Weakly infinite-dimensional compactifications and countable-dimensional compactifications

Weakly infinite-dimensional compactifications and countable-dimensional compactifications

Takashi Kimura, Chieko Komoda

Abstract. In this paper we give a characterization of a separable metrizable space having a metrizable S-weakly infinite-dimensional compactification in terms of a special met- ric. Moreover, we give two characterizations of a separable metrizable space having a metrizable countable-dimensional compactification.

Keywords: S-weakly infinite-dimensional, countable-dimensional, compactification Classification: Primary 54D35, 54F45

1. Introduction

We assume that all spaces are separable and metrizable. By a compactification of a space X, we mean a compact metrizable space containing X as a dense subspace. We refer the reader to [3] for notions and terminology not explicitly given.

Borst [2] gave a characterization of spaces having a S-weakly infinite-dimen- sional compactification in terms of a special base. In [6], we obtained a result concerningC-spaces, which is similar to Borst’s one.

On the other hand, in [1], Borst gave a characterization of spaces having a com- pactification which is aC-space in terms of a special metric. In this paper we give an alternative characterization of spaces having a S-weakly infinite-dimensional compactification in terms of a special metric, which is similar to Borst’s one.

It is known that the class of C-spaces contains the class of countable-dimen- sional spaces.

Next, we give a characterization of spaces having a countable-dimensional com- pactification. The following theorem is well-known.

1.1 Theorem ([3, Theorem 7.2.21]). A space X has a countable-dimensional compactification if and only if X has small transfinite dimensiontrind.

However, by using Borst’s method, we give two characterizations of spaces having a countable-dimensional compactification.

For a collection A of subsets of a spaceX and forY ⊂X we write A|Y for {A∩Y :A∈ A}, SAforS{A:A∈ A},TAfor T{A:A∈ A}and [A]^<ω for {B:Bis a finite subcollection ofA}.

We denote by ( ˜X,d) the completion of a metric space (X, d).˜

For a point x of a metric space (X, d) and for a positive number ε, the set B(x;ε) ={y∈X : d(x, y)< ε}is called theε-ball aboutx. For a setA⊂X and a positive numberε, by theε-ball aboutAwe meanB(A;ε) =S{B(x;ε) :x∈A}.

Let Γ be an index set. A collectionτ ={(A_i, B_i) :i∈Γ} of pairs of disjoint closed subsets of X is called essential if for every {Li : i ∈ Γ}, where Li is a partition in X betweenAi and Bi for everyi ∈Γ, we have T

i∈ΓLi 6=∅; ifτ is not essential then it is calledinessential.

A collectionAof subsets of a spaceX is calledclosure-distributive if for every finite subcollection{A₁, A2,· · · , An}ofA, the equality Cl(A₁∩A2∩ · · · ∩An) = ClA1∩ClA2∩ · · · ∩ClAn holds.

1.2 Lemma([7, Lemma 3.2]). LetV be a closure-distributive finite collection of open subsets of a space X and (F, U) be a pair of subsets of X such that F is closed,U is open andF ⊂U. Then there exists an open subsetV of X such that F ⊂V ⊂ClV ⊂U andV ∪ {V} is closure-distributive.

The following lemma will play an important role in the proof of our main theorem.

1.3 Lemma([10], cf. [5]). Every ˇCech-complete spaceX has a compactification αX such thatαX−X is strongly countable-dimensional.

2. Spaces having a S-weakly infinite-dimensional compactification We consider a characterization of spaces having a S-weakly infinite-dimensional compactification in terms of a special metric.

2.1 Definition. A space X is µ-S-weakly infinite-dimensional if there exists a totally bounded metricdonX satisfying the following condition:

(∗) For every collection {(A_i, B_i) :i < ω} of pairs of disjoint closed subsets of X withd(A_i, B_i)>0 for everyi < ω, there exists a collection{L_i:i < ω}

of subsets ofX such thatL_iis a partition inX betweenA_iandB_ifor every i < ωandT

i<nL_i =∅ for somen < ω.

Obviously, every S-weakly infinite-dimensional space is µ-S-weakly infinite- dimensional and everyµ-S-weakly infinite-dimensional compact space is S-weakly infinite-dimensional.

A space Y is a Cech-complete extensionˇ of a space X if Y contains X as a dense subspace andY is ˇCech-complete.

2.2 Lemma. Everyµ-S-weakly infinite-dimensional space has aµ-S-weakly infi- nite-dimensional ˇCech-complete extension.

Proof: Let X be a µ-S-weakly infinite-dimensional space and d be a totally bounded metric onX satisfying the condition (∗) in Definition 2.1.

Take an arbitrary countable baseU for ˜X which is closed under finite unions.

Note that ˜X is compact. Let us set A=

(

(U, U^′;V, V^′) : U, U^′, V, V^′ ∈ U,Cl_X˜U ⊂U^′,Cl_X˜V ⊂V^′ and Cl_X˜U^′∩Cl_X˜ V^′=∅

) .

We enumerateAas A={(U_i, U_i^′;V_i, V_i^′) :i < ω}. Let us set

D={∆∈[ω]^<ω:{(Cl_X˜U_n^′ ∩X,Cl_X˜V_n^′ ∩X) :n∈∆} is inessential inX}.

Consider an element ∆ ∈ D. We can take a partition L(∆, n) in X between Cl_X˜U_n^′ ∩X and Cl_X˜V_n^′ ∩X for everyn∈∆ such thatT

n∈∆L(∆, n) =∅. For everyn∈∆, we take a partition ˜L(∆, n) in ˜X between ClX˜Unand ClX˜Vnsuch that ˜L(∆, n)∩X⊂L(∆, n). For every ∆∈Dthe set

T_∆= \

n∈∆

L(∆, n)˜

is closed in ˜X and disjoint fromX. Thus the space Y = ˜X−[

{T_∆: ∆∈D}

is a ˇCech-complete extension of X. Let d_Y be the restriction of ˜d to Y. It suffices to show thatd_Y satisfies the condition (∗) in Definition 2.1. Consider a collection{(Ai, Bi) :i < ω} of pairs of closed subsets of Y withd_Y(Ai, Bi)>0 for everyi < ω. Since ˜d(Cl_X˜A_i,Cl_X˜B_i) = d_Y(A_i, B_i)> 0, we have Cl_X˜A_i∩ Cl_X˜B_i =∅. TakeUⁱ, U^′i, Vⁱ, V^′i ∈ U such that Cl_X˜A_i ⊂Uⁱ ⊂Cl_X˜Uⁱ ⊂U^′i, Cl_X˜B_i⊂Vⁱ ⊂Cl_X˜Vⁱ ⊂V^′i, and Cl_X˜U^′i∩Cl_X˜V^′i=∅ for everyi < ω. Since d(Cl˜ _X˜U^′i,Cl_X˜V^′i)>0, we haved(Cl_X˜ U^′i∩X,Cl_X˜V^′i∩X)>0. Thus there exists a partitionLⁱ inX between Cl_X˜U^′i∩X and Cl_X˜V^′i∩X for everyi < ω such that T

i<mLⁱ = ∅ for somem < ω. Since {(Cl_X˜U^′i∩X,Cl_X˜V^′i∩X) : i < m} is inessential in X, we have {(Uⁱ, U^′i;Vⁱ, V^′i) : i < m} ∈ A; thus (Uⁱ, U^′i, Vⁱ;V^′i) = (U_n(i), U_n(i)^′ ;V_n(i), V_n(i)^′ ) for somen(i)< ω. Letting

∆ ={n(i) :i < m}, we have ∆∈D. For everyi < m, letting

L_i= ˜L(∆, n(i))∩Y,

L_i is a partition inY betweenA_i andB_i. For everyi≥mwe take a partitionL_i inY betweenA_i andB_i. We have

\

i<m

Li = \

i<m

( ˜L(∆, n(i))∩Y ) = \

n(i)∈∆

L(∆, n(i))˜ ∩ Y

=T_∆∩Y ⊂ T_∆∩( ˜X−T_∆) =∅,

thusd_Y satisfies the condition (∗) in Definition 2.1. HenceY isµ-S-weakly infinite-

dimensional.

2.3 Lemma. Every ˇCech-completeµ-S-weakly infinite-dimensional spaceX has a S-weakly infinite-dimensional compactification.

Proof: SinceX is ˇCech-complete, by Lemma 1.3, there exists a compactification αXsuch that the remainderαX−X is strongly countable-dimensional. We shall prove thatαX is S-weakly infinite-dimensional.

Let{(A_i, B_i) :i < ω}be a collection of pairs of disjoint closed subsets ofαX.

For every i < ω, we take two open subsets U_2i+1 and V_2i+1 of αX such that A_2i+1⊂U_2i+1,B_2i+1⊂V_2i+1 and Cl_αXU_2i+1∩Cl_αXV_2i+1=∅. SinceαX−X is A-weakly infinite-dimensional, there exists a partitionL_2i+1inαX−Xbetween Cl_αXU_2i+1∩(αX−X) and Cl_αXV_2i+1∩(αX−X) for every i < ω such that T

i<ωL_2i+1=∅. For everyi < ωwe take a partitionL^′_2i+1inαXbetweenA_2i+1 and B2i+1 such thatL^′_2i+1∩(αX−X)⊂L2i+1. Let us set K =T

i<ωL^′_2i+1. SinceKis S-weakly infinite-dimensional, there exists a partitionL2iinKbetween A_2i∩K and B_2i∩K for everyi < ω such that T

i<nL_2i =∅ for some n < ω.

For every i < ω we take a partitionL^′_2i in αX between A_2i and B_2i such that L^′_2i ∩K ⊂ L_2i. Obviously, we have T

i<ωL^′_i =∅. This implies thatαX is A- weakly infinite-dimensional and hence since αX is compact it is also S-weakly

infinite-dimensional.

2.4 Lemma. Every spaceX having a S-weakly infinite-dimensional compactifi- cationαX isµ-S-weakly infinite-dimensional.

Proof: Take an arbitrary metricdonαX. Let dX be the restrictiondto X. It is easy to show thatdX satisfies the condition (∗) in Definition 2.1. Hence X is

µ-S-weakly infinite-dimensional.

We now come to our main theorem.

2.5 Theorem. A spaceX has a S-weakly infinite-dimensional compactification if and only if X isµ-S-weakly infinite-dimensional.

Proof: The theorem follows from Lemmas 2.2, 2.3 and 2.4.

2.6 Problem. Does Lemma 2.2 remain true if we replace ‘a totally bounded metric onX’ in Definition 2.1 by ‘a metric onX’ ?

3. Spaces having a countable-dimensional compactification

In this section we consider characterizations of spaces having a countable- dimensional compactification.

A collectionAof subsets of a spaceX isstrongly point-finiteif for every infinite subcollectionA^′ ofAthere existsA^′′∈[A^′]^<ω such that∩A^′′=∅.

We need the following theorem to prove our main theorems.

3.1 Theorem([5, Theorem 1]). A spaceX has small transfinite dimensiontrind if and only if X has a base Bsuch that{BdB:B ∈ B}is strongly point-finite.

On the other hand, the following theorem is well-known.

3.2 Theorem ([8], [9]). A space X is countable-dimensional if and only if for every collection{(A_i, B_i) :i < ω} of pairs of disjoint closed subsets of X, there exists a collection{L_i :i < ω} of subsets of X such thatL_i is a partition in X betweenA_i andB_i for everyi < ωand {L_i:i < ω}is point-finite.

A collectionAof subsets of a space X is separating in X if for every x∈ X and every closed setF⊂X withx /∈F there existA1, A2 ∈ Asuch thatx∈A1, F ⊂A2 andA1∩A2=∅. Obviously, every separating collection of open subsets of a spaceX is a base forX.

3.3 Definition. A spaceXissmall countable-dimensional if there exists a count- able separating collectionBof open subsets ofXsatisfying the following condition:

(∗) For every collection {(B_i1, B_i2) : i < ω} of pairs of elements of B with ClB_i1∩ClB_i2=∅ for everyi < ω, there exists a collection{L_i :i < ω}of subsets ofX such thatLi is a partition inX between ClB_i1and ClB_i2 for everyi < ω and{Li:i < ω} is strongly point-finite.

We now come to our main theorem.

3.4 Theorem. A spaceX has a countable-dimensional compactification if and only if X is small countable-dimensional.

Proof: LetX be small countable-dimensional andU be a countable separating collection of open subsets ofX satisfying the condition (∗) in Definition 3.3. Let us set

A={(U, U^′) :U, U^′ ∈ U with Cl_X˜U∩Cl_X˜U^′ =∅}.

We enumerate A as A={(U_i, U_i^′) : i < ω}. Take a partitionLi between ClUi

and ClU_i^′ for every i < ω such that {L_i : i < ω} is strongly point-finite. We can take disjoint open subsets Bi and B_i^′ such that ClUi ⊂Bi,ClU_i^′ ⊂B_i^′ and X−Li =Bi∪B_i^′. It is easy to show that the set B ={Bi : i < ω} is a base for X. Since {Li : i < ω} is strongly point-finite, so is {BdBi : i < ω}. From Theorem 1.1,X has small transfinite dimension trind. By Theorem 3.1,X has a countable-dimensional compactification.

Now let αX be a countable-dimensional compactification of X and U be a countable baseU forαX. Let us set

A={(U, U^′) :U, U^′∈ U with Cl_αXU ⊂U^′}.

We enumerateAasA={(U_i, U_i^′) :i < ω}. For everyi < ω, inductively, we shall construct two open subsetsV_i andV_i^′ ofαX satisfying the following conditions:

Cl_αXU_i⊂V_i ⊂Cl_αXV_i⊂αX−Cl_αXV_i^′⊂αX−V_i^′⊂U_i^′ and V={V_i:i < ω} ∪ {V_i^′ :i < ω} is closure-distributive.

Assume that for every k < i (> 0) we have constructed two open subsets V_k and V_k^′ of αX satisfying the following conditions: Cl_αXU_k ⊂V_k ⊂ Cl_αXV_k ⊂ αX −Cl_αXV_k^′ ⊂ αX −V_k^′ ⊂ U_k^′ and V_i = {V_k : k < i} ∪ {V_k^′ : k < i} is closure-distributive. By Lemma 1.2, there exists open subsetsV_i^′ and V_i^′ of αX such that ClαXUi ⊂ Vi ⊂ ClαXVi ⊂ αX −ClαXV_i^′ ⊂ αX −V_i^′ ⊂ U_i^′ and V_i+1 = Vi∪ {Vi, V_i^′} is closure-distributive. It is easily seen that V is closure- distributive. Let us set

B=V|X.

We shall prove that B is a countable separating collection of open subsets of X satisfying the condition (∗) in Definition 3.3. First we shall show that B is separating. Consider a point x ∈ X and a closed subset F of X with x /∈ F. The collectionU being a base for αX, we can takeU, U^′ ∈ U such that x∈U ⊂ Cl_αXU ⊂U^′⊂Cl_αXU^′ ⊂αX−Cl_αXF. Since (U, U^′)∈ A, (U, U^′) = (Un, U_n^′) for some n < ω. We have x ∈ Un ⊂ Cl_αXUn ⊂ Vn; thus x ∈ Vn∩X ∈ B.

Since αX −V_n^′ ⊂ U_n^′ ⊂ Cl_αXU_n^′ ⊂ αX −Cl_αXF, we have Cl_αXF ⊂ V_n^′; thus F = ClαXF ∩X ⊂ V_n^′ ∩X ∈ B. Obviously, (V_n∩X)∩(V_n^′ ∩X) = ∅.

Thus B is separating. Next, we shall show that B satisfies the condition (∗) in Definition 3.3. Consider a collection{(B_i1, B_i2) : i < ω} of pairs of elements of B with Cl_XB_i1∩Cl_XB_i2 = ∅ for every i < ω. For every i < ω we can take B_i1^′ , B^′_i2∈ V such that

B_i1=B_i1^′ ∩X and B_i2=B_i2^′ ∩X.

Then we have

ClαXB_i1^′ ∩ClαXB_i2^′ = ClαX(B_i1^′ ∩B_i2^′ ) = ClαX(B_i1^′ ∩B_i2^′ ∩X)

= Cl_αX(B_i1∩B_i2)⊂ Cl_αX(Cl_XB_i1∩Cl_XB_i2) =∅.

Since αX is countable-dimensional, we can take a collection {L^′_i : i < ω} of subsets ofαX such thatL^′_i is a partition inαX between ClαXB_i1^′ and ClαXB^′_i2, and {L^′_i : i < ω} is strongly point-finite. ThenLi =L^′_i∩X is a partition in X between Cl_XB_i1 and Cl_XB_i2. Obviously, {Li : i < ω} is strongly point-finite;

thusB satisfies the condition (∗) in Definition 3.3. HenceX is small countable-

dimensional.

3.5 Problem. Does Theorem 3.4 remain true if we replace ‘a countable separat- ing collection of open subsets of a spaceX’ in Definition 3.3 by ‘a countable base forX’ ?

Next we consider a characterization of spaces having a countable-dimensional compactification in terms of a special metric.

3.6 Definition. A space X is µ-countable-dimensional if there exists a totally bounded metricdonX satisfying the following condition:

(∗) For every collection{(A_i, B_i) :i < ω} of pairs of disjoint closed subsets of X withd(A_i, B_i)>0 for everyi < ω, there exists a collection{L_i:i < ω}

of subsets ofX such thatL_iis a partition inX betweenA_iandB_ifor every i < ωand{L_i:i < ω}is strongly point-finite.

3.7 Theorem. A spaceX has a countable-dimensional compactification if and only if X isµ-countable-dimensional.

Proof: LetX beµ-countable-dimensional anddbe a totally bounded metric on X satisfying the condition (∗) in Definition 3.6. The completion ( ˜X,d) of (X, d)˜ is compact. Take an arbitrary countable baseU for ˜X. Let us set

A={(U, U^′) :U, U^′∈ U with Cl_X˜U ⊂U^′}.

We enumerate A as A = {(U_i, U_i^′) : i < ω}. For every i < ω, since Cl_X˜Ui∩ ( ˜X−U_i^′) =∅,ε_i= ˜d(Cl_X˜U_i,X˜−U_i^′)>0. Thus we can take a partitionL_iin X between Cl_X˜B(Cl_X˜U_i;ε_i/3)∩X and Cl_X˜B( ˜X−U_i^′;ε_i/3)∩X for everyi < ω such that{L_i:i < ω}is strongly point-finite. For everyi < ωwe take a partition L˜_i in ˜X between Cl_X˜U_i and ˜X−U_i^′ such that ˜L_i∩X ⊂L_i. Let us set

D={∆∈[ω]^<ω: \

n∈∆

Ln=∅}.

For every ∆∈Dthe set

T_∆= \

n∈∆

L˜n

is closed in ˜X and disjoint fromX. The set Y = ˜X−[

{T_∆: ∆∈D}

is a ˇCech-complete extension of X. Now, for every i < ω, we can take disjoint open subsetsV_i andV_i^′ ofY such that Cl_X˜U_i∩Y ⊂V_i,( ˜X−U_i^′)∩Y ⊂V_i^′ and Y −( ˜Li∩Y) =Vi∪V_i^′. Let us set

V ={V_i:i < ω}.

It is easily seen thatV is a base for Y. We shall show that{Bd_Y V_i :i < ω} is strongly point-finite. Obviously, Bd_Y V_i ⊂L˜_i∩Y for everyi < ω. It suffices to show that {L˜i∩Y : i < ω} is strongly point-finite. Consider an infinite subset Λ of ω. The collection {L_i : i < ω} being strongly point-finite, we can take

∆ ∈ [Λ]^<ω such that T

n∈∆Ln = ∅; thus ∆ ∈ D. We have T

n∈∆( ˜Ln∩Y) = T_∆∩Y⊂T_∆∩( ˜X−T_∆) =∅. Thus{L˜_i∩Y :i < ω}is strongly point-finite. By Theorem 3.1,Y has a countable-dimensional compactificationαY. ThenαY is a compactification ofX.

Now letαX be a countable-dimensional compactification ofX. Take an arbi- trary metric d onαX. Let d_X be the restriction of d to X. It is easy to show that d_X satisfies the condition (∗) in Definition 3.6. Hence X is µ-countable- dimensional.

3.8 Problem. Does Theorem 3.7 remain true if we replace ‘a totally bounded metric onX’ in Definition 3.6 by ‘a metric onX’ ?

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Department of Mathematics, Faculty of Education, Saitama University, Sakura, Saitama, 338-0825, Japan

E-mail: [email protected]

Department of Health Science, School of Health & Sports Science, Juntendo Uni- versity, Inba, Chiba, 270-1695, Japan

E-mail: chieko [email protected]

(Received July 6, 2007)