-Continuity of Maps in the Product Space and Some Versions of Separation Axioms

TMJM

THE MINDANAWAN Journal of Mathematics

The Mindanawan Journal of Mathematics

Official Journal of theDepartment of Mathematics and Statistics Mindanao State University-Iligan Institute of Technology ISSN: 2094-7380 (Print) | 2783-0136 (Online)

Vol. 4 (2022), no. 1, pp. 1–12

.

θ

sw

-Continuity of Maps in the Product Space and Some Versions of Separation Axioms

Jan Alejandro C. Sasam¹ and Mhelmar A. Labendia^2,∗

1,2Department of Mathematics and Statistics

MSU-Iligan Institute of Technology, 9200 Iligan City, Philippines

janalejandro.sasam@g.msuiit.edu.ph, mhelmar.labendia@g.msuiit.edu.ph

Received: 23rd May 2021 Revised: 12th February 2022

Abstract

In this study, the concept ofθ_sw-open set is introduced and its relationship to the other well-known concepts such as the classical open,θ-open, and ωθ-open sets is described. The concepts ofθsw-interior and θsw-closure of a set is also defined and investigated. Related concepts such asθsw-open andθsw-closed functions,θsw-continuous function,θsw-connected, and some versions of separation axioms are defined and characterized. Finally, the concept of θsw-continuous function from an arbitrary topological space into the product space is investigated further.

1 Introduction and Preliminaries

The first attempt to substitute concept in topology with concept possessing either weaker or stronger property was done by N. Levine [24] when he introduced the concepts of semi-open set, semi-closed set, and semi-continuity of a function. Several mathematicians then became interested in introducing other topological concepts which can replace the concept of open set, closed set, and continuity of a function.

In 1968, Velicko [27] introduced the concepts ofθ-continuity between topological spaces and subsequently defined the concepts of θ-closure and θ-interior of a subset of topological space.

The concept of θ-open set and its related topological concepts had been deeply studied and investigated by numerous authors, see [1,7,8,15,16,20,21,22,25,26].

Let (X,T) be a topological space and A ⊆ X. The θ-closure and θ-interior of A are, respectively, denoted and defined by

Cl_θ(A) ={x∈X :Cl(U)∩A6=∅for every open setU containingx}

and

Int_θ(A) ={x∈X:Cl(U)⊆A for some open setU containing x},

whereCl(U) is the closure of U inX. A subsetA ofX isθ-closed ifCl_θ(A) =A andθ-open if Intθ(A) =A. Equivalently, A isθ-open if and only if X\A is θ-closed.

In 1971, Hoyle and Gentry [19] introduced the class of somewhat-continuous functions and somewhat-open functions. The somewhat-continuous functions, which are generalization of

∗Corresponding author

2020 Mathematics Subject Classification: 54A10, 54A05

Keywords and Phrases:θsw-open,θsw-closed,θsw-connected,θsw-continuous,θsw-Hausdorff,θsw-regular,θsw-normal This research has been supported by Department of Science & Technology-Accelerated Science and Technology Human

continuity requiring nonempty inverse images of open sets to have nonempty interiors instead of being open, have proved to be very useful in topology. Since then, the concepts ofsomewhat- interior andsomewhat-closure of a subset of a topological space have been subsequently defined and the concept of somewhat-open and somewhat-closed sets have been used to characterize somewhat-continuity, see [5,6].

A subsetU of a spaceXis said to besomewhat-open ifU =∅or if there existsx∈U and an open setV such thatx∈V ⊆U. A set is calledsomewhat-closed if its complement issomewhat- open. Let A be a subset of a spaceX. The somewhat-closure andsomewhat-interior of A are, respectively, denoted and defined byswCl(A) =∩{F :F is somewhat-closed and A⊆F} and swInt(A) =∪{U ⊆A:U issomewhat-open}.

In 1982, Hdeib [18] introduced the concepts ofω-open andω-closed sets andω-closed map- pings on a topological space. He showed thatω-closed mappings are strictly weaker than closed mappings and also showed that the Lindel¨of property is preserved by counter images ofω-closed mappings with Lindel¨of counter image of points. The concepts ofω-open sets and its correspond- ing topological concepts had been studied in several papers, see [2,3,4,9,10,11,12,13,14,23].

In 2010, Ekici et al. [17] introduced the concepts of ωθ-open and ωθ-closed sets on a topo- logical space. They showed that the family of all ω_θ-open sets in a topological spaceX forms a topology on X. They also introduced the notions ofω_θ-interior andω_θ-closure of a subset of a topological space.

A pointx of a topological spaceX is called a condensation point of A⊆X if for each open set G containing x, G∩A is uncountable. A subset B of X is ω-closed if it contains all of its condensation points. The complement ofB isω-open. Equivalently, a subsetU ofX is ω-open (resp.,ω_θ-open) if and only if for eachx∈U, there exists an open setO containingxsuch that O\U (resp.,O\Int_θ(U)) is countable. A subsetB ofX isω_θ-closed if its complement X\B isω_θ-open.

A topological space X is said to be somewhat-connected (resp., θ-connected, ω-connected, ω_θ-connected) if X cannot be written as the union of two nonempty disjoint somewhat-open (resp.,θ-open, ω-open,ωθ-open) sets.

Otherwise,Xissomewhat-disconnected (resp.,θ-disconnected,ω-disconnected,ω_θ-disconnected).

Let A be an indexing set and {Y_α : α ∈ A} be a family of topological spaces. For each α∈A, letTα be the topology onY_α. The Tychonoff topology on Π{Y_α :α∈A}is the topology generated by a subbase consisting of all sets p⁻¹_α (U_α), where the projection map p_α : Π{Y_α : α∈A} →Yα is defined bypα(hy_βi) =yα,Uα ranges over all members ofTα, andα ranges over all elements of A. Corresponding to U_α ⊆ Y_α, denote p⁻¹_α (U_α) by hU_αi. Similarly, for finitely many indicesα₁, α₂, . . . , α_n,and sets U_α1 ⊆Y_α1,U_α2 ⊆Y_α2, . . . , U_αn ⊆Y_αn,the subset

hU_α1i ∩ hU_α2i ∩ · · · ∩ hU_αni=p⁻¹_α1(Uα1)∩p⁻¹_α2(Uα2)∩ · · · ∩p⁻¹_αn(Uαn)

is denoted by hU_α1, U_α2, . . . , U_αni. We note that for each open set U_α subset of Y_α, hU_αi = p⁻¹_α (Uα) =Uα×Πβ6=αYβ. Hence, a basis for the Tychonoff topology consists of sets of the form hB_α1, Bα2, ..., Bα_ki, where Bαi is open inYαi for everyi∈K={1,2, ..., k}.

Now, the projection map pα : Π{Y_α : α ∈ A} → Yα is defined by pα(hy_βi) = yα for each α ∈A. It is known that every projection map is a continuous open surjection. Also, it is well known that a functionf from an arbitrary spaceX into the Cartesian product Y of the family of spaces{Y_α:α∈A} with the Tychonoff topology is continuous if and only if each coordinate functionp_α◦f is continuous, wherep_α is theα-th coordinate projection map.

In this paper, given a topology on X, we define a new type of topology onX which is finer than the topology formed by the collection of θ-open sets, but coarser than the given topology on X.

2 θ

sw

-Open and θ

sw

-Closed Functions

In this section, we shall determine the connection ofθsw-open set to the classical open, θ-open, andω_θ-open sets. We shall also define and characterize the concepts ofθ_sw-open andθ_sw-closed functions.

Definition 2.1. Let X be a topological space. A subset A of X is said to be θsw-open if for everyx∈A, there exists an open set U containingxsuch that swCl(U)⊆A. A subsetF ofX is called θ_sw-closed ifX\F isθ_sw-open.

Remark 2.2. The following diagram holds for a subset of a topological space.

ω_θ-open ω-open

θ-open θsw-open open somewhat-open We remark that the above diagram is also true for their respective closed sets.

The following examples show that the implications in the above diagram (with respect to θ_sw-open set) are not reversible. We note that since ω_θ-open and open are two independent notions [17, Example 5], ω_θ-open does not imply θsw-open.

Example 2.3. (i) Let X = {a, b, c, d, e} with topology T = {∅, X, {a}, {c}, {d}, {a, c}, {a, d}, {c, d}, {a, b, c}, {a, c, d}, {c, d, e}, {a, b, c, d}, {a, c, d, e}}. Then {a, b, c} is θ_sw- open but not θ-open.

(ii) LetRbe the real line with topology T={∅,R,Q}. ThenQis open but notθ_sw-open.

(iii) Let R be the real line with topologyT={∅,R,R\(1,2),{1.5},R\(1,2)∪ {1.5}}. Then R\(1,2) isθ_sw-open but notω_θ-open.

Before showing that the collection ofθ_sw-open sets forms a topology, we shall consider first the following remark.

Remark 2.4. Let X be a topological space and A, B⊆X. Then (i) swInt(A) issomewhat-open and swInt(A)⊆A;

(ii) swCl(A) is somewhat-closed and A⊆swCl(A);

(iii) swInt(A) is the largestsomewhat-open set contained in A;

(iv) IfA⊆B, thenswInt(A)⊆swInt(B);

(v) x ∈ swInt(A) if and only if there exists a somewhat-open set U containing x such that U ⊆A;

(vi) A issomewhat-open if and only ifA=swInt(A);

(vii) swInt(swInt(A)) =swInt(A);

(viii) swInt(A∩B)⊆swInt(A)∩swInt(B);

(ix) swCl(A) is the smallest somewhat-closed set containing A;

(x) A⊆B implies thatswCl(A)⊆swCl(B);

(xi) swCl(swCl(A)) =swCl(A);

(xii) swCl(A)∪swCl(B)⊆swCl(A∪B);

(xiii) swInt(X\A) =X\swCl(A);

(xiv) A issomewhat-closed if and only ifA=swCl(A); and

(xv) x∈swCl(A) if and only if for every somewhat-open set U containing x,U ∩A6=∅.

Theorem 2.5. Let Tθsw be a family of allθsw-open subsets of topological space X. Then, Tθsw

forms a topology on X.

Proof. It is not difficult to see that ∅, X ∈ Tθsw. Now, let{A_i : i∈A} be a family members of Tθsw. Let x ∈ [

i∈A

Ai. Then x ∈ Aj for some j ∈ A. Since Aj is θsw-open, there exists an open set U containing x such that swCl(U) ⊆ Aj ⊆ [

i∈A

Ai. Hence, [

i∈A

Ai is θsw-open. Next, let A₁, A₂ ∈ Tθsw. Let x ∈ A₁ ∩A₂.Then there exist open sets U₁ and U₂, both containing x, such that swCl(U1) ⊆ A1 and swCl(U2) ⊆ A2. Note that U1 ∩U2 is open containing x.

Hence, swCl(U₁ ∩U₂) ⊆ swCl(U₁)∩swCl(U₂) ⊆ A₁∩A₂. Therefore, A₁ ∩A₂ is θ_sw-open.

Consequently, Tθsw is a topology onX.

Definition 2.6. Let X be topological space and A⊆X. Theθ_sw-interior ofA is defined and denoted by Int_θsw(A) =S{U :U is an θ_sw-open set and U ⊆ A}. We note that by Theorem 2.5,Intθsw(A) is the largest θsw-open set contained in A. Moreover, x∈Intθsw(A) if and only if there exists a θ_sw-open set U containing x such thatU ⊆A.

Definition 2.7. Let X be topological space andA ⊂X. The θ_sw-closure of A is defined and denoted by Cl_θsw(A) =T{F :F is an θ_sw-closed set andA ⊆ F}. We note that by Theorem 2.5,Clθsw(A) is the smallest θsw-closed set containing A.

Remark 2.8. Let X be a topological space and A, B⊆X. Then (i) If A⊆B, thenIntθsw(A)⊆Intθsw(B);

(ii) A isθ_sw-open if and only if A=Int_θsw(A);

(iii) Intθsw(A∩B) =Intθsw(A)∩Intθsw(B);

(iv) x∈Cl_θsw(A) if and only if for everyθ_sw-open subsetU containingx,U∩A6=∅; (v) A⊆B implies thatClθsw(A)⊆Clθsw(B);

(vi) A isθ_sw-closed if and only ifCl_θsw(A) =A;

(vii) Cl_θsw(Cl_θsw(A)) =Cl_θsw(A);

(viii) Cl_θsw(A)∪Cl_θsw(B) =Cl_θsw(A∪B);

(ix) Int_θsw(X\A) =X\Cl_θsw(A);

(x) Cl_θsw(X\A) =X\Int_θsw(A);

(xi) A isθ_sw-open if and only if for everyx∈A, there exists a basic open setB containing x such thatswCl(B)⊆A;

(xii) x∈Int_θsw(A) if and only if there exists an open setO containingx,swCl(O)⊆A; and (xiii) x∈Clθsw(A) if and only if for every open setU containingx,swCl(U)∩A6=∅.

Next, we introduce and characterize the concepts ofθ_sw-open and θ_sw-closed functions.

Definition 2.9. LetX and Y be topological spaces. A function f :X→Y isθ_sw-open (resp., θsw-closed) iff(G) isθsw-open (resp.,θsw-closed) for every open (resp., closed) setGinX.

In view of Remark 2.2, we have the following results.

Remark 2.10. Letf :X →Y be a function of topological spaces.

(i) If f is θ_sw-open (resp.,θ_sw-closed) onX, then f is open (resp., closed) onX.

(ii) Iff is θ-open (resp.,θ-closed) on X, thenf isθsw-open (resp.,θsw-closed) onX.

Remark 2.11. The converse of Remark2.10(i) and (ii) do not necessarily hold.

(i) Consider R with topologies T1 = {∅,R,Q} and T2 = {∅,R,Q,Q\ {0}}. Define f : (R,T1) → (R,T2) by f(x) =x for all x ∈R. Obviously, f is open on (R,T1). Next, we will show that there exists an open set in (R,T1) such that its image is not θ_sw-open in (R,T2). Note that Q^c is not somewhat open since for every x∈Q^c the only open set in (R,T2) containing x is Rand R * Q^c. Now, if swCl(Q) =Q, then Qis somewhat close, a contradiction. Thus,Q⊂swCl(Q). Note also thatswCl(R) =R * Q. Hence,Qis not θ_sw-open in (R,T2). This means that f(Q) =Q is not θ_sw-open in (R,T2). Therefore,f is notθsw-open on (R,T1). Since f is bijective,f is closed but not θsw-closed on (R,T1).

(ii) Consider the topological spacesX =Y ={a, b, c, d, e} with the corresponding topologies Tx ={∅, X,{a},{c},{d},{a, c},{a, d},{c, d},{a, b, c},{a, c, d},{a, b, c, d}} and Ty =T in Example2.3(i). Letf : (X,Tx)→(Y,Ty) be a function defined byf(x) =xfor allx∈X.

By definition of f and θ_sw-open set, f is θ_sw-open on (X,Tx). Next, we will show that there exists an open set in (X,Tx) such that its image is notθ-open in (X,Ty). Consider the open setA ={a, b, c} in (X,Tx). Thenf(A) = A. We claim that A is θsw-open but notθ-open in (X,Ty). By Example2.3 (i),A is θ_sw-open but not θ-open. Thus,f is not θ-open on (X,Tx). Since f is bijective,f is θsw-closed but not θ-closed on (X,Tx).

Theorem 2.12. Let X and Y be topological spaces and f : X → Y be a function. Then the following statements are equivalent:

(i) f isθsw-open on X.

(ii) f(Int(A))⊆Int_θsw(f(A)) for everyA⊆X.

(iii) f(B) is θsw-open for every basic open set B in X.

(iv) For each x∈X and every open set U in X containing x, there exists an open setV in Y containing f(x) such that swCl(V)⊆f(U).

Proof. (i)⇒(ii): Let A⊆X. Note that f(Int(A))⊆f(A) and f(Int(A)) is θsw-open. In view of Definition 2.6,f(Int(A))⊆Int_θsw(f(A)).

(ii)⇒(iii): LetB be a basic open set inX. Thenf(B) =f(Int(B))⊆Int_θsw(f(B))⊆f(B).

By Remark2.8 (ii), f(B) isθsw-open.

(iii)⇒(iv): Let x ∈X and U be an open set containing x. Then there exists a basic open setB containing xsuch thatB⊆U, which impliesf(x)∈f(B)⊆f(U). By assumption, there exists an open setV containingf(x) such thatswCl(V)⊆f(B)⊆f(U).

(iv)⇒(i): Let U be an open set inX and let y∈f(U). Then there exists x∈U such that f(x) =y. By assumption, there exists an open set V containing y such thatswCl(V)⊆f(U).

Hence,f(U) is θsw-open inY. Therefore,f is θsw-open on X.

Theorem 2.13. Let X and Y be topological spaces and f : X → Y be a function. Then the following are equivalent:

(i) f isθ_sw-closed on X.

(ii) Clθsw(f(A))⊆f(Cl(A)), for every A⊆X.

Proof. (i)⇒(ii): Let A ⊆X. Note that f(A) ⊆f(Cl(A)) and f(Cl(A)) is θ_sw-closed. In view of Definition 2.7,Clθsw(f(A))⊆f(Cl(A)).

(ii)⇒(i): LetF be closed in X. By assumption, f(F) ⊆Cl_θsw(f(F))⊆f(Cl(F)) =f(F).

By Remark2.8 (vii),f is θ_sw-closed onX.

Remark 2.14. Let X and Y be topological spaces and f : X → Y be a bijective function.

Then f isθ_sw-open on X if and only if f is θ_sw-closed on X.

3 θ

_sw

-Continuity of Functions in the Product Space

In this section, we provide a definition of a θ_sw-continuous function and its characterization from an arbitrary topological space into the product space.

Definition 3.1. A functionf :X→Y isθ_sw-continuous iff⁻¹(G) is θ_sw-open for every open set Gof Y.

In view of Remark 2.2, the following remark holds.

Remark 3.2. Letf :X →Y be a function of topological spaces. If f isθ_sw-continuous on X, thenf is continuous onX.

Remark 3.3. The converse of Remark3.2 does not necessarily hold.

To see this, considerRwith topologyT={∅,R,Q}. Definef : (R,T)→(R,T) byf(x) =x for all x ∈ R. Obviously, f is continuous on R. Next, we will show that there exists an open set in (R,T) such that its inverse image is not θsw-open in (R,T). Consider the open set Qin (R,T). We will show that it is notθsw-open in (R,T). By Example 2.3 (ii), Q is not θsw-open in (R,T). That is,f⁻¹(Q) = Q is not θ_sw-open in (R,T). Therefore, f is continuous but not θsw-continuous on (R,T).

The proofs of the following results are standard, hence omitted.

Theorem 3.4. Let X and Y be topological spaces and f : X → Y be a function. Then the following statements are equivalent:

(i) f isθ_sw-continuous onX.

(ii) f⁻¹(F) is θ_sw-closed in X for each closed subset F of Y.

(iii) f⁻¹(B) is θsw-open inX for each (subbasic) basic open setB in Y.

(iv) For every p∈X and every open set V of Y containing f(p), there exists a θ_sw-open set U of X containing p such thatf(U)⊆V.

(v) f(Cl_θsw(A))⊆Cl(f(A))for each A⊆X.

(vi) Cl_θsw(f⁻¹(B))⊆f⁻¹(Cl(B))for each B ⊆Y.

Theorem 3.5. Let X be any topological space and χ_A:X → D the characteristic function of a subset A of X. Then χA is θsw-continuous if and only if A is both θsw-open and θsw-closed.

In the following results, if Y = Π{Y_α:α∈A} is a product space and A_α ⊆ Y_α for each α∈A, we denoteAα1× · · · ×Aαn×Π{Y_i :α /∈K}byhA_α1,· · ·, Aαni, whereK={α₁,· · · , αn}.

If Y = Π{Y_i: 1≤i≤n}is a finite product, denote Aα1 × · · · ×Aαn byhA_α1,· · · , Aαni.

Theorem 3.6. Let Y = Π{Y_α :α ∈ A} be a product space. Let S ={α₁, . . . , α_n} be a finite subset of A and ∅6=O_α ⊆Y_α for each α∈A. ThenO =hO_α1, . . . , O_αni is somewhat-open in Y if and only if each Oαi is somewhat-open in Yαi.

Theorem 3.7. Let Y = Π{Y_α :α∈A} be a product space and A_α⊆Y_α for eachα ∈A. Then swCl(Π{A_α:α∈A})⊆Π{swCl(A_α) :α∈A}.

Proof. Let x = ha_αi ∈/ Π{swCl(A_α) : α ∈ A}. Then a_β ∈/ swCl(A_β) for some β ∈ A. This means that there exists a somewhat-open set G_β containing a_β such that G_β∩A_β = ∅. By Theorem3.6, hG_βi is somewhat-open containing x. Hence, hG_βi ∩Π{A_α :α ∈A}=∅. Thus, x /∈swCl(Π{A_α :α∈A}).

Theorem 3.8. Let Y = Π{Y_α :α∈A} be a product space and A_α⊆Y_α for eachα ∈A. Then Clθsw(Π{A_α:α∈A})⊆Π{Cl_θsw(Aα) :α ∈A}.

Proof. Let x = ha_αi ∈ Cl_θsw(Π{A_α : α ∈ A}). Then for every open set O containing x, swCl(O)∩Π{A_α : α ∈ A} 6= ∅. Suppose that there exists β ∈ A such that a_β ∈/ Cl_θsw(A_β).

Then there exists an open setUβ containingaβ such thatswCl(Uβ)∩Aβ =∅. By Theorem3.7, swCl(hU_βi)⊆ hswCl(U_β)i. It follows thatx=ha_αi ∈ hU_βiandswCl(hU_βi)∩Π{A_α:α∈A}=

∅, a contradiction. Thus,x∈Π{Cl_θsw(A_α) :α∈A}.

Theorem 3.9. Let Y = Π{Y_i : 1 ≤ i ≤ n} be a (finite) product space and Ai ⊆ Yi for each i= 1,2, . . . , n. Then Π{Int_θsw(A_i) : 1≤i≤n} ⊆Int_θsw(Π{A_i : 1≤i≤n}).

Proof. Let x = ha₁, a2, . . . , ani ∈ Π{Int_θsw(Ai) : 1 ≤ i ≤ n}. Then ai ∈ Intθsw(Ai) for all i= 1,2, . . . , n. This means that for alli= 1,2, . . . , n, there exists an open setOi containingai

such thatswCl(O_i)⊆A_i. LetO=hO₁, O₂, . . . , O_ni, which is an open set inY containingx. By Theorem 3.7, swCl(O) = swCl(hO₁, O₂, . . . , O_ni) ⊆ hswCl(O₁), swCl(O₂), . . . , swCl(O_n)i ⊆ hA₁, A2, . . . , Ani. Hence,x∈Int_θsw(Π{A_i : 1≤i≤n}).

Theorem 3.10. Let X be a topological space and Y = Π{Y_α: α∈A} a product space. A function f : X → Y is θ_sw-continuous on X if and only if p_α◦f is θ_sw-continuous on X for every α∈A.

Proof. Suppose that f is θ_sw-continuous onX. Letα∈Aand U_α be an open set inY_α. Since pα is continuous, pα−1(Uα) is open in Y. Hence, f⁻¹(pα−1(Uα)) = (pα◦f)⁻¹(Uα) is θsw-open set in X. Therefore, pα◦f isθsw-continuous for every α∈A.

Conversely, suppose that each coordinate functionpα◦f isθsw-continuous. LetGα be open inY_α. Then,hG_αiis a subbasic open set inY and (p_α◦f)⁻¹(G_α) =f⁻¹(p_α⁻¹(G_α)) =f⁻¹(hG_αi) isθsw-open inX. Therefore, f isθsw-continuous.

Corollary 3.11. Let X be a topological space, Y = Π{Y_α : α∈A} a product space, and fα : X →Y_α a function for each α∈A. Let f :X→ Y be the function defined by f(x) =hf_α(x)i.

Then f is θ_sw-continuous onX if and only if each f_α isθ_sw-continuous on X for each α∈A. Theorem 3.12. Let Y = Π{Y_α :α ∈A} be a product space, S = {α₁, α2, . . . , αn} ⊆ A, and

∅6=O_αi ⊆Y_αi for each α_i ∈S. If eachO_αi is θ_sw-open inY_αi, then O:=hO_α1, O_α2, . . . , O_αni is θsw-open in Y.

Proof. Let x = ha_αi ∈ O. Then a_αi ∈ O_αi for every α_i ∈ S. This means that for every αi ∈ S, there exists an open set Uαi containing aαi such that swCl(Uαi) ⊆ Oαi. Let U = hU_αi, Uα2, . . . , Uαni. Then x∈U and by Theorem3.7,

swCl(U)⊆ hswCl(U_α1), swCl(U_α2), . . . , swCl(U_αn)i ⊆O.

Thus,O is θsw-open inY.

Theorem 3.13. Let X = Π{X_α :α ∈A} and Y = Π{Y_α :α ∈A} be product spaces and for each α ∈ A, let fα : Xα → Yα be a function. If each fα is θsw-continuous on Xα, then the functionf :X→Y defined by f(hx_αi) =hf_α(x_α)i is θ_sw-continuous onX.

Proof. Let hV_αi be a subbasic open set in Y. Then f⁻¹(hV_αi) = hf_α⁻¹(Vα)i. Since each fα

is θ_sw-continuous, f_α⁻¹(V_α) is θ_sw-open in X_α. Let x = hx_βi ∈ f⁻¹(hV_αi) = hf_α⁻¹(V_α)i. Then x_α ∈f_α⁻¹(V_α). Hence, there exists an open setU_αcontainingx_αsuch thatswCl(U_α)⊆f_α⁻¹(V_α).

Note that hU_αi is open in X contining x. By Theorem 3.7, swCl(hU_αi) ⊆ hswCl(U_α)i ⊆ hf_α⁻¹(V_α)i=f⁻¹(hV_αi). This means thatf⁻¹(hV_αi) isθ_sw-open inX. Thus,f isθ_sw-continuous on X.

4 θ

_sw

-Connectedness and Some Versions of Separation Axioms

In this section, we define and characterize the concepts of θ_sw-connected, θ_sw-Hausdorff, θ_sw- regular, and θsw-normal spaces.

Definition 4.1. A topological space X is said to be θ_sw-connected if it is not the union of two nonempty disjoint θsw-open sets. Otherwise, X is θsw-disconnected. A subset B of X is θ_sw-connected if it isθ_sw-connected as a subspace of X.

Theorem 4.2. Let X be a topological space. Then the following statements are equivalent:

(i) X isθ_sw-connected.

(ii) The only subsets ofX that are both θsw-open and θsw-closed are ∅ andX . (iii) Noθ_sw-continuous function from X to D is surjective.

Theorem 4.3. A topological spaceX is connected if and only if it is θ-connected.

Proof. In view of Remark 2.2, connectedness impliesθ-connectedness.

Conversely, assume thatX isθ-connected. IfX were disconnected, then there exist disjoint nonempty open setsGand Hsuch that X=G∪H. This implies thatGandH are also closed sets, henceCl(G) =G⊆Gand Cl(H) =H ⊆H. Thus,G and H are θ-open sets. ThusX is θ-disconnected, contrary to our assumption.

In view of Remark 2.2and Theorem 4.3, the following result holds.

Theorem 4.4. A topological spaceX is θ_sw-connected if and only if it is θ-connected.

Remark 4.5. The following diagram holds for a subset of a topological space.

ω_θ-connected ω-connected

θ-connected θsw-connected connected somewhat-connected Definition 4.6. A topological spaceX is said to be

(i) θ_sw-Hausdorff if given any pair of distinct points p, q inX there exist disjoint θ_sw-open setsU and V such thatp∈U and q∈V;

(ii) θ_sw-regular if for each closed set F and each point x /∈ F, there exist disjoint θ_sw-open setsU and V such thatx∈U and F ⊆V; and

(iii) θ_sw-normal if for every pair of disjoint closed sets E and F of X, there exist disjoint θ_sw-open setsU and V such thatE⊆U and F ⊆V.

In view of Remark 2.2, everyθ_sw-Hausdorff (resp., θ_sw-regular, θ_sw-normal) space is Haus- dorff (resp., regular, normal).

Theorem 4.7. Let X be a topological space. Then the following are equivalent:

(i) X isθsw-Hausdorff.

(ii) Let x∈X. Fory 6=x, there exists aθ_sw-open setU containingx such that y /∈Cl_θsw(U).

(iii) For each x∈X, C :=∩{Cl_θsw(U) :U is an θsw-open set with x∈U}={x}.

Proof. (i)⇒(ii): For every disjoint pointsx, y ∈X, there exist disjoint θ_sw-open sets U and V such thatx∈U and y∈V. By Remark2.8(v), y /∈Clθsw(U).

(ii)⇒(iii): Letx ∈X. Then x∈ C. Hence, for every y 6=x, there exists a θ_sw-open set U containing xsuch that y /∈Clθsw(U). Thus,y /∈C. Since y is arbitrary,C={x}.

(iii)⇒(i): Let x, y ∈ X such that x 6= y. By assumption, there exists a θ_sw-open set U containing x such that y /∈ Cl_θsw(U). By Remark 2.8 (v), there exists a θ_sw-open set V containing y such thatU ∩V =∅. Hence,X is θsw-Hausdorff.

Theorem 4.8. Let X be a topological space. Then the following are equivalent:

(i) X isθsw-regular.

(ii) For each x ∈ X and open set U with x ∈ U, there exists a θ_sw-open set V with x ∈ V such that V ⊆Clθsw(V)⊆U.

(iii) For each x ∈ X and closed set F with x /∈ F, there exists a θ_sw-open set V with x /∈ V such that Clθsw(V)∩F =∅.

Proof. (i)⇒(ii): Let x ∈ X and U be an open set containing x. Then X \U is closed and x /∈ X \U. By assumption, there exist disjoint θsw-open sets V and W such that x ∈ V and X\U ⊆ W. By Remark 2.8 (vi), (vii), Cl_θsw(V) ⊆Cl_θsw(X\W) =X \W. Moreover, Cl_θsw(V)∩X\U ⊆Cl_θsw(V)∩W =∅so that Cl_θsw(V)⊆U. Thus, V ⊆Cl_θsw(V)⊆U.

(ii)⇒(iii): Let x∈X and F be a closed set with x /∈F. ThenX\F is open containing x.

By assumption, there exists a θ_sw-open set V containing x such that V ⊆Cl_θsw(V)⊆ X\F.

This means that Cl_θsw(V)∩F =∅.

(iii)⇒(i): Let x ∈ X and F be a closed set with x /∈ F. By assumption, there exists a θ_sw-open set V with x∈ V such that Cl_θsw(V)∩F =∅. Note that X\Cl_θsw(V) is θ_sw-open and F ⊆X\Clθsw(V). Furthermore, V ∩X\Clθsw(V) =∅. Hence,X isθsw-regular.

Theorem 4.9. Let X be a topological space. Then the following are equivalent:

(i) X isθsw-normal.

(ii) For each closed set A and each open set U containing A, there exists a θ_sw-open set V containing A such that Clθsw(V)⊆U.

(iii) For each pair of disjoint closed sets A and B, there exists θ_sw-open set U containing A such that Clθsw(U)∩B =∅.

Proof. (i)⇒(ii): LetA be closed and U be open containing A. ThenA and X\U are disjoint closed sets in X. By assumption, there exist disjoint θ_sw-open setsV and W such thatA⊆V andX\U ⊆W or (X\W ⊆U). By Remark2.8(vi), (vii), Clθsw(V)⊆Clθsw(X\W) =X\W so that Cl_θsw(V)⊆Cl_θsw(X\W) =X\W ⊆U.

(ii)⇒(iii): Let A and B be two disjoint closed sets. Then,A ⊆X\B and X\B is open.

By assumption, there exists aθsw-open set U containingA such that Clθsw(U)⊆X\B. This means that Cl_θsw(V)∩B =∅.

(iii)⇒(i): LetAandB be disjoint closed sets. By assumption, there exists aθ_sw-open setU containingA such thatCl_θsw(U)∩B=∅. ThenB ⊆X\Cl_θsw(U). SinceU and X\Cl_θsw(U) are disjoint θ_sw-opens,X isθ_sw-normal.

A topological space X is said to be a T₁-space if for each p, q ∈X with p6= q, there exist open sets U and V such that p∈U and q /∈U, and q∈V and p /∈V.

Theorem 4.10. Let X be a T1-space. Then

(i) If X isθ_sw-regular, then X is θ_sw-Hausdorff; and (ii) If X isθsw-normal, then X is θsw-regular.

Proof. (i): Suppose that X is θsw-regular. For each x, y ∈ X with x 6=y, there exist disjoint open sets U and V such that x ∈ U and y /∈ U, and y ∈ V and x /∈ V. This implies that x /∈X\U andy /∈X\V. Note thatX\U is closed. SinceXisθsw-regular, there exists disjoint θsw-open setsA and B such that x∈A and X\U ⊆B. Note that y∈X\U. Hence, y∈B.

Thus,X is θ_sw-Hausdorff.

We can prove (ii) by following the same argument used in (i).

5 Conclusion and Recommendations

The paper has introduced the concept ofθsw-open set and described its connection to the other well-known concepts such as the classical open, θ-open, and ω_θ-open sets. The paper has also defined and characterized the concepts of θsw-open and θsw-closed functions, θsw-continuous function, andθsw-connected,θsw-Hausdorff,θsw-regular, andθsw-normal spaces. Moreover, the paper has formulated a necessary and sufficient condition forθ_sw-continuity of a function from an arbitrary space into the product space. A worthwhile direction for further investigation is to establish versions of Urysohn’s Lemma and Tietze Extension Theorem for θsw-open sets.

One may also try to investigate θ_sw-open and θ_sw-closed sets, and other related concepts in a generalized topological space.

Acknowledgement

The authors would like to thank the anonymous referee for his valuable comments for the improvement of this paper.

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