 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

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

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 presemi 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

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

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 semipre 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 semipre homeomorphism (IFGSPHM for short) if f is both an IFGSP continuous mapping and an IFGSP closed mapping.


Generalized PreSemi Homeomorphisms in Intuitionistic Fuzzy Topological Spaces
In this paper we have introduced intuitionistic fuzzy generalized presemi 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, (Âµ,
Âµ), (, ) 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, (Âµ, Âµ), (, ) 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 presemi interior of A (gpsint(A) for short) and generalized presemi closure of A (gpscl(A) for short) are defined by

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

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 f1: (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 f1 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:

f is an IFGPS open mapping.

f is an IFGPSHM.

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.


iGeneralized PreSemi Homeomorphisms in Intuitionistic Fuzzy Topological Spaces
In this paper we have introduced intuitionistic fuzzy igeneralized presemi 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 igeneralized 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.

f is an IFiGPSOM.

f is an IFiGPSHM.

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.
REFERENCES

K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20, 1986, 8796.

C. L. Chang, Fuzzy topological spaces, J.Math.Anal.Appl. 24, 1968, 182190.

D. Coker, An introduction to intuitionistic fuzzy topological space, Fuzzy Sets and Systems, 88, 1997, 8189.

T. Sampoornam, Gnanambal Ilango and K. Ramesh, Generalized PreSemi Continuous and Irresolute Mappings in Intuitionistic Fuzzy Topological Space, International Journal of Innovative Research in Science Engineering and Technology, Vol. 2, Issue 8, Aug 2013, 4034 4040.

T. Sampoornam, Gnanambal Ilango and K. M. Arifmohammed, Generalized Presemi Closed Mappings in Intuitionistic Fuzzy Topological Spaces.(submitted)

H. Gurcay, Es. A. Haydar and D. Coker, On fuzzy continuity in [11] R. Santhi and D. Jayanthi, Intuitionistic Fuzzy Generalized SemiPre
intuitionistic fuzzy topological spaces, J.Fuzzy Math.5 (2) , 1997, 365 378.
Continuous Mappings, Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 30, 1455 1469.

K. Ramesh, M. Thirumalaiswamy and S. Kavunthi, Generalized [12] Santhi and D. Jayanthi, Intuitionistic fuzzy generalized semipre closed
semipre regular continuous and irresolute mappings in intuitionistic fuzzy topological spaces, Vol. 2 Issue 8, Aug 2013.
mappings, Journal of Informatics and Mathematical Sciences, NIFS 16(2010), 3, 2839.

K. Ramesh and M. Thirumalaiswamy, On GSPR Closed Mappings and [13] R. Santhi and D. Jayanthi, Intuitionistic fuzzy generalized semipre
GSPR Homeomorphisms in Intuitionistic Fuzzy Topological Spaces.(accepted)

K. Sakthivel, Alpha generalized homeomorphism in intuitionistic fuzzy topological space, Notes IFS 17(2011) 3036.

T. Sampoornam, Gnanambal Ilango and K. Ramesh, On Generalized PreSemi Closed Sets in Intuitionistic Fuzzy Topological Spaces.(submitted)
homeomorphisms, Journal of Pure Mathematics.(accepted)

Seok Jong Lee and Eun Pyo Lee, The category of intuitionistic fuzzy topological spaces, Bull. Korean Math. Soc. 37, No. 1, 2000, pp. 6376.

L. A. Zadeh, Fuzzy sets, Information and control, 8, 1965, 338353.