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ON Hg-HOMEOMORPHISMS in TOPOLOGICAL SPACES Proyecciones Vol. 28, No 1, pp. 1—19, May 2009. Universidad Cat´olica del Norte Antofagasta - Chile ON g-HOMEOMORPHISMS IN TOPOLOGICAL SPACES e M. CALDAS UNIVERSIDADE FEDERAL FLUMINENSE, BRASIL S. JAFARI COLLEGE OF VESTSJAELLAND SOUTH, DENMARK N. RAJESH PONNAIYAH RAMAJAYAM COLLEGE, INDIA and M. L. THIVAGAR ARUL ANANDHAR COLLEGE Received : August 2006. Accepted : November 2008 Abstract In this paper, we first introduce a new class of closed map called g-closed map. Moreover, we introduce a new class of homeomor- phism called g-homeomorphism, which are weaker than homeomor- phism.e We prove that gc-homeomorphism and g-homeomorphism are independent.e We also introduce g*-homeomorphisms and prove that the set of all g*-homeomorphisms forms a groupe under the operation of composition of maps. e e 2000 Math. Subject Classification : 54A05, 54C08. Keywords and phrases : g-closed set, g-open set, g-continuous function, g-irresolute map. e e e e 2 M. Caldas, S. Jafari, N. Rajesh and M. L. Thivagar 1. Introduction The notion homeomorphism plays a very important role in topology. By definition, a homeomorphism between two topological spaces X and Y is a 1 bijective map f : X Y when both f and f − are continuous. It is well known that as J¨anich→ [[5], p.13] says correctly: homeomorphisms play the same role in topology that linear isomorphisms play in linear algebra, or that biholomorphic maps play in function theory, or group isomorphisms in group theory, or isometries in Riemannian geometry. In the course of generalizations of the notion of homeomorphism, Maki et al. [9] intro- duced g-homeomorphisms and gc-homeomorphisms in topological spaces. Recently, Devi et al. [2] studied semi-generalized homeomorphisms and generalized semi-homeomorphisms. In this paper, we first introduce g-closed maps in topological spaces and then we introduce and study g-homeomorphisms, which are weaker than homeomorphisms. We prove that gc-homeomorphisme and g-homeomorphism are independent. We also introducee g -homeomorphisms. It turns out that ∗ the set of all g*-homeomorphisms forms a group under thee operation com- position of functions. e e 2. Preliminaries Throughout this paper (X, τ)and(Y,σ) represent topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of a space (X, τ), cl(A), int(A)andAc denote the closure of A, the interior of A and the complement of A in X, respectively. We recall the following definitions and some results, which are used in the sequel. Definition 1. A subset A of a space (X, τ) is called: (i) semi-open [6] if A cl(int(A)), (ii) α-open [11] if A ⊂int(cl(int(A))), (iii) regular open [17]⊂ if A = int(cl(A)), (iv) pre-closed [10] if cl(int(A)) A. ⊂ Thesemi-closure[1](resp. α-closure [11]) of a subset A of X,denoted by sclX(A)(resp. αclX (A)) briefly scl(A), (resp. αcl(A)) is defined to be the intersection of all semi-closed (resp. α-closed) sets containing A. On g-homeomorphisms in topological spaces 3 e Definition 2. A subset A of a space (X, τ) is called a: (i) generalized closed (briefly g-closed) set [7] if cl(A) U whenever A U and U is open in (X, τ). ⊂ ⊂ (ii) generalized α-closed (briefly gα-closed) set [8] if αcl(A) U whenever A U and U is α-open in (X, τ). ⊂ (iii)⊂ g-closed set [19] if cl(A) U whenever A U and U is semi-open in (X, τ). ⊂ ⊂ (iv) bg-closed set [20] if cl(A) U whenever A U and U is g-open in (X, τ∗). ⊂ ⊂ (v) #g-semi-closed (briefly #gs-closed) set [21] if scl(A) U bwhenever A U and U is g-open in (X, τ). ⊂ (vi)⊂ g-closed set∗ [4] if cl(A) U whenever A U and U is #gs-open in (X, τ). ⊂ ⊂ e Where g-open (resp. g-open, gα-open, rg-open, g-open, #gs-open, g- open), are defined as the complement of g-closed (resp.∗ g-closed, gα-closed, rg-closed, bg-closed, #gs-closed, g-closed). e ∗ b Definition 3. A function f :(X,e τ) (Y,σ) is called: 1 → (i) g-continuous [18] if f − (V ) is g-closed in (X, τ) for every closed set V in (Y,σ). 1 (ii) g-continuous [19] if f − (V ) is g-closed in (X, τ) for every closed set V in (Y,σ). 1 (iii)bg-continuous [15] if f − (V ) is bg-closed in (X, τ) for every closed set V in (Y,σ). 1 (iv) ge-irresolute [13] if f − (V ) is g-closede in (X, τ) for every g-closed set V in (Y,σ). 1 (v) gce -irresolute [18] if f − (V ) iseg-closed in (X, τ) for each ge-closed set V of (Y,σ). 1 (vi) #gs-irresolute [21] if f − (V ) is #gs-closed in (X, τ) for each #gs- closed set V of (Y,σ). (vii) strongly g-continuous [13] if the inverse image of every g-open set in (Y,σ) is open in (X, τ). e e Definition 4. A function f :(X, τ) (Y,σ) is called: (i) g-open [18] if f(V ) is g-open in (Y,σ→ ) for every g-open set V in (X, τ). (ii) g-open [19] if f(V ) is g-open in (Y,σ) for every g-open set V in (X, τ). Defibnition 5. A bijectiveb function f :(X, τ) (Y,σb ) is called a: (i) generalized homeomorphism (briefly g-homeomorphism)→ [9] if f is both 4 M. Caldas, S. Jafari, N. Rajesh and M. L. Thivagar g-continuous and g-open, 1 (ii) gc-homeomorphism [9] if both f and f − are gc-irresolute maps, (iii) g-homeomorphism [19] if f is both g-continuous and g-open. Defibnition 6. [14] Let (X, τ) be a topologicalb space and Eb X.Wedefine the g-closure of E (briefly g-cl(E)) to be the intersection of⊂ all g-closed sets containing E.i.e., g-cl(eE)= A : E A ande A GC(X, τ) . e ∩{ ⊂ ∈ } Propositione 2.1. [13] If f :(X,e τ) (Y,σ) is g-irresolute, then it is g- continuous. → e e Proposition 2.2. [14] Let (X, τ) be a topological space and E X.The following properties are satisfied: ⊂ (i) g-cl(E)) is the smallest g-closed set containing E and (ii) E is g-closed if and only if g-cl(E)) = E. e e Propositione 2.3. [14] For anye two subsets A and B of (X, τ), (i) If A B,theng-cl(A)) g-cl(B)), (ii) g-cl(⊂A B)) g-cl(A))⊂ g-cl(B)). ∩ ⊂ ∩ e e Propositione 2.4. e[4] Every ge-closed set is gα-closed and hence pre-closed. Proof. Let A be g-closedin(e X, τ)andU be any α-open set containing A. Since every α-open set is #gs-open [21] and since αcl(A) cl(A)for ⊂ every subset A of X,e we have by hypothesis, αcl(A) cl(A) U and so A is gα-closed in (X, τ). ⊂ ⊂ In [8], it has been proved that every gα-closed set is pre-closed. There- fore, every g-closed set is pre-closed. Theorem 2.5.e [4] Suppose that B A X, B is a g-closed set relative to A and that A is open and g-closed in⊂ (X,⊂ τ).ThenB is g-closed in (X, τ). e Corollary 2.6. [4] If A ise a g-closed set and F is a closede set, then A F is a g-closed set. ∩ e Theoreme 2.7. [4] A set A is g-open in (X, τ) if and only if F int(A) whenever F is #gs-closed in (X, τ) and F A. ⊂ ⊂ e Definition 7. [14] Let (X, τ) be a topological space and E X.We define the g-interior of E (briefly g-int(E)) to be the union of⊂ all g-open sets contained in E. e e e On g-homeomorphisms in topological spaces 5 e Lemma 2.8. [14] For any E X, int(E) g-int(E) E. ⊂ ⊂ ⊂ Proof. Since every open set is g-open, thee proof follows immediately. Definition 8. A topological spacee (X, τ) is a, (i) T1/2 space [7] if every g-closed subset of (X, τ) is closed in (X, τ), (ii) Tg space [16] if every g-closed subset of (X, τ) is closed in (X, τ), (iii) semi-T1/2 space [19] if every g-closed subset of (X, τ) is closed in (X, τ). e e 3. g-closed maps b Malghane [12] introduced the concept of generalized closed maps in topo- logical spaces. In this section, we introduce g-closed maps, g-open maps, g*-closed maps, g*-open maps and obtain certain characterizations of these maps. e e e e Definition 9. Amapf :(X, τ) (Y,σ) is said to be g-closed if the image of every closed set in (X, τ) is g-closed→ in (Y,σ). e Example 3.1. (a) Let X = eY = a, b, c , τ = ,X, a and σ = ,Y, b .Define a map f :(X, τ){ (}Y,σ) by {f∅(a)={ }}b, f(b)=a and{∅ f{(c)=}} c.Thenf is a g-closed map.→ (b) Let X = Y = a, b, c , τ = ,X, a , b , a, b and σ = ,Y, a, b .
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