Ordinal products of topological spaces by

144 (1994)

Ordinal products of topological spaces

by

V. A. C h a t y r k o (Moscow)

Abstract. The notion of the ordinal product of a transfinite sequence of topological spaces which is an extension of the finite product operation is introduced. The dimensions of finite and infinite ordinal products are estimated. In particular, the dimensions of ordinary products of Smirnov’s [S] and Henderson’s [He1] compacta are calculated.

Introduction. The necessary information about the notions and nota- tions we use can be found in [A-Pa], [E1], [E2], [K-M] and in the appendix.

One of the main questions in transfinite dimension theory is

The problem of product dimension (PPD). Let DIM be a transfinite dimension function, for example: ind, Ind, dim_w, dim_c, D, and suppose U is a fixed class of topological spaces. What can be said about the dimension DIM of the product of two spaces X, Y from the class U if this dimension is defined for the factors?

Let us give some possible concretizations of (PPD):

(1) Does DIM X × Y exist?

(2) Is there an (optimal) transfinite function Φ = Φ(α, β) of two trans- finite variables such that

DIM X × Y ≤ Φ(DIM X, DIM Y ) ?

(The function Φ(α, β) is called optimal if for every pair of transfinite numbers α, β there are spaces X = X(α, β) and Y = Y (α, β) in U such that DIM X = α, DIM Y = β, and DIM X × Y = Φ(α, β).)

(3) What is the value of DIM X × Y ?

In this paper we will be interested in questions (2), (3) and their gener- alizations. In the introduction we discuss the case of metric compacta unless otherwise stated.

1991 Mathematics Subject Classification: Primary 54F45.

For the traditional transfinite dimensions ind and Ind (see [A-Pa]) the inequality

(∗) DIM X × Y < ω₁

is well known, and it is equivalent to the existence of DIM X × Y for these dimensions. The analogous statement is true for dim_c, the transfinite exten- sion of the Lebesgue covering dimension dim to compact C-spaces ([B3] and [Ha-Y]). A more delicate result for ind has been obtained by Toulmin [T]:

ind X × Y ≤ (ind X(+) ind Y ) + n, where n is a finite non-negative integer which depends on the inductive dimensions of X and Y , and (+) is the natural sum of Hessenberg [Hes].

For any metrizable compactum Z we have Ind Z ≤ ω · ind Z [Le], which leads to a more precise estimate for Ind than (∗). Note that an improvement of (∗) for Ind for a certain class of topological spaces has also been stated by Polkowski [Po]. For the dimension D the inequality D(X ×Y ) ≤ DX(+)DY has been proved by Henderson [He2].

So PPD(2) for the dimensions indicated above reduces either to ob- taining an optimal estimating function Φ or to the proof of optimality of a given one. Note that for dim_w, another transfinite extension of dim to weakly infinite-dimensional compacta [B1], even the rough estimate (∗) has not been obtained yet. This is the well-known problem of the weak infinite-dimensionality of the product of weakly infinite-dimensional com- pacta. Note that PPD(2) coincides with PPD(3) in the part which deals with optimality—one has to calculate the dimension of the product of the chosen pair of compacta. As far as I know the calculation of the dimension of the product of two infinite-dimensional compacta has not been made yet.

In [S] Smirnov constructed compacta S^α with Ind S^α = α, α < ω₁, and from this he deduced that there are no universal spaces in the class of countable-dimensional metric compacta. Smirnov’s construction turned out to be very useful. Using its modification Henderson [He1] constructed AR-compacta H^α←- S^αwith Ind H^α= α, α < ω₁, and defined for them the notion of an essential mapping. He also proved that DS^α= α [He2]. In [B1], [B2] Borst, having extended the covering dimension dim to ordinals, proved a transfinite analog of Aleksandrov’s theorem on essential mappings for locally compact metric spaces, namely: dim_wX ≥ α iff X × C has an essential mapping onto H^α, where C is the Cantor middle thirds set (dimw is the above mentioned transfinite extension of dim). He also proved the equalities dim_wS^α = dim_wH^α = α for α < ω₁, from which one sees directly that there is no universal space in the class of weakly infinite-dimensional metric compacta (the weak infinite-dimensionality of a metric compactum X is equivalent to the inequality dimwX < ω1[B1]). Note that the non-existence of universal spaces in the class of weakly infinite-dimensional compacta also

follows from an earlier result of Pol [P] connected with Smirnov compacta.

Namely:

If a complete space X topologically contains every Smirnov compactum S^α, then X topologically contains the Hilbert cube. Hence X is not weakly infinite-dimensional.

Naturally PPD(3) arises for useful and easily constructed Smirnov and Henderson compacta. In this paper this question is completely solved. It turns out that

DIM X^α× X^β = α(+)β ,

where DIM is Ind, Id (to be defined below), dimw, or D and X^γ, γ < ω1, are either the Smirnov compacta S^γ or the Henderson compacta H^γ.

The paper consists of three parts. In the first part, starting from Smir- nov’s construction we suggest the definition of an infinite product of topolog- ical spaces—the ordinal ℵ0-product (Definition 1) for which, in contrast to Tikhonov products, there are non-trivial solutions of the natural extension of PPD to an infinite number of non-zero-dimensional factors (Theorem 4).

Let, for example, S = {Xγ, γ < β} be a set of compacta indexed by ordinals < β (such sets will be called β-sequences). Then the compactum

Y_ω,ord

γ<β

Xγ =





















point if β = 0 ;

 Yω,ord γ<β−1

Xγ



× Xβ−1 if β is a non-limit ordinal;

Aleksandrov compactification of the free sum

 (+)

δ<β

 Y_ω,ord

γ<δ

Xγ



× N if β is a limit ordinal, where N are the natural numbers, is the ordinal ℵ₀-product of the β-sequence S. If all the X_γ are homeo- morphic to X, then Q_ω,ord

γ<β Xγ is called the β-ordinal ℵ0-power and is de- noted by S_β^ω(X). In particular, if β < ω₁ and I is the interval [0,1] then S^β ,→ S_β^ω(I) ,→ S^β, where S^β is the Smirnov compactum (,→ denotes closed embedding).

The new product, just as the Tikhonov product, is an extension of the notion of a finite topological product to an infinite number of factors, but in contrast to the latter, it essentially depends on the order of the indexed set of factors. For example, for two different countable ordinals α and β the α- and β-ordinal ℵ0-powers of the interval are not homeomorphic because Ind S^α= α 6= β = Ind S^β. Let us state one of the main results of the paper which explains why ℵ₀-products are called products.

Theorem 1. Let X be an arbitrary topological space and α, β be count- able ordinals. Then

S^ω_α(X) × S^ω_β(X) = S_α(+)β^ω (X) .

From Theorem 1 which reminds the main property of the power, one directly obtains

Corollary 2. Let Φ be a numerical function on topological spaces, monotone on closed subsets, for example a dimension (ind, Ind, dim_w, D or others). Then

Φ(S_α^ω(X) × S_β^ω(X)) = Φ(S_α(+)β^ω (X)) . In particular , for Smirnov compacta one has:

(a) DIM S^α× S^β = α(+)β, where DIM is dim_w, Ind, Id or D;

(b) ind S^α× S^β = ind S^α(+)β;

(c) S^α× S^β can be continuously mapped into [0, 1] so that every point of the interval has a finite-dimensional preimage.

One can easily see that an upper estimate of the dimension of S^α× S^β is obtained from the inclusion S^α× S^β ,→ S^α(+)β, which is not true for Henderson compacta. For these, the necessary estimate is deduced from the second part of the paper, where for the transfinite extension of the finite dimension Id, introduced by Pasynkov [Pa1], we give the optimal solution of PPD(2), namely:

Let X, Y be compacta for which Id is defined. Then Id X × Y ≤ Id X(+) Id Y .

This inequality, obtained as a corollary of the more general Theorem 3, makes it possible to give an optimal upper bound for the dimensions Ind, dimw of products of compacta under natural additional assumptions (Corol- lary 6). Note that this inequality has been independently obtained by Vino- gradov. Also note that the obtained estimate for Id, just as Henderson’s inequality for D, is optimal (use Smirnov compacta).

In the third part questions connected with the dimension of infinite or- dinal products are discussed. In particular, we prove the following trans- finite generalization to ordinal ℵ0-products of the Brower theorem on the n-dimensionality of the cube Iⁿ.

Theorem 5. Let DIM be Ind, Id or dim_w, and let X_γ, γ < β, be one- dimensional metric compacta. Then

DIMYω,ord γ<β

X_γ = β .

1. Ordinal products of topological spaces (a special case). Let Xα, α ∈ A, |A| ≥ ℵ0, be a family of topological spaces. The one-point Aleksandrov extension of the free sum of the spaces X_α, α ∈ A, is the space X = {∗} ∪ (+)α∈AXα, formed from (+)α∈AXα by adding the point ∗ with the following topology:

• Xα is clopen in X for all α ∈ A;

• the sets X\{Xα₁(+) . . . (+)Xα_k}, αi∈ A, i = 1, . . . , k, k ∈ N, generate the base of the topology at ∗.

Obviously, if all X_α, α ∈ A, are compact, then the one-point Aleksandrov extension coincides with the one-point Aleksandrov compactification.

Let us list some elementary properties of the one-point Aleksandrov ex- tension. The notation Z ' Y will mean that the spaces Z and Y are homeo- morphic. Let X = {∗} ∪ (+)_α∈AX_α. Then

• if V is an open subset of X\{∗} such that B = {α ∈ A : (X\V ) ∩ Xα

6= ∅} has cardinality ≥ ℵ₀, then X\V = {∗} ∪ (+)_α∈B(X_α\V ) ,→ X;

• if Y_α,→ X_α for all α ∈ A, then {∗} ∪ (+)_α∈AY_α,→ X;

• if for every α ∈ A the space Xα is Hausdorff (Tikhonov, pseudo- compact, normal, paracompact, compact, S-weakly infinite-dimensional, a C-space), then so is X;

• if |A| = ℵ0 and for every α ∈ A the space Xα is perfectly normal (metrizable, with a countable base), then so is X;

• β({∗} ∪ (+)_α∈AX_α) = {∗} ∪ (+)_α∈AβX_α where β denotes the ˇCech–

Stone compactification.

A family S = {X_γ, γ < β} of topological spaces indexed by ordinals < β will be called a β-sequence of topological spaces.

Definition 1. The ordinal product (resp. ordinal ℵ₀-product) of a β- sequence S = {X_γ, γ < β} of topological spaces is, respectively, the topo- logical space

Yord γ<β

X_γ =















point if β = 0;

Y_ord

γ<δ

Xγ



× Xδ if β = δ + 1;

{∗} ∪ (+)

δ<β

Y_ord

γ<δ

Xγ



if β is a limit ordinal, and

Y_ω,ord

γ<β

Xγ =















point if β = 0;

Yω,ord γ<δ

Xγ



× Xδ if β = δ + 1;

{∗} ∪ (+)nYω,ord γ<δ

Xγ



i: δ < β, i < ω o

if β is limit.

Here (Q_ω,ord

γ<δ X_γ)_i'Q_ω,ord

γ<δ X_γ, i < ω. The notationQ_(ω),ord

γ<δ X_γ will mean eitherQ_ω,ord

γ<δ X_γ orQ_ord

γ<δX_γ.

Let us list some elementary properties of ordinal products. Let X = Q_(ω),ord

γ<β Xγ. Then

• if Y_γ ,→ X_γ for every γ < β, then Q_(ω),ord

γ<β Y_γ ,→ X;

• if δ < β, thenQ_(ω),ord

γ<δ X_γ ,→ X;

• if for every γ < β the space X_γ is a Hausdorff (Tikhonov, compact, compact C-) space, then so is X;

• if β < ω1 and for every γ < β the space Xγ is metrizable (with a countable base), then so is X;

• if X is pseudocompact, then

βX = Y_(ω),ord

γ<β

βXγ.

Recall that the product of two compact C-spaces is a compact C-space [Ha-Y], and if X × Y is pseudocompact, then β(X × Y ) = βX × βY [E1].

2. Ordinal power of a topological space (a special case). Let X be a topological space and S = {Xγ, γ < β} be a β-sequence of topological spaces such that X_γ ' X for every γ < β.

Definition 2. The β-ordinal power (resp. ℵ0-power) of X is the space Sβ(X) =Q_ord

γ<βXγ (resp. S_β^ω(X) =Q_ω,ord

γ<β Xγ).

The notation S_β^(ω)(X) will mean either S_β^ω(X) or S_β(X).

It is clear that, if 1 ≤ β < ω1, then

• Sβ(I) is the Smirnov compactum S^β;

• C ,→ S_β(C) ,→ C, where C is the Cantor set.

Let us state some elementary properties of ordinal powers:

• if X ,→ Y , then S_β^(ω)(X) ,→ S_β^(ω)(Y );

• if β < α, then S_β^(ω)(X) ,→ Sα^(ω)(X).

The following statement will often be used below.

Lemma 1. Let α < ω₁. Then

(a) if {α_i}^∞_i=1 is a sequence of ordinals such that α_i< α_i+1 and sup_iα_i= α, then

S_α^(ω)(X) ,→ {∗} ∪(+)^∞

i=1

S_α^(ω)_i (X) ,→ S_α^(ω)(X) ; (b) S_α(X) ,→ S_α^ω(X) ,→ S_α(X);

(c) if α is a limit ordinal, then S_α^ω(X) ' S_α^ω(X)\{a finite number of terms of the free sum defining S_α^ω(X)}.

The proof is obvious.

We now prove the finite multiplicativity of ℵ0-powers for countable or- dinals.

Theorem 1. Let α, β < ω1. Then

S_α^ω(X) × S_β^ω(X) = S_α(+)β^ω (X)

(here α(+)β is the natural sum of the ordinals α and β; see appendix).

P r o o f. We use induction. Let β = 1, and let α be an ordinal < ω1. By the definition we have

S_α^ω(X) × X = S^ω_α(+)1(X) .

Assume that for β < ν and for all α < ω₁ our statement is true, and let β = ν. Suppose β = ε + 1. Then by the definition and the inductive assumption one can easily check that

S_α^ω(X) × S_β^ω(X) = S_α^ω(X) × S_ε^ω(X) × X

= S_α+1^ω (X) × S_ε^ω(X) = S_(α+1)(+)ε^ω (X) = S_α(+)β^ω (X) . Let now β be a limit ordinal. We now use induction on α. For α = 1 the statement is obvious. Suppose that for every α < µ and the fixed limit β the statement is true, and let α = µ. Assume that α = ε + 1. Then by the definition and the inductive assumption we have

S_α^ω(X) × S_β^ω(X) = S_ε^ω(X) × S_β^ω(X) × X = S_ε(+)β^ω (X) × X = S_α(+)β^ω (X) . Let now α be a limit ordinal. By Definition 1 one has

S_α^ω(X) = {∗₁} ∪ (+){(S_δ^ω(X))_i: δ < α, i < ω} .

Note that for a fixed δ < α the space S_δ^ω(X) appears in the free sum countably many times. Let us number all spaces of the free sum by positive integers:

S_α^ω(X) = {∗₁} ∪(+)^∞

i=1

X_i, and the same for S_β^ω(X):

S^ω_β(X) = {∗₂} ∪(+)^∞

i=1

Y_i.

We also number by positive integers all different ordinals of the form δ(+)η, where δ < α and η < β:

{δ(+)η : δ < α, η < β} = {γ₁, γ₂, . . .} .

By the inductive assumption the product X_k×Y_l, k, l < ω, is homeomorphic to S_ξ^ω, where ξ = δ(+)η for some δ < α and η < β, and X_k = S_δ^ω(X), Yl = S_η^ω(X). Consider an increasing sequence {m(i)}^∞_i=0 of positive integers with m(0) = 1 such that for every i < ω the spaces S_γj(X), j = 1, . . . , i, occur in the free sum

(+){X_k× Y_l: m(i) ≤ k, l < m(i + 1)} .

Hence in the free sum

(+){X_k× Y_l: m(i) ≤ k, l < m(i + 1), i < ω}

there are countably many spaces Sγ_j(X), for j = 1, 2, . . . Clearly, S_α^ω(X) × S_β^ω(X) = {∗} ∪ (+){X_k× Y_l

(+)X_k× (S_β^ω(X)\(+){Y_p: p < m(i + 1)}) (+)(S_α^ω(X)\(+){X_p: p < m(i + 1)}) × Y_l :

m(i) ≤ k, l < m(i + 1), i < ω} . By Lemma 1(c) for every i < ω one has

S_α^ω(X)\(+){Xp: p < m(i + 1)} = S_α^ω(X) , S_β^ω(X)\(+){Y_p: p < m(i + 1)} = S_β^ω(X) . Hence

S_α^ω(X) × S_β^ω(X)

= {∗} ∪ (+){X_k× Y_l(+)X_k× S_β^ω(X)(+)S_α^ω(X) × Y_l :

m(i) ≤ k, l < m(i + 1), i < ω} . Recall that Xk = S_δ^ω(X) and Yl = S_η^ω(X) for some δ < α and η < β, k, l < ω. Hence by the inductive assumption one has

X_k× S_β^ω(X) = S_δ^ω(X) × S_β^ω(X) = S^ω_δ(+)β(X) , S_α^ω(X) × Yl= S_α^ω(X) × S_η^ω(X) = S^ω_α(+)η(X) .

Moreover, by Lemma A1 of the appendix for every γ < α(+)β there exist or- dinals α₁, β₁such that γ = α₁(+)β₁, α₁≤ α, β₁≤ β. So S_α^ω(X) × S_β^ω(X) = {∗} ∪ (+){(S^ω_ν(X))_i : ν < α(+)β, i < ω}, where (S_ν^ω(X))_i ' S_ν^ω(X), i < ω. By Definitions 1, 2 one finally has S_α^ω(X) × S_β^ω(X) = S^ω_α(+)β(X).

The theorem is proved.

Question 1. Can the assumption α, β < ω1be omitted in Theorem 1?

R e m a r k 1. Sω(I) × Sω(I) 6' S_ω(·)2(I), because in Sω(I) × Sω(I) there is one isolated point and in S_ω(·)2(I) there are two isolated points.

Note, however, that for the ordinal powers there is a relation very close to equality:

Corollary 1. Let α, β < ω1. Then

(a) S_α(+)β(X) ,→ Sα(X) × Sβ(X) ,→ S_α(+)β(X);

(b) Sα^(ω)(X × Y ) ,→ S^(ω)α (X) × Sα^(ω)(Y ) ,→ Sp(α)(·)2+n(α)^(ω) (X × Y ).

P r o o f. (a) follows directly from Theorem 1 by using Lemma 1(b). Let us prove (b). We shall examine the case of ordinal products and use induction.

Let α < ω. It is clear that Sα(X × Y ) = Sα(X) × Sα(Y ). Moreover, p(α)(·)2 + n(α) = α.

Let us prove the embedding Sα(X × Y ) ,→ Sα(X) × Sα(Y ) for α ≥ ω.

Assume that for α < ν ≥ ω the statement is true, and let α = ν. Suppose that α = ε + 1. By the definition and the inductive assumption one easily has

S_α(X) × S_α(Y ) = S_ε(X) × S_ε(Y ) × X × Y

←- S_ε(X × Y ) × (X × Y ) = S_α(X × Y ) .

Let now α be a limit ordinal. Then there exists an increasing sequence {α_i}^∞_i=1of ordinals such that sup_iα_i= α. Consider the chain of embeddings:

S_α(X) × S_α(Y )

←-



{∗1} ∪(+)^∞

i=1

Sα_i(X)



×



{∗2} ∪(+)^∞

i=1

Sα_i(Y )



(by Lemma 1)

←- {∗} ∪(+)^∞

i=1

S_αi(X) × S_αi(Y )

←- {∗} ∪(+)^∞

i=1

S_αi(X × Y ) (by the inductive assumption)

←- S_α(X × Y ) (by Lemma 1) . Let us prove the inverse embedding

S_α(X) × S_α(Y ) ,→ Sp(α)(·)2+n(α)(X × Y ) .

Assume that for α < ν ≥ ω the statement is true, and let α = ν. Suppose α = ε + 1. Clearly,

S_α(X) × S_α(Y ) ,→ Sp(ε)(·)2+n(ε)(X × Y ) × (X × Y ) = Sp(α)(·)2+n(α)(X × Y ) . Suppose now that α is a limit ordinal. Then p(α)(·)2+n(α) = α(·)2. Clearly,

S_α(X) × S_α(Y ) ,→ S_α(X × Y ) × S_α(X × Y ) ,→ S_α(+)α(X × Y ) (by (a))

= S_α(·)2(X × Y ) . The corollary is proved.

Now we get the following statement.

Corollary 2. Let Φ be a numerical function on topological spaces, monotone on closed sets, for example a dimension (ind, Ind, dim_w, D or others). Then

Φ(S_α^(ω)(X) × S_β^(ω)(X)) = Φ(S_α(+)β^(ω) (X)) .

In particular , for Smirnov compacta S^γ, γ < ω1, one has

(a) DIM S^α× S^β = α(+)β, where DIM is dim_w, Ind, Id or D;

(b) ind S^α× S^β = ind S^α(+)β;

(c) S^α× S^β can be continuously mapped into I so that every point of I has a finite-dimensional preimage.

P r o o f. Recall (see the introduction) that dimwS^α = Ind S^α = DS^α = Id S^α= α for α < ω₁ (for Id see Corollary 5 in §3).

R e m a r k 2. Since the product H^α×H^β cannot be continuously mapped into the interval with every point preimage finite-dimensional, and H^γ can be mapped in such a way, the inclusion H^α× H^β ,→ H^γ is not true for any infinite α, β, γ < ω1. Therefore for the upper estimate of the dimension of H^α×H^βwe need Corollary 5 of §3, namely: Id H^α×H^β ≤ Id H^α(+) Id H^β = α(+)β.

Since S^α(+)β ,→ S^α× S^β ,→ H^α× H^β, the lower estimate is immediate;

recall that dim X ≤ Ind X ≤ Id X for every compactum X. In the case of the dimension D, we need the following statements mentioned in the introduction: DS^α = α, α < ω1, D(H^α× H^β) ≤ DH^α(+)DH^β and the equalities DH^γ = γ, γ < ω₁, which one easily checks by induction using the following properties of D in the class of metrizable spaces [He2]:

• if either Ind X or DX is finite then they are equal;

• D(X × Y ) ≤ DX(+)DY ;

• if X is the union of a locally finite collection of closed subsets each with D-dimension ≤ β, then DX ≤ β;

• if F is a closed subset of the space X then DX ≤ D(X\F ) + DF . (For the definition of Henderson’s compacta H^γ, γ < ω₁, see the proof of Corollary 5.)

One finally has DIM H^α× H^β = α(+)β, where DIM is Ind, Id, dimw or D, and α, β < ω₁.

In order to simplify formulas we write Q_ord

γ<βX_γ and S_β(X) for ordinal ℵ0-products and ordinal ℵ0-powers. The notation (+)γ<βαγ denotes the inductive extension of the natural sum to infinite sequences of ordinals (see appendix).

Theorem 2. Let X be an arbitrary topological space and let ordinals β and α_γ ≥ 1, γ < β, be countable. Then

(∗) S_(+)γ<βαγ(X) ,→ Y_ord

γ<β

S_αγ(X) ,→ S_(+)γ<βαγ(X) .

P r o o f. We use induction. Let β < ω. Then (∗) is obvious by Corollary 1.

Suppose that the statement is true for all β < ν ≥ ω, and let β = ν ≥ ω.

Assume that β = ε + 1. Then by the inductive assumption one has

(∗∗) S_(+)γ<εαγ(X) ,→ Yord

γ<ε

S_αγ(X) ,→ S_(+)γ<εαγ(X) .

Multiply (∗∗) by Sα_ε(X). By the definition one has ((+)γ<εαγ)(+)αε = (+)_γ<βα_γ. Now we only need to use either Corollary 1 or Theorem 1.

Let now β be a limit ordinal. Then there exists a sequence {βi}^∞_i=1 of ordinals such that sup_iβ_i = β. Set σ_i = (+)_γ<βiα_γ. It is clear that (+)_γ<βα_γ= sup_iσ_i. By the inductive assumption for every i < ω one has

S_σi(X) ,→ Yord γ<βi

S_αγ(X) ,→ S_σi(X) . Moreover, obviously

Y_ord

γ<β

Sα_γ(X) ⊂ {∗} ∪(+)^∞

i=1

Y_ord

γ<βi

Sα_γ(X) ,→ Y_ord

γ<β

Sα_γ(X) and

S_(+)γ<βαγ(X) ,→ {∗} ∪(+)^∞

i=1

S_σi(X) ,→ S_(+)γ<βαγ(X) .

Our statement follows from these three chains of embeddings. The theorem is proved.

R e m a r k 3. S_ω^ω(S₂^ω(I)) 6= S_ω^ω(I), though 2 × ω = ω.

Corollary 3. Let α, β < ω₁. Then

Sα×β(X) ,→ Sβ(Sα(X)) ,→ Sα×β(X) .

P r o o f. Let us only note that if we set αγ = α for all γ < β, then by the definition from the appendix we have α × β = (+)_γ<βα_γ, and by the definition of the powerQ_ord

γ<βS_αγ(X) = S_β(S_α(X)).

Corollary 4. Let Φ be a numerical function on topological spaces, monotone on closed sets, for example a dimension (ind, Ind, dimw, D or others), and let α_γ, β < ω₁. Then

Φ Yord γ<β

S_αγ(X)



= Φ(S_(+)γ<βαγ(X)) .

Note that the product Q_ord

γ<βS_αγ(I) can be continuously mapped into an interval with all point preimages finite-dimensional. For the product Q_ord

γ<βH_αγ(I) there is no such mapping.

3. Solution of the product problem for the inductive dimension functions id and id_p. Let X be a normal space.

Definition 3. Let Id X = −1 iff X = ∅. Let Id X ≤ α, where α is an ordinal, if there are collections σ₋₁, σ₀, . . . , σ_δ, δ ≤ α, of closed subsets of X such that

(a) σ₋₁= {∅}, σ_δ3 X, σ_β ⊆ σ_γ for all β, γ with −1 ≤ β ≤ γ ≤ δ;

(b) for all 0 ≤ γ ≤ δ and F ∈ σ_γ and every pair of disjoint closed subsets A and B of X there exist β < γ and ψ ∈ σβ such that ψ ⊂ F and ψ is a partition in F between A ∩ F and B ∩ F ;

(c) for every γ ≤ δ and every pair F₁, F₂ of elements of σ_γ there exists F ∈ σγ such that F ←- F1∪ F2.

Let us put Id X = min{α : Id X ≤ α}.

If Id X ≤ α for no ordinal α, then we define Id X = ∞ (which means that Id X does not exist).

Note that for every F ∈ σ_γ, γ ≤ δ, we have Id F ≤ γ.

If one of the sets from Definition 3, say A, is a single point, we get the definition of the dimension id.

If both A, B are singletons, we get the definition of the dimension id_p. If both A, B are compacta we get the definition of the dimension cId.

Clearly, idpX ≤ id X ≤ Id X and idpX ≤ cId X for every space X.

R e m a r k 4. (a) The definitions of the finite dimensions id and Id and the statements given below for id and Id in the finite-dimensional case are due to Pasynkov [Pa1].

(b) The transfinite extensions of id and Id were independently considered by Vinogradov. He has independently proved the elementary properties of these dimensions given below, as well as Theorem 3 (for id) and Corollar- ies 5, 6.

Let DM be idp, id, Id or cId, and DIM be indp, ind, Ind or cInd (in the definition of ind_p the partitions are taken between points and in the definition of cInd—between compacta [Ha]).

Statement 1. DIM X ≤ α iff there exist collections σ₋₁, . . . , σ_δ, δ ≤ α, of closed subsets of X satisfying conditions (a), (b) of Definition 3.

P r o o f. Let DIM X = δ ≤ α. Put σ_γ = {Y ⊆ X : Y is closed in X and DIM Y ≤ γ}, γ = −1, . . . , δ. It is clear that (a) and (b) are satisfied. Let us prove the converse. We use induction. Let α = −1; then X = ∅ and, hence, DIM X = −1. Suppose that for α < ν the statement is true, and let α = ν.

By (a), X ∈ σ_δ, δ ≤ ν, and by (b) for every pair A, B of disjoint closed subsets of X there exist β < δ ≤ ν and ψ ∈ σβ such that ψ is a partition in X between A and B. Consider the collections σ¹_γ = {C ∩ ψ : C ∈ σ_γ}, γ ≤ β, of subsets of ψ. It is clear that they satisfy conditions (a) and (b).

By the inductive assumption we have DIM ψ ≤ β < ν. Hence, DIM X ≤ ν.

A collection B of closed subsets of X is called monotone if every closed subset of any element of B also belongs to B. A collection B of closed subsets of X is called additive if for any A, B ∈ B we have A ∪ B ∈ B.

Statement 2. DM X ≤ α iff there exist collections σ−1, . . . , σδ, δ ≤ α, of closed subsets of X satisfying conditions (a), (b) of Definition 3 and the condition

(c)¹σ_γ is monotone and additive for every γ ≤ δ.

P r o o f. The “if” part is clear. Let us prove the “only if” part. Let DM X ≤ α. Then there exist collections σ−1, . . . , σδ, δ ≤ α, of closed subsets of X satisfying conditions (a), (b) and (c) of Definition 3. Put

σ_γ¹= {A ⊆ B : A is closed in X and B ∈ σ_γ}, γ ≤ δ . It is clear that these collections satisfy (a), (b) and (c)¹.

Note that in Statement 2 one can always put δ = α. From Statements 1 and 2 we get

Statement 3. If DM X exists, then DIM X exists. Moreover , DIM X ≤ DM X.

Statement 4. (i) For every compactum X we have id_pX = id X = Id X = cId X.

(ii) For every normal space X we have cId X = id_pX.

P r o o f. (i) Let idpX ≤ α. There exist collections σ−1, σ0, . . . , σα of closed subsets of X, satisfying (a), (b) and (c)¹(with A and B being points).

By additivity and monotonicity (Statement 2 ), the same systems σγ, γ ≤ α, satisfy (a), (b) and (c)¹for A and B arbitrary closed disjoint sets. Thus (i) is proved. The proof of (ii) is analogous.

Statement 5. (i) If F is closed in X, then DM F ≤ DM X.

(ii) Suppose DIM X exists and the finite sum theorem for DIM holds in X, for example, X is a metric compactum with Ind X ≤ ω. Then DM X exists and DM X = DIM X.

(iii) Let X_α be a compactum with DM X_α = β_α, α ∈ A, |A| ≥ ω, and sup_α∈Aβα = β. Let X = {∗} ∪ (+)α∈AXα be the one-point Aleksandrov compactification of the free sum of the spaces X_α, α ∈ A. Then DM X exists and DM X = β.

P r o o f. (i) is clear.

(ii) Let DIM X = α. Then for γ = −1, 0, . . . , α we put σγ = {Y ⊆ X : Y is closed in X and DIM Y ≤ γ} .

By the monotonicity of DIM on closed sets and the finite sum theorem one can easily check that the collections σ_γ, γ ≤ α, satisfy conditions (a), (b)

and (c)¹. So DM X ≤ DIM X. From Statement 3 one finally has DM X = DIM X.

(iii) Let σγ^(α), γ ≤ β, be collections of closed subsets of X satisfying (a), (b) and (c)¹, for α ∈ A. Put

σ_γ = {A₁∪ . . . ∪ A_k : A_i∈ σ^(α_γ ⁱ⁾, α_i∈ A, k ∈ N}, γ < β ,

and σ_β = {A ⊆ X : A is compact}. It is clear that for σ₋₁, . . . , σ_β conditions (a), (b) and (c)¹ are satisfied.

Hence, DM X ≤ β. Since DM Xα ≤ DM X for every α ∈ A, one finally obtains DM X = β.

Questions. 2. Does DM X exist if DIM X exists?

3. It is known that for Smirnov compacta with Ind > ω the finite sum theorem for Ind does not hold [Le]. Does there exist a compactum X with Ind X > ω in which the finite sum theorem for Ind holds?

4. By Filippov’s [F] and Pasynkov’s [Pa1] results there exists a com- pactum X with Ind X = 2 and Id X ≥ 3. How large may be the difference between Ind and Id for infinite-dimensional spaces (for DIM and DM)?

Lemma 2 (B. A. Pasynkov). Let X = F₁∪ . . . ∪ F_n be a normal space, where F_i, i = 1, . . . , n, are closed in X. Let A and B be two disjoint closed subsets of X, and Ci be a partition in Fi between A ∩ Fi and B ∩ Fi, i = 1, . . . , n. Then there exists a partition C in X between A and B such that

C ⊆

[ⁿ

i=1

C_i



∪[

i<j

(F_i∩ F_j) .

The notation id_(p)will mean either id_p or id. The main statement of this section is

Theorem 3. Let X1× X2 be a normal space and suppose id_(p)Xi exists for i = 1, 2. Then

id_(p)X₁× X₂≤ id_(p)X₁(+) id_(p)X₂. P r o o f. Let us show the inequality

(#) id_pX₁× X₂≤ id_pX₁(+) id_pX₂.

For id the proof of the corresponding inequality is analogous.

Let id_pX₁ = ξ, id_pX₂ = ζ and let σ_−1,1 ⊆ . . . ⊆ σ_ξ,1 be collections of closed subsets of X₁, and σ_−1,2 ⊆ . . . ⊆ σ_ζ,2 collections of closed subsets of X2satisfying (a), (b) and (c)¹ for A and B being points. Put

Σ = ξ(+)ζ, σ₋₁= {∅} ,

σ_γ¹= {A × B : A ∈ σα,1, B ∈ σβ,2 and α(+)β ≤ γ} , σ_γ²= {C₁∪ . . . ∪ C_k: C_i∈ σ¹_γ, i = 1, . . . , k, k ∈ N} ,

σ_γ = {P ⊆ C : P is closed in X₁× X₂ and C ∈ σ²_γ}, γ ≤ Σ . Obviously, the collections σ_γ, γ ≤ Σ, satisfy (a) and (c)¹. Let us show that (b) holds. Let P ∈ σγ and A, B be a pair of different points in P . We have to show that there exist ν < γ and a set C ∈ σ_ν with C ⊆ P which is a partition in P between A and B.

T h e m a i n c a s e. Let P ⊆ D1× D2, where D1∈ σα,1, D2∈ σβ,2 and α(+)β ≤ γ. From the monotonicity of the collections σ_γ, γ ≤ Σ, one can easily check that it is only necessary to consider the case P = D1× D2.

Let π : X₁× X₂ → X₁ be the projection. Since A = B, without loss of generality one can assume that πA = πB. There exists a partition C1 in D₁ between the points πA, πB such that C₁ ∈ σ_λ,1, λ < α. Then clearly π₁⁻¹C1is a partition in P between A and B, and π₁⁻¹C1∈ σ_λ(+)β. Note that λ(+)β < α(+)β (see appendix).

T h e g e n e r a l c a s e. Without loss of generality, by monotonicity and additivity of σ_γ, γ ≤ Σ, and by Lemma 2 one can assume that P = (D⁽¹⁾₁ × D⁽¹⁾₂ ) ∪ (D₁⁽²⁾× D₂⁽²⁾), where D⁽ⁱ⁾₁ ∈ σ_αi,1, D₂⁽ⁱ⁾ ∈ σ_βi,2 and α₁(+)β₁ ≤ γ, α₂(+)β₂≤ γ. Two cases are possible:

(I) α1(+)β1 < α2(+)β2 ≤ γ. By the main case there exists a partition C1 in D₁⁽²⁾× D⁽²⁾₂ between A and B such that C1∈ σµ for some µ < γ. By Lemma 2 one can choose a partition C in P between A and B such that C ⊆ C₁∪ (D⁽¹⁾₁ × D₂⁽¹⁾). Let ν = max(α₁(+)β₁, µ) < γ. Then C ∈ σ_ν.

(II) α1(+)β1= α2(+)β2. The following subcases are possible.

(II)₁Let α₁= α₂= α. Then (see appendix) β₁= β₂= β. By additivity, D1= D₁⁽¹⁾∪ D₁⁽²⁾∈ σα,1, D2= D₂⁽¹⁾∪ D₂⁽²⁾∈ σβ,1 and P ⊆ D1× D2. So the conditions of the main case are satisfied.

(II)₂ Let α₁< α₂. Then (see appendix) β₁ > β₂. In this case by mono- tonicity one has D⁽¹⁾₁ ∩ D₁⁽²⁾∈ σ_α1,1, D⁽¹⁾₂ ∩ D₂⁽²⁾∈ σ_β2,2 and

L = (D₁⁽¹⁾× D⁽¹⁾₂ ) ∩ (D⁽²⁾₁ × D₂⁽²⁾) = (D₁⁽¹⁾∩ D⁽²⁾₁ ) × (D₂⁽¹⁾∩ D⁽²⁾₂ ) . Moreover, α₁(+)β₂ < α₁(+)β₁ = α₂(+)β₂ ≤ γ and L ∈ σ_α1(+)β2. By the main case there exists a partition C_i in D⁽ⁱ⁾₁ × D₂⁽ⁱ⁾ between A and B such that Ci ∈ σµ_i for some µi < γ, i = 1, 2. By Lemma 1 there ex- ists a partition C in P between A and B such that C ⊆ L ∪ C₁∪ C₂. Let ν = max(α1(+)β2, µ1, µ2) < γ. By the additivity and monotonicity of σγ, γ ≤ Σ, we get C ∈ σ_ν. The theorem is proved.

In the sequel, dim stands for Borst’s [B1] transfinite extension dimw of the Lebesgue covering dimension dim. Recall that dim X ≤ Ind X for every normal space X [B1].

Corollary 5. (i) Let X₁, X₂ be compacta for which Id exists. Then dim X1× X2≤ Ind X1× X2≤ Id X1× X2≤ Id X1(+) Id X2. In particular , if Id X₂= n < ω, for example, X₂= Iⁿ, then

dim X₁× X₂≤ Ind X₁× X₂≤ Id X₁× X₂≤ Id X₁+ n . (ii) Id S^α= α for α < ω₁.

(iii) Id H^α= α for α < ω1. P r o o f. (i) is evident.

(ii) One can easily check by induction using (i) and Statement 5(iii) that Id S^α= α for α < ω₁ (recall that Ind S^α= α for α < ω₁, see [S]).

(iii) Recall Henderson’s description of H^α. For α < ω₁, we define H^α and pα as follows: H⁰= {0}, H¹ = [0, 1] = I and p1 = 0; H^α+1 = H^α× I and p_α+1= p_α× {0}; if α is a limit ordinal, for β < α let A^β_αbe a half-open arc with H^β ∩ A^β_α = {p_β}; then H^α = {∗} ∪ (+)_β<α(H^β ∪ A^β_α) is the one point-compactification of the free sum where pα= ∗ is the compactification point. Since S^α,→ H^αand Id S^α= α for α < ω₁(see (ii)), we need to prove that Id H^α≤ α for α < ω₁. If α is a non-limit ordinal we can use (i).

Let now α be a limit ordinal and suppose that Id H^β ≤ β for all β < α.

By Statement 4, Id can be replaced by idp in the above inequality. Let σγ^(β), γ ≤ α, be collections of closed subsets of H^β satisfying (a), (b) and (c)¹(see Statement 2 for id_p) for β < α. Put

M = {∅} ∪ n

finite subsets of [

{A^β_α: β < α}

o

, σ₋₁= {∅} ,

σγ = {A1∪ . . . ∪ Ak∪ P : Ai∈ σ^(β_γ ⁱ⁾, βi< α, k ∈ N, P ∈ M }, γ < α , and σ_α = {A ⊆ X : A is compact}. Obviously, σ₋₁, . . . , σ_α satisfy (a), (b) and (c)¹.

Corollary 6. Let X1 and X2 be compacta for which Ind exists and in which the finite sum theorem for Ind holds. Then

dim X₁× X₂≤ Ind X₁× X₂≤ Ind X₁(+) Ind X₂.

In particular , if Ind X2= n < ω, for example, X2= Iⁿ, then dim X1×X2≤ Ind X₁× X₂≤ Ind X₁+ n.

Question 5. Are there two metric compacta X₁ and X₂ such that Ind X1 × X2 > Ind X1(+) Ind X2? The same question may be asked for dim_w, dim_c and ind.