Operator Compactification of Topological Spaces
Compactificaci´ on de Espacios Topol´ ogicos mediante Operadores Ennis Rosas ([email protected])
Universidad de Oriente Departamento de Matem´aticas
Cuman´a, Venezuela.
Jorge Vielma (email:[email protected])
Universidad de los Andes Departamento de Matem´aticas
Facultad de Ciencias M´erida, Venezuela.
Abstract
The concepts ofα-locally compact spaces andα-compactification are introduced. It is shown that everyα-T2 andα-locally compact space has anα-compactification.
Key words and phrases:α-locally compact,β-compactification.
Resumen
Se introducen los conceptos de espacioα-localmente compacto yα- compactificaci´on de un espacio y se muestra que todo espacioα-T2,α localmente compacto, tiene unaα-compactificaci´on.
Palabras y frases clave: α-localmente compacto,β-compactificaci´on.
In this paper, we try to show that everyα-T2,α-locally compact space has an α-compactification which generalizes the Alexandroff compactification of locally compact Hausdorff spaces.
We recall the definitions of operator associated to a topology,α-open set, α-T2space andα-compact space.
Recibido 1999/10/26. Revisado 2000/05/23. Aceptado 2000/07/03.
MSC (2000): Primary 54D35; Secondary 54D30, 54D45.
Definition 1 ([3]). Let (X,Γ) be a topological space,B a subset ofX and α an operator from Γ to P(X), i.e., α : Γ → P(X). We say that α is an operator associated with Γ if (O)U ⊆α(U) for allU ∈Γ.
We say that the operatorαassociated with Γ isstablewith respect toBif (S)αinduces an operatorαB: ΓB→P(B) such thatαB(U∩B) =α(U)∩B for every U in Γ, where ΓB is the relative topology onB.
Definition 2 ([1]). Let (X,Γ) be a topological space and α an operator associated with Γ. A subset A of X is said to be α-open if for each x ∈A there exists an open set U containingxsuch thatα(U)⊂A. A subsetB is said to be α-closed if its complement isα-open.
We observe that the collection ofα-open sets, in general, is not a topology, but if α is considered to be regular (see [1], [3] for the definition) then this collection is a topology.
Definition 3 ([1]). Let (X,Γ) be a topological space and α an operator associated with Γ. We say that a subsetAofX isα-compact if for every open covering Π ofAthere exists a finite subcollection{C1, C2, . . . , Cn} of Π such that A⊂Sn
i=1α(Ci).
Properties of α-compact spaces has been investigated in [1, 3]. The fol- lowing theorems were given in [3].
Theorem 1 ([3]). Let (X,Γ) be a topological space,αan operator associated with Γ, A⊂X andK ⊂A. If Ais α-compact andK isα-closed, thenK is α-compact.
Theorem 2 ([3]). Let (X,Γ) be a topological space andαa regular operator on Γ. IfX isα-T2 (see [1,3]) andK⊂X isα-compact, then K isα-closed.
Lemma 1. The collection of α-compact subsets of X is closed under finite unions. If αis a regular operator and X is an α-T2 space then it is closed under arbitrary intersections.
Proof. Trivial.
Definition 4. Let (X,Γ) be a topological space andαan operator associated with Γ. The operatorαissubadditive if for every collection of open sets{Uβ}, α(S
Uβ)⊆S α(Uβ).
Theorem 3. Let (X,Γ) be a topological space and α a regular subadditive operator associated with Γ. If Y ⊂X is α-compact, x ∈ X\Y and (X,Γ) is α-T2, then there exist open sets U and V with x ∈ U, Y ⊂ α(V) and α(U)∩α(V) =∅.
Proof. For each y ∈ Y, let Vy and Vxy be open sets such that α(Vy) ∩ α(Vxy) = ∅, with y ∈ Vy and x ∈ Vxy. The collection {Vy : y ∈ Y} is an open cover of Y. Now, since Y is α-compact, there exists a finite subcollection {Vy1, . . . , Vyn} such that {α(Vy1), . . . , α(Vyn)} covers Y. Let U =Tn
i=1Vxyi and V =Sn
i=1Vyi. Since U ⊂Vxyi for every i∈ {1,2, . . . , n}, then α(U)∩α(Vyi) =∅for everyi∈ {1,2, . . . , n}. Thenα(U)∩α(V) =∅. AlsoY ⊂α(V).
Now we give the definition ofβ-compactification of a topological space.
Definition 5. Let (X,Γ) and (Y,Ψ) be two topological spaces, α and β operators associated with Γ and Ψ, respectively. We say that (Y,Ψ) is a β-compactificationofX if
1. (Y, Ψ) isβ-compact.
2. (X,Γ) is a subspace of (Y,Ψ).
3. The operatorβ/X is equal toα.
4. The Ψ-closure ofX isY.
Now we give the definition ofα-locally compact space.
Definition 6. Let (X,Γ) be a topological space andαan operator associated with Γ. The space (X,Γ) isα-locally compact at the pointx if there exists an α-compact subset C ofX and an α-open neighborhoodU ofxsuch that α(U)⊂C. The space (X,Γ) is said to beα-locally compact if it is α-locally compact at each of its points.
Clearly everyα-compact space isα-locally compact. Observe thatRwith the usual topology and αdefined asα(U) =U (the closure ofU) isα-locally compact but not compact.
Now we give the main theorem.
Theorem 4. Let (X,Γ) be a space, where α is regular, monotone (see [4]), subadditive, stable with respect to all α-closed subsets of (X,Γ) and satisfies the additional condition that α(∅) = ∅. If (X,Γ) is α-locally compact, not α-compact andα-T2, then there exist a space (Y,Ψ) and an operator β on Ψ such that:
1. (Y,Ψ) is aβ-compactification of (X,Γ).
2. |Y \X|= 1.
3. (Y,Ψ) isβ-T2.
Proof. DefineY =X∪ {∞}, where∞is an object not inX. OnY, we define a topology as follows:
1. IfU ∈Γ, thenU ∈Ψ.
2. IfCis anα-compact subset ofX, thenX\C∪ {∞} ∈Ψ.
Let us show that Ψ is a topology onY. In fact, the empty set is a set of type (1) and beingα-compact, it makesY to be a set of type (2). IfU1and U2belongs to Ψ, checking that U1∩U2 belongs to Ψ involves three cases:
(a) IfU1andU2belong to Γ, then U1∩U2 belongs to Γ and so to Ψ.
(b) IfU1=Y\C1 andU2=Y \C2, whereC1andC2areα-compact subsets ofX, we get thatU1∩U2=Y\(C1∪C2), which is of type (2), sinceC1∪C2 is anα-compact subset ofX.
(c) If U1 ∈Γ andU2 =Y \C2, whereC2 is anα-compact subset ofX, then U1∩U2=U1∩(Y \C2) =U1∩(X\C2) which is a set of type (1).
Similarly we check that the union of any collection{Uβ}of elements of Ψ belongs to Ψ. Again we consider three cases:
(a) If eachUβ∈Γ then∪Uβ∈Γ and so it belongs to Ψ.
(b) If each Uβ =Y \Cβ, where eachCβ isα-compact, we have that ∪Uβ = Y \(∩Cβ) which is of type (2), since∩Cβ is anα-compact subset ofX.
(c) If someUβ are of type (1) and some are of type (2), the problem reduces to the case in which one is of type (1) and the other is of type (2). So we need to show that every set of the formU ∪(Y \C), whereU ∈Γ andC is α-compact, is an element of Ψ. In fact U∪(Y \C) =Y \(C\U).
To prove thatC\U isα-compact we go as follows: let {Wi}be a Γ open covering of C\U. Then the collection{Wi} ∪ {U} is an open covering ofC and, sinceC isα-compact, there exist finitely many indicesi1, i2, . . . , ik such that C⊂Sk
j=1α(Wij)∪α(U). Now,
C\U ⊂[k
j=1
α(Wij)∩(C\U)
∪(α(U)∩(C\U)).
Sinceαis stable with respect to allα-closed subsets ofX and sinceC\U isα-closed we have thatα(U)∩(C\U) =αC\U(U∩(C\U)) =αC\U(∅) =∅, by hypothesis. Therefore C\U ⊂Sk
j=1α(Wij). This implies thatC\U is α-compact. So Ψ is a topology onY.
Define an operatorβ : Ψ−→P(Y) as follows:
β(V) =
α(V) if V∈Γ
α(X\C)∪ {∞} ifV =X\C∪ {∞},whereCisα-compact.
Let us show that (Y,Ψ) is β-compact.
Let {Ui : i ∈ Λ} be a Ψ-open cover of Y. This collection must have at least one element of type (2), sayUi0 =X\C∪ {∞}, whereCisα-compact.
Let Vi =Ui∩X fori6=i0. Then {Vi:i∈Λ} is a Γ-open cover of C. Since C isα-compact, there exists a finite subcollection{i1, . . . , in} of Λ such that C ⊆ Sn
j=1α(Vij). Now Y ⊂ (Sn
j=1β(Vij)∪β(Y /C)). Therefore (Y,Ψ) is β-compact.
Proving thatX is a subspace ofY is trivial and, sinceXis notα-compact, thenX is dense inY. Also, sinceαis subadditive and stable with respect to allα-closed subsets ofX, it is easy to prove that (Y,Ψ) isβ-T2.
References
[1] Kasahara, S., Operation-Compact spaces, Mathematica Japonica, 24 (1979), 97–105.
[2] Ogata, H., Operation on Topological Spaces and Associated Topology, Mathematica Japonica36(1) (1991), 175–184.
[3] Rosas, E., Vielma, J. Operator-Compact and Operator-Connected Spaces, Scientiae Mathematicae1(2) (1998), 203–208.
[4] Carpintero, C., Rosas, E., Vielma, J. Operadores Asociados a una Topolog´ıa Γ sobre un Conjunto X y Nociones Conexas, Divulgaciones Matem´aticas,6(2) (1998), 139–148.
