On rgwα-Homeomorphism in topological spaces

A bijection f: (X, 𝜏) → (Y, σ) is called regular generalized weakly ⍺- homeomorphism if f and f-1 are rgw⍺-continuous. Also we introduce the new class of maps, namely rgw⍺*-homeomorphisms which form a subclass of rgw⍺-homeomorphisms. The class of rgw⍺- homeomorphism and rgw⍺*- homeomorphism maps are closed under composition of maps. We prove that the set of all rgw⍺- homeomorphism and rgw⍺*-homeomorphisms forms a group under the operation composition of maps.

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Introduction

The homeomorphism in topological space plays a significant role in the study of topology. First we provide basic requirement for better understanding of rgw⍺ -Homeomorphism in topological space. In this paper, we define new class of homeomorphism called rgw⍺-Homeomorphism and study the properties of this map and also compare this homeomorphism with well-known homeomorphisms w-homeomorphisms, g-homeomorphisms, rgαhomeomorphisms and rwg-homeomorphism etc. Also we introduce new class of maps rgw⍺*homeomorphisms which form a subclass of rgw⍺-homeomorphisms. The class of rgw⍺homeomorphism and rgw⍺*-homeomorphism maps are closed under composition of maps.We prove that the set of all rgw⍺-homeomorphism and rgw⍺*-homeomorphism maps forms a group under the operation composition of maps.

Preliminaries

Let us recall the following definition which we shall require later.

Definition 2.1:

A subset A of a space (X, τ) is called i. Regular open [18] if A= int(cl(A)) and regular closed if A= cl(int(A)). ii. Pre-open [16] if A ⊂ int(cl(A)) and pre-closed if cl(int(A)) ⊂ A. iii. Semi-open [9] if A ⊂ cl(int(A)) and semiclosed if int(cl(A)) ⊂ A. iv. α-open [13] if A ⊂ int(cl(int(A))) and α-closed if cl(int(cl(A))) ⊂ A. v. Semi-pre-open [10,24] if A ⊂ cl(int(cl(A))) and a semi-preclosed if int(cl(int(A))) ⊂ A. vi. Regular α-open [4] if there is a regular open set U such that U ⊂ A ⊂ αcl(U). vii. Regular semi open [8] if there is a regular open set U such that U ⊂ A ⊂ cl(U).

Definition 2.2:

A subset A of a space (X, τ) is called i. Generalized closed set (briefly, g-closed) [22] if cl(A) ⊂ U whenever A ⊂ U and U is open in X. ii. Generalized α-closed set (briefly, gα-closed) [3] if αcl(A) ⊂ U whenever A ⊂ U and U is α-open in X. iii. α-generalized closed set (briefly, αg-closed) [2] if αcl(A) ⊂ U whenever A ⊂ U and U is open in X. iv. Regular generalized closed set (briefly, rg-closed) [14] if cl(A) ⊂ U whenever A ⊂ U and U is regular open in X.

v. Generalized pre -regular closed set (briefly, gpr-closed) [12] if pcl(A) ⊂ U whenever A ⊂ U and U is regular open inX. vi. Weakly generalized closed set (briefly, ωg-closed) [9,14] if clint(A) ⊂ U whenever A ⊂ U and U is open in X. vii. Weakly closed sets (briefly, ω-closed) [7,11] if cl(A) ⊂ U whenever A ⊂ U and U is semi open in X. viii. Regular weakly generalized closed set (briefly, rωgclosed) [1] if cl(int(A)) ⊂ U whenever A ⊂ U and U is regular open in X. ix. Regular generalized α-closed set (briefly, rgα-closed) [4] if αcl(A) ⊂ U whenever A ⊂ U and U is regular α-open in X. x. Regular ω-closed (briefly, rω-closed) [11] if cl(A) ⊂ U whenever A ⊂ U and U is regular semi open in X. xi. Regular generalized weakly α-closed set (briefly rgwαclosed) [19] if rαcl (A) ⊆U whenever A⊆U & U is weak α-open set in X.

Definitions 2.3:

A map f : (X, τ) → (Y, σ) is said to be i. continuous [15] if f -1 (V) is closed in X, for every closed set V in Y. ii. ω-continuous [1] if f -1 (V) is ω-closed in X, for every closed set V in Y. iii. α-continuous [11] if f -1 (V) is α-closed in X, for every closed set V in Y. iv. gα-continuous [3] if f -1 (V) is gα-closed in X, for every closed set V in Y. v. rgα-continuous [4] if f -1 (V) is rgα-closed in X, for every closed set V in Y. vi. swg-continuous [11] if f -1 (V) is swg-closed in X, for every closed set V in Y. vii. rwg-continuous [11] if f -1 (V) is rwg-closed in X, for every closed set V in Y. viii. rgw⍺-continuous [20] if f -1 (V) is rgw⍺-closed in X, for every closed set V in Y.

Definitions 2.4:

A map f :(X, τ) → (Y, σ) is said to be i. irresolute [21] if f -1 (V) is semi-open in X, for every semiopen set V in Y. ii. ω-irresolute [13] if f -1 (V) is ω-closed in X, for every ωclosed set V in Y. iii. rgα-irresolute [5] if f -1 (V) is rgα-closed in X, for every rgα-closed set V in Y. iv. rgw⍺-irresolute [20] if f -1 (V) is rgw⍺-closed in X, for every rgw⍺-closed set V in Y.

Definition 2.5:

A bijective function f: (X, τ) → (Y, σ) is called (i) generalized homeomorphism (g-homeomorphism) [9] if both f and f -1 are g-continuous, (ii) w-homeomorphism [13] if both f and f -1 are w-continuous.

(iii) w*-homeomorphism [13] if both f and f -1 are w-irresolute, (iv) gc-homeomorphism [9] if both f and f -1 are gc-irresolute, (v) rwg-homeomorphism [16] if both f and f -1 are rwgcontinuous, (vi) rgα-homeomorphism [16] if both f and f -1 are rgαcontinuous.

(vii) wgr⍺-homeomorphism [23] if both f and f -1 are wgrαcontinuous

rgwα-homeomorphisms in Topological Spaces

We introduce the following definition

Theorem 3.5: For any topological space (X, τ), h(X, τ) ⊆ rgw⍺-h(X, τ).

Proof: Let us consider f ∈ h(X, τ), then by the definition of homeomorphism f and f -1 are continuous. Since every continuous function is rgw⍺-continuous so f and f -1 are rgw⍺continuous map. Now by the definition of rgw⍺homeomorphism we can say that f ∈ rgw⍺-h(X, ).

Theorem 3.6:

A bijection f: (X, ) → (Y, ) be rgw⍺homeomorphism and (X, ) and (Y, ) are rgw⍺-space, then f is homeomorphism. Proof: It follows from fact that in rgw⍺-space, every rgw⍺closed set is closed.

Theorem 3.7: Every ⍺-homeomorphism is a rgw⍺homeomorphism but not conversely. Proof: Let f: (X, τ) → (Y, σ) be a ⍺-homeomorphism. Then f and f -1 are ⍺-continuous and f is bijection. As every ⍺continuous function is rgw⍺-continuous, we have f and f -1 is rgw⍺-continuous. Therefore f is rgw⍺-homeomorphism. The converse of the above Theorem is not true in general as seen from the following example.

Example 3.8:

Consider X=Y={a,b,c,d,e}, τ ={X,ϕ,{a},{d},{e},{a,d},{a,e},{d,e},{a,d,e}} and σ ={Y, ϕ, {a},{d},{e}, {a,d}, {a,e},{d,e},{a,d,e }. Let map f: X→Y defined by f(a)=c, f(b)= a, f(c)=b, f(d)=d, f(e)=e, then f is rgw⍺-homeomorphism but not α-homeomorphism, as closed set F= {a,c,e} in Y, then f -1 (F)={a,b,e} in X which is not ⍺closed set in X. Theorem 3.9: Every w-homeomorphism (resp. rhomeomorphism, β-homeomorphism, rw-homeomorphism, rs-homeomorphism, r⍺-homeomorphism, w⍺homeomorphism, g⍺-homeomorphism, rg⍺homeomorphism) is a rgw⍺-homomorphism but not conversely. Proof: Proof similar to the theorem 3.7.

Example

Example

3.10:

Let X=Y={a,b,c,d,e}, τ ={X, ϕ,{a},{d},{e},{a,d},{a,e},{d,e},{a,d,e}} and σ ={Y, ϕ, {a},{d},{e}, {a,d}, {a,e},{d,e},{a,d,e }. Let map f: X→Y defined by f(a)=c, f(b)= a, f(c)=b, f(d)=d, f(e)=e, then f is rgw⍺-homeomorphism but not w-homeomorphism rhomeomorphism, β-homeomorphism, rw-homeomorphism, rs-homeomorphism, r⍺-homeomorphism, w⍺homeomorphism, g⍺-homeomorphism, rg⍺-homeomorphism as closed set F= {b,c,d,e} in Y, then f -1 (F)={a,c,d.e} in X which is rgw⍺-closed but not r-closed, w-closed, β-closed, rsclosed and r⍺-closed set in X. and closed set F={b,c,d} in Y f - 1 (F) ={a,b,d} which is not rw-closed, w⍺-closed, g⍺-closed, rg⍺-closed set in X. Theorem 3.11: Every rgw⍺-homeomorphism is a gβhomeomorphism but not conversely. Proof: Let f: (X, τ) → (Y, σ) be rgw⍺-homeomorphism. Then f and f -1 are rgw⍺-continuous and f is bijection. Since every rgw⍺-continuous function is gβ-continuous, we have f and f -1 are gβ-continuous. Therefore f is gβ-homeomorphism. The converse of the above Theorem is not true in general as seen from the following example.

Example 3.12: Consider X=Y= {a,b,c}, τ={X, ϕ,{a},{b,c}} and σ ={Y, ϕ, {a}}. Let map f: X→Y defined by f(a)=b, f(b)= a, f(c)=c, then f is gβ-homeomorphism but not rgw⍺homeomorphism as a closed set F={b,c} in Y, f -1 (F)= f - 1 {b,c}= {a,c} which is not rgw⍺-closed set. Remark 3.13: rgw⍺-homeomorphism and g-homeomorphism are independent as seen from the following example.

Example 3.14: Let X=Y= {a,b,c}, τ={X, ϕ,{a},{b,c}} and σ ={Y, ϕ, {a}}. Let map f: X→Y defined by f(a)=b, f(b)= a, f(c)=c, then f is g-homeomorphism but not rgw⍺homeomorphism, as a closed set F={b,c} in Y, f -1 (F)= f -1 {b,c} ={a,c} is not rgw⍺-closed set. ) → (Z, ) are rgw⍺-homeomorphism, then composition of these two function as gof : (X, τ) → (Z, ) is also rgw⍺homeomorphism.

Therefore, gof is continuous. Similarly we can show that (gof) -1 is also continuous. Hence, by the definition gof is rgw⍺-homeomorphism. By the above result we can proof that Theorem 3.21: The rgw⍺-homeomorphism is an equivalence relation in the family of all topological spaces. Proof: It follows from above theorem. Theorem 3.22: Let (X, τ) be a topological space, then the collection rgw⍺-h(X, τ) forms a group under the composition of functions. Proof: Let a binary operation o': rgw⍺-h(X, τ) × rgw⍺-h (X, τ) → rgw⍺-h(X, τ) defined by f o' g = gof for all f, g ∈ rgw⍺h(X, τ), where gof : X → X is composite maps of f and g such that (gof)(x) = g(f (x)) for all x in X. By the theorem 3.8, gof ∈ rgw⍺-h(X, τ). The following properties hold by the collection rgw⍺-h(X, τ).

(1) Associativity: Since the composition of two rgw⍺homeomorhism is again rgw⍺-homeomorhism, with this easily we can prove it is associative.

(2) Existence of identity: Since the identity map iX: X → X is also rgw⍺-homeomorphism, then easily we can say that for all element f ∈ rgw⍺-h(X, τ), there exists an element iX such that f o' iX = iX o' f = iX (3) Existence of inverse: We know that the composition of maps is associative and the identity map iX : (X, τ) → (X, τ) belonging to rgw⍺-h(X, τ) servers as the identity element. If f ∈ rgw⍺-h(X, τ), then f -1 ∈ rgw⍺-h(X, τ) such that f of -1 = f -1 of = i and so inverse exists for each element of rgw⍺-h(X, τ). Therefore, (rgw⍺-h(X, τ), o) is a group under the operation of composition of maps.

Theorem 3.23: If f: (X, τ) → (Y,

) is rgw⍺homeomorphism, then f induces an isomorphism between rgw⍺-h(X, τ) and rgw⍺-h(Y, ) i.e. rgw⍺-h(X, ) ≅ rgw⍺h(Y, ). Proof: Using the map f, we define a map ψf : rgw⍺-h(X, τ) → rgw⍺-h(Y, ) by ψf (h) = f ohof 1 for every h ∈ rgw⍺-h(X, τ1). Then ψf is bijection. Further, for all h1, h2 ∈ ψf, ψf (h1oh2) = f o(h1 oh2)of -1 = (f oh1of -1 )o(f oh2 of -1 ) = ψf (h1)oψf(h2). Therefore ψf is a homeomorphism and so it is an isomorphism induced by f. Definition 3.24: A bijection f: (X, τ) → (Y, σ) is said to be rgw⍺*-homeomorphism if both f and f -1 are rgw⍺-irresolute. We say that spaces (X, τ) and (Y, σ) are rgw⍺*homeomorphic if there exists a rgw⍺*-homeomorphism from (X, τ) onto(Y, σ). We denote the family of all rgw⍺*-homeomorphism of a topological space (X, τ) onto itself by rgw⍺*-h(X, τ). Theorem 3.25: Every rgw⍺*-homeomorphism is a rgw⍺homeomorphism but not conversely. Proof: Let f: (X, τ) → (Y, σ) be an rgw⍺*-homeomorphism. Then f and f -1 are rgw⍺-irresolute and f is bijection. By Theorem 5.3 in [20], f and f -1 are rgw⍺-continuous. Therefore f is rgw⍺-homeomorphism. The converse of the above Theorem is not true in general as seen from the following example. Example 3.31: Let X=Y= {a,b,c}, τ={X, ϕ,{a},{b,c}} and σ ={Y, ϕ, {a}}. Let map f: X→Y defined by f(a)=b, f(b)= a, f(c)=c, then f is w*-homeomorphism but not rgw⍺-homeomorphism, as a closed set F={b,c} in Y, f -1 (F)= f -1 {b,c} ={a,c} is not rgw⍺-closed set. Theorem 3.32: Let f : (X, τ) → (Y, σ) and g : (Y, σ) → (Z, η) are rgw⍺*-homeomorphisms, then their composition g • f : (X, τ) → (Z, η) is also rgw⍺*-homeomorphism.

Theorem 3.33: The set rgw⍺*-h(X, τ) is a group under the composition of maps. Proof: Define a binary operation ψ: rgw⍺*-h(X, τ) × rgw⍺*h(X, τ) → rgw⍺*-h(X, τ) by f * g = g • f for all f, g ∈ rgw⍺*h(X, τ) and • is the usual operation of composition of maps. Then by Theorem 3.7, g • f ∈ rgw⍺*-h(X, τ). We know that the composition of maps is associative and the identity map I: (X, τ) → (X, τ) belonging to rgw⍺*-h(X, τ) serves as the identity element. If f ∈ rgw⍺*-h(X, τ), then f -1 ∈ rgw⍺*-h(X, τ) such that f • f -1 = f -1 • f = I and so inverse exists for each element of rgw⍺*h(X, τ). Therefore (rgw⍺*-h(X, τ), •) is a group under the operation of composition of maps. Theorem 3.34: Let f: (X, τ) → (Y, σ) be a rgw⍺*homeomorphism. Then f induces an isomorphism from the group rgw⍺*-h(X, τ) onto the group rgw⍺*-h(Y, σ). Proof: Using the map f, we define a map Ψf : rgw⍺*-h(X, τ) → rgw⍺*-h(Y, σ) by Ψf (h) = f • h • f -1 for every h ∈ rgw⍺*h(X, τ). Then Ψf is a bijection. Further, for all h1, h2 ∈ rgw⍺*h(X,τ), Ψf (h1

Therefore Ψf is a homeomorphism and so it is an isomorphism induced by f. Theorem 3.35: If f : (X, τ) → (Y, ) is homeomorphism, then there exists isomorphism between rgw⍺-h(X, τ) and rgw-h(Y, ) i.e. rgw⍺-h(X, ) ≅ rgw⍺ -h(Y, ). Proof: Using the map f, we define a map ψf : rgw⍺-h(X, τ) → rgw⍺-h(Y, ) by ψf (h) = f ohof 1 for every h ∈ rgw⍺h(X, τ). Then ψf is bijection. Further, for all h1, h2 ∈ ψf, ψf (h1oh2) = f o(h1 oh2)of -1 = (f oh1of -1 )o(f oh2 of -1 ) = ψf (h1)oψf(h2). Therefore ψf is a homeomorphism and so it is an isomorphism induced by f. Definition 3.36: A function f: (X, τ) → (Y, ) is said to contra rgw⍺-irresolute if f -1 (V) is rgw⍺-closed in (X, τ) for every rgw⍺-open set V of (Y, ). Lemma 3.37: Let two function f : (X, τ) → (Y, ) and g : (Y, ) → (Z, ) defined on topological spaces (X, τ) and (Y, ) respectively, then (1) If functions f and g are contra rgw⍺-irresolute, then the composition gof is also rgw⍺-irresolute.

(2) If function f is rgw⍺-irresolute (resp. contra rgw⍺irresolute) and g are contra rgw⍺-irresolute (resp. rgw⍺irresolute), then the composition function gof is contra rgw⍺irresolute. Definition 3.38: A bijection f: (X, τ) → (Y, σ) is called contra regular generalized weakly α-homeomorphism (briefly, conra rgw⍺-homeomorphism) if f is contra rgw⍺-irresolute and f -1 are rgw⍺-irresolute. The family of all contra rgw⍺-homeomorphism of a topological space (X, τ) onto itself is denoted by con-rgw⍺h(X, τ). i. e. Con-rgw⍺-h(X, τ) = {f: f (X, τ) → (X, τ) is a contra rgw⍺irresolute bijection and f -1 is rgw⍺-irresolute} For a topological space (X, τ), we construct alternative groups, say rgw⍺-h(X, τ) ∪ con-rgw⍺-h(X, τ). Theorem 3.39: If (X, τ) be a topological space, then union of two collections, rgw⍺-h(X, τ) ∪ con-rgw⍺-h(X, τ), forms a group under the composition of functions. Proof: Let us BX = rgw⍺-h(X, τ) ∪ con-rgw⍺-h(X, τ). A binary operation wX : BX ×BX → BX is well defined by wX (a, b) = boa, where boa : X → X is the composite function of the functions a and b. Indeed, let (a, b) ∈ BX ; if a ∈ rgw⍺-h(X, τ) and b∈ con-rgw⍺-h(X, τ), then boa : (X, τ)→ (X, τ) a contra rgw⍺-irresolute bijection and (boa) -1 is also contra rgw⍺irresolute and so wX (a, b) = boa ∈ rgw⍺-h(X, τ) ⊂ BX, if a ∈ rgw⍺-h(X, τ) and b ∈ rgw⍺-h(X, τ) then boa : (X, τ) → (X, τ) is a rgw⍺-irresolute bijection and so wX (a, b) = boa ∈ rgw⍺-h(X, τ) ⊆ BX, if a ∈ con -rgw⍺-h(X, τ) and b ∈ con -rgw⍺-h(X, τ), then boa: (X, τ) → (X, τ) is a rgw⍺irresolute bijection and (boa) -1 is also rgw⍺-irresolute and so wX (a, b) = boa ∈rgw⍺-h(X, τ) ⊂ BX is a ∈ con-rgw⍺-h(X, τ) and b ∈ rgw⍺-h(X, τ) then boa: (X, τ) → (X, τ) is a contra rgw⍺-irresolute bijection and (boa) -1 is also rgw⍺-irresolute and and so wX (a, b) = boa ∈ con -rgw⍺-h(X, τ) ⊆ BX. By the similar arguments, it is claimed that the binary operation wX: BX × BX → BX satisfies the axiom of group; for the identity element e of BX, e = 1X : (X, τ) → (X, τ). Thus the pair (BX, wX) forms a group under the composition of functions, i.e., rgw⍺-h(X, τ) ∪ con -rgw⍺-h(X, τ) is a group.

Theorem 3.40:

The homeomorphism group h(X, τ) is a subgroup of rgw⍺-h(X, τ) ∪ con -rgw⍺-h(X, τ). Proof: By Theorem 3.11, it can be show that h(X, τ) is a subgroup of rgw-h(X, τ) ∪ con -rgw-h(X, τ). Theorem 3.41: Let f : (X, τ) → (Y, ) is rgw⍺-continuous and collection of functions f is defined as Gg (f) = {(x, y) ∈ X × Y : y = f (x)}, where X × Y is product topological space and Gg (f) is called rgw⍺-graph f. Then the following properties are satisfied: (1) Gg (f), as a subspace of X × Y, rgw⍺-homeomorphism to X.

(2) If Y is rgw⍺-Housdrofff space, then Gg (f) is rgw⍺-closed in X × Y. Proof:

(1) Consider the function g: X → Gg (f) is defined by g(x) = (x, f (x)) for each x ∈ X is rgw⍺-continuous and g 1 is also rgw⍺-continuous. It is obvious that g is an injective function. Let P and Q are an arbitrary neighbourhood x ∈ X and (x, f (x)) in Gg(f) respectively. So, there exists two rgw⍺-open sets U and V in X and Y respectively containing x and f (x) for which (U × V) ∩ G g (f) ⊂ E and U ⊂ P and f (U) ⊂ V. Let N = (U × V) ∩ Gg (f), then (x, f (x)) ∈ N and x ∈ g -1 (N) ⊂ U ⊂ P. This shows that g -1 is rgw⍺-continuous. Therefore, g(U) ⊂ (U × V) ∩ Gg (f) ⊂ Q. Hence g is rgw⍺continuous which means that g is a rgw⍺-homeomorphism.

(2) Let (x, y) ∉ Gg (f). Then y1 = f (x) = y. By hypothesis, there exist disjoint rgw⍺-open sets V1 and V in Y such that y1 ∈V1, y ∈ V. Since f is rgw⍺-continuous, there exists an open set U in X containing x such that f (U) ⊂ V1. Then g(U) ⊂ U × V1. It follows from this and the fact that V1 ∩ V = φ that (U × V) ∩ Gg (f) = φ.

Conclusion

In this paper we have introduced and studied the properties of rgw⍺-Homeomorphism and rgw⍺*-Homeomorphisms. Our future extension is to study Locally rgw⍺-Homeomorphism and Locally rgw⍺*-Homeomorphism in Topological Spaces.

Acknowledgement

The Authors would like to thank the referees for useful comments and suggestions.

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