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 Authors : A. Pushpalatha, R. Nithyakala
 Paper ID : IJERTV2IS120389
 Volume & Issue : Volume 02, Issue 12 (December 2013)
 Published (First Online): 30122013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
sc*gHomeomorphisms in Topological Spaces
A. Pushpalatha
Department of Mathematics, Government Arts College
Udumalpet642 126, Tirupur District Tamil Nadu, India
R. Nithyakala
Department of Mathematics Vidyasagar College of Arts and Science Udumalpet642 126, Tirupur District Tamil Nadu, India
Abstract
In this paper, we introduced a new class sc*g Homeomorphisms in Topological space.
Key words: Homeomorphism – strongly ghomeomorphism sc*g – homeomorphism.
1. Introduction
Several mathematicians have generalized homeomorphisms in topological spaces. Biswas[18], Crossely and Hildebrand[19], Gentry and Hoyle[20] and Umehara and Maki[21] have introduced and investigated semihomeomorphisms somewhat homeomorphisms and gA homeomorphisms Crossely and Hildebrand defined yet another semihomeomorphism which is also a generalization of homeomorphisms. Sundaram[6] introduced ghomeomorphisms and gc homeomorphisms in topological spaces.
In the section, we introduce the concept of sc*g homeomorphisms and study some of their properties.
Definition: 2.7.1 A bijection f : (X, ) (Y, ) from a topological space X into a topological space Y is called a sc*generalized homeomorphism (sc*g homeomorphism) if f is both sc*gopen and sc*gcontinuous.
Theorem: 2.7.2 Every homeomorphism is a sc*ghomeomorphism.
Proof : Since every continuous function is sc*g continuous and every open map is sc*gopen, the proof follows.
The converse of the above theorem need not be true as seen from the following example.
Example: 2.7.3 Consider the topological spaces X = Y= {a, b, c} with topologies ={ ,X,{a, b}} and = { ,Y,{a}, {a, b}}. Then the identity
map f : (X, ) (Y, ) is a sc*g homeomorphism but not a homeomorphism. Since for the open set {a} in Y is not open in X.
Theorem:2.7.4 Everystrongly g homeomorphism is a sc*ghomeomorphism but not conversely.
Proof : Let f : X Y be a strongly g homeomorphism. Then f is strongly g continuous and strongly g open. Since every strongly g – continuous function is sc*gcontinuous and every strongly g open map is sc*gopen, f is sc*g continuous and sc*gopen. Hence f is a sc*g homeomorphism.
The converse of the above theorem need not be true as seen from the following example.
Example : 2.7.5 Let X=Y={a, b, c} with
={ ,X,{a}}and = { ,Y,{a, b}} respectively.
Then the identity map f : (X, ) (Y, ) is sc*ghomeomorphism but not a strongly g homeomorphism. Since for the open set {a, b} in Y is not a strongly gopen set in X.
Next we shall characterize the sc*g homeomorphism and sc*gopen maps.
Theorem : 2.7.6 For any bijection f : X Y the following statements are equivalent.
(a)f 1: Y X is sc*gcontinuous. (b)f is a sc*gopen map.
(c)f is a sc*gclosed map.
Proof : (a) (b) Let G be any open set in X. Since f 1 is sc*gcontinuous, the inverse image of G under f 1, namely f(G) is sc*gopen in Y and so f
is sc*gopen map.

(c) Let f be any closed set in X. Then Fc is open in X. Since f is sc*gopen, f(Fc) is
sc*gopen in Y. But f(Fc) = Y f(F) and so f(F) is
sc*gclosed in Y. Therefore f is a sc*gclosed map.

(a) Let F be any closed set in X. Then
(f 1)1(F) = f(F) is sc*gclosed in Y . Since f is a sc*gclosed map. Therefore f 1 is sc*gcontinuous.
Theorem : 2.7.7 Let f : (X, ) (Y, ) be a bijective and sc*gcontinuous map, the following statements are equivalent.
(a)f is a sc*gopen map.

f is a sc*ghomeomorphism.

f is a sc*gclosed map.
Proof : (a)(b) By assumption, f is bijective and sc*gcontinuous and sc*gopen. Then by definition, f is sc*g homeomorphism.
(b)(c) By assumption, f is sc*gopen and bijective. By theorem 2.7.4 f is sc*gclosed map. (c)(a) By assumption, f is sc*gclosed and bijective. By theorem 2.7.4 f is sc*gopen map.
The following example shows that the composition of two sc*ghomeomorphisms need not be sc*g homeomorphism.
Example : 2.7.8 Consider the topological spaces X = Y= Z= {a, b, c} with topologies
1={ , X, {a}, {a, b}}, 2={ , Y, {a}}and
3={ , Z, {b, c}} respectively. Let f and g identity maps such that f : X Y and g : Y Z. Then f and g are sc*ghomeomorphisms but their
composition g f : X Z is not a sc*g
homeomorphism. For the open set {a, b} in X, g(f{a, b}) = {a, b} is not sc*gopen set in Z. Definition :2.7.9 A bijection f : (X, ) (Y, ) is said to be a (sc*g)*homeomorphism if f and its inverse f 1are sc*girresolute maps. Notation : Let family of all (sc*g)* homeomorphisms from (X, ) onto itself be denoted by (sc*g)*h(X, ) and family of all sc*g homeomorphisms from (X, ) onto itself be
denoted by sc*g h(X, ). The family of all homeomorphisms form from (X, ) onto itself be denoted by h(X, ).
Theorem :2.7.10 Let X be a Topological space.
Then

The set (sc*g)*h(X) is a group under composition of maps.
(ii)h(X) is a subgroup of (sc*g)*h(X). (iii)(sc*g)*h(X) sc*g h(X).
Proof: (i) Let f, g (sc*g)*h(X). Then g h (sc*g)*h(X) and so (sc*g)*h(X) is closed under the composition of maps. The composition of maps is associative. The identity map i : xx is a (sc*g)* homeomorphism and so i (sc*g)*h(X). Also f . i =
i. f = f for every f (sc*g)*h(X). If f (sc*g)*h(X),
then f 1 (sc*g)*h(X) and f. f 1 = f 1. f = i. Hence (sc*g)*h(X) is a group under the
composition of maps.

Let f : X Y be a homeomorphism. Then by theorem 2.6.5 both f and f 1 are
(sc*g)*irresolute and so f is a (sc*g)* homeomorphism. Therefore every homeomorphism is a (sc*g)*homeomorphism and so h(X) is a subset of (sc*g)*h(X). Also h(X) is a group under the composition of maps. Therefore h(X) is a subgroup of the group (sc*g)*h(X).

Since every (sc*g)*irresolute map is sc*g continuous, (sc*g)*h(X) is a subset of sc*gh(X).
From the above observations we get the following diagram:
Homeomorphism strongly ghomeomorphism sc*g – homeomorphism
References

Levine , N., Semiopen sets and semicontinuity in topologicalspaces, Amer.Math.Monthly,70(1963),36 41.

Levine ,N.,Generalized closed sets in topology, Rend. Circ. Mat. Palermo, 19(1970),8996.

Maki., Devi, R., and Balachandran, K., Generalized closed sets in topology, Bull. Fukuoka Univ. Ed. Part III, 42(1993), 1321.

Hatir, E., Noiri, T., and Yuksel, S., A decomposition of continuity, ActaMath. Hungar.,70(1996),145150.

Rajamani, M., Studies on decompositions of generalized continuous maps in topological spaces, Ph.D Thesis, Bharathiar University, Coimbatore,(2001).

Sundaram, P., Studies on generalizations of continuous maps in topological spaces, Ph.D Thesis, Bharathiar University, Coimbatore,(1991).

Sundaram, P., On Ccontinuous maps in topological spaces, Proc. of 82nd session of the Indian Science Congress, Calcutta (1995),4849.

Tong, J., On decomposition of continuity in topological spaces, ActaMath.Hungar.,54(1989),5155.

Bhattachraya and Lahiri, B.K., Semigeneralized closed sets in topology, Indian J.Math., 29(3)375 382(1987)

Nagaveni, N., Studies on Generalizations of Homeomorphisms in Topological spaces,Ph.D Thesis, Bharathiar University, Coimbatore,(1999).

Andrijevic,D., Semipreopen sets, Mat. Vesnik,38(1986),2432.

Dontchev, J., On generalizing semipre open sets, Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 16(1995), 35 48.

Nithyakala, R., scg, sc*g, sc(s)gclosed sets in Topological spaces, Proc. Of 99th Indian Science Congress, Bhubaneswar(2012), 89.

Balachandran, K., Sundaram, P., and Maki, H., On generalized continuous maps in Topological spaces, Mem. Fac. Kochi Univ. Ser. A. Math., 12(1991), 513.

Maki, H., Devi, R., and Balachandran, K., Associated topologies of generalized – closed sets and – generalized closed sets, Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 15 (1994), 5163.

Crossley, S.G., and Hildebrand, S.K., Semi closure, Texas J. Sci., 22 (1971), 99112.

Malghan, S.R., Generalized closed maps, J.Karnataka Univ.Sci., 27(1982), 8288.

Biswas,N., On some mappings in Topological spaces, Bull.Cal.Math.Soc., 61(1969),127135

Crossley, S. G. and Hildebrand, S.K., Semi topological properties, Fund.Math., 74(1972), 233254.

Gentry, K. R. And Hoyle, H. G., Somewhat continuous functions, Czechoslovak Math. J., 21(1971), 512.

Umehara, J. and Maki, H., A note on the homeomorphic image of a T v space, Mem. Fac. Sci. Kochi Univ. (Math)., 10(1989), 3945