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.
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(SU
β)⊆Sα(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
Proof. For each y ∈ Y, let Vy and Vxy be open sets such that α(Vy) ∩
α(Vy
x) = ∅, 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=1V
yi
x and V =
Sn
i=1Vyi. Since U ⊂V yi
x 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.
Proof. DefineY =X∪ {∞}, where∞is an object not inX. OnY, we define
U2belongs to Ψ, checking that U1∩U2 belongs to Ψ involves three cases:
(a) IfU1andU2belong to Γ, thenU1∩U2 belongs to Γ and so to Ψ.
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.
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.
