such that ind Z

145 (1994)

Cantor manifolds in the theory of transfinite dimension

by

Wojciech O l s z e w s k i (Warszawa)

Abstract. For every countable non-limit ordinal α we construct an α-dimensional Cantor ind-manifold, i.e., a compact metrizable space Z

such that ind Z

= α, and no closed subset L of Z

with ind L less than the predecessor of α is a partition in Z

. An α-dimensional Cantor Ind-manifold can be constructed similarly.

1. Introduction. Unless otherwise stated all spaces considered are metr- izable and separable. Our terminology and notation follow [2] and [4] with the exception of the boundary, closure, and interior of a subset A of a topo- logical space X, which are denoted by bd A, cl A, and int A, respectively.

We denote by I the unit closed interval, and by I

the standard n- dimensional cube, i.e., the Cartesian product of n copies of I. By an arc we mean every space homeomorphic to I, and by an n-dimensional cube every space homeomorphic to I

; we identify every such space with I

by a

“canonical” homeomorphism. This allows us to apply geometrical notions, e.g., broken line or parallelism, to spaces homeomorphic to I

. In the sequel, we assume that “canonical” homeomorphisms are always defined in a natural way, and they are not described; we will simply apply geometrical notions to the cubes.

We denote by a

b any arc with endpoints a and b, i.e., an arc J such that h(0) = a and h(1) = b, where h : I → J is the “canonical” homeomorphism.

A partition in a space X between a pair of disjoint sets A and B is a closed set L such that X − L = U ∪ V , where U and V are disjoint open sets with A ⊆ U and B ⊆ V .

The small transfinite dimension ind and the large transfinite dimen- sion Ind are the extension by transfinite induction of the classical Menger–

Urysohn dimension and the classical Brouwer– ˇ Cech dimension:

• ind X = −1 as well as Ind X = −1 means X = ∅,

1991 Mathematics Subject Classification: Primary 54F45.

[39]

• ind X ≤ α (resp. Ind X ≤ α), where α is an ordinal, if and only if for every x ∈ X and each closed set B ⊆ X such that x 6∈ B (resp. for every pair A, B of disjoint closed subsets of X), there exists a partition L between x and B (resp. a partition L between A and B) such that ind L < α (resp.

Ind L < α),

• ind X is the smallest ordinal α with ind X ≤ α if such an ordinal exists, and ind X = ∞ otherwise,

• Ind X is the smallest ordinal α with Ind X ≤ α if such an ordinal exists, and ind X = ∞ otherwise.

The transfinite dimension ind was first discussed by W. Hurewicz [6]

and the transfinite dimension Ind by Yu. M. Smirnov [12]; a comprehensive survey of the topic is given by R. Engelking [3].

The small transfinite dimension of a space X at a point x does not exceed an ordinal α (written briefly ind

X ≤ α) if for each closed set B ⊆ X not containing x there exists a partition L between x and B such that ind L < α.

The small transfinite dimension of a space X at a point x, denoted by ind

X, is the smallest ordinal α such that ind

X ≤ α if such an ordinal exists, and

∞ otherwise.

If X 6= ∅, then ind X and Ind X are either countable ordinals or equal to infinity (see [6] and [12], or [3], Theorems 3.5 and 3.8). Obviously, ind X ≤ Ind X, but the reverse inequality does not hold; there exists a compact space X such that ind X < Ind X (see [9]).

For a long time all known compact spaces X with ind X = α ≥ ω

had the property that ind

X = α only for some distinguished points x;

B. A. Pasynkov asked whether there exist compact spaces X with ind

X = α for every x ∈ X, or even with a stronger property that X is an α- dimensional Cantor manifold (see [1]).

1.1. Definition. Let α = β+1 be a non-limit ordinal. A compact metriz- able space X such that ind X = α (resp. Ind X = α) is an α-dimensional Cantor ind-manifold (resp. Ind-manifold) if no closed set L ⊆ X with ind L < β (resp. Ind L < β) is a partition in X between any pair of points.

For α = n < ω

, the above notions and the classical notion of a Cantor

manifold (see [2], Definition 1.9.5) are equivalent; the n-dimensional cube

I

is an example of an α-dimensional Cantor manifold. Of course, every

α-dimensional Cantor ind-manifold has the property that ind

X = α for

each x ∈ X. In [1], V. A. Chatyrko gave examples of a non-metrizable α-

dimensional ind-manifold and a non-metrizable α-dimensional Ind-manifold

for every non-limit ordinal α such that ω

< α < ω

; he also constructed,

for every infinite α < ω

, a compact metrizable space X

with ind X

< ∞,

and a compact metrizable space Y

with Ind Y

< ∞ such that ind L ≥ α

for every partition L in X

between any pair of points, and Ind L ≥ α for every partition L in Y

between any pair of points.

In the present paper, we construct a metrizable α-dimensional Cantor ind-manifold Z

for every non-limit infinite ordinal α < ω

. Slightly modify- ing this construction, one can also define metrizable Cantor Ind-manifolds.

However, we will restrict the discussion to the small transfinite dimension, and in the sequel “Cantor manifold” will mean “Cantor ind-manifold”.

Acknowledgements. The paper contains some of the results of my Ph.D. thesis written under the supervision of Professor R. Engelking whom I wish to express my thanks for his active interest in the preparation of the thesis.

2. Henderson’s spaces. Yu. M. Smirnov defined a sequence {S

: α < ω

} of compact metrizable spaces with the property that Ind S

= α for every α < ω

(see [12]). Slightly modifying his construction, D. W. Hen- derson defined a sequence {H

: α < ω

} of absolute retracts with the same property (see [5]).

Let us recall the definitions of S

and H

.

We apply induction on α; we simultaneously distinguish a point p

∈ H

and, for α > 0, a covering C

of H

by cubes of positive dimensions.

Let S

= H

= {p

} be the one-point space, S

= H

= I, p

= 0, and C

= {I}. Assume that S

, H

, p

, and C

are defined for every β < α.

If α = β +1 for some β, then set S

= S

×I, H

= H

×I, p

= (p

, 0), C

= {C × I : C ∈ C

}.

If α is a limit ordinal, then let S

be the one-point compactification of the topological sum L

{S

: β < α}. In order to define H

take a half-open arc A

with endpoint p

such that H

∩ A

= {p

}, and set K

= H

∪ A

for every β < α. Let H

be the one-point compactification of the topological sum L

{K

: β < α}; let p

stand for the unique point of the remainder.

Set A

= A

∪ {p

} and K

= K

∪ {p

} for β < α. Let C

= {A

: β < α} ∪ [

{C

: 0 < β < α} .

For simplicity of notation, we will identify the spaces H

= I

and H

× I

, where H

= {p

}.

The spaces H

and H

are exhibited in Fig. 2.1.

Observe that S

is embeddable in H

for every α < ω

, and that if β < α, then S

is embeddable in S

and H

is embeddable in H

.

Henderson’s spaces play an important role in our considerations, whereas Smirnov’s spaces will only be used in the proof of Theorem 2.1.

Both Henderson’s and Smirnov’s spaces are also a source of examples of

compact metrizable spaces with given small transfinite dimension.

Fig. 2.1

Indeed, one proves that

Ind X ≤ ω

· ind X for every hereditarily normal compact space X (see [8]),

ind(X × I) ≤ ind X + 1 for every metrizable space X (see [13]).

From the theorems mentioned above and the easy equality (2.1) ind H

= sup{ind H

: α < λ} ,

where λ is any countable limit ordinal, it follows that for every ordinal α < ω

, there exists an ordinal β ≥ α such that ind H

= α.

The least ordinal β with ind H

= α will be denoted by β(α). Note that β(α) > α for some α (see [9]), and ind H

is unknown for some α (see [3], Problems 2.3 and 2.4). From (2.1) it follows immediately that

(2.2) if α is a non-limit ordinal, then so is β(α).

The remaining part of this section is devoted to some notions and results concerning the spaces H

.

Every ordinal α can be uniquely represented as the sum λ + n of a limit ordinal λ or λ = 0 and a natural number n. From the construction of H

it follows that H

= H

× I

. Let B

denote the set {p

} × I

; we call it

the base of H

. Sometimes, we will identify the base B

and the cube I

; in

particular, we will write H

= H

× B

. Thus for every ordinal α < ω

, the

base B

of H

is a finite-dimensional cube, and it has positive dimension whenever α is a non-limit ordinal.

In the sequel, we need the following two theorems. The first theorem for ind is a consequence of a theorem of G. H. Toulmin ([13], or [3], Theorem 5.16), and for Ind it is a consequence of a theorem obtained independently by M. Landau and A. R. Pears ([7] and [11], or [3], Theorem 5.17); the theorem for Ind also follows from a theorem of B. T. Levshenko ([8], or [3], Theorem 5.15). The second theorem was established by G. H. Toulmin ([13]).

2.A. Theorem ([13], resp. [7] and [11]). If a hereditarily normal space X can be represented as the union of closed subspaces A

and A

such that ind A

≤ α ≥ ω

(resp. Ind A

≤ α ≥ ω

) for i = 1, 2, and A

∩ A

is finite-dimensional, then ind X ≤ α (resp. Ind X ≤ α).

2.B. Theorem ([13]). If a hereditarily normal space X can be represented as the union of closed subspaces A

and A

with the property that there is a homeomorphism h : A

→ A

such that f (x) = x for every x ∈ A

∩A

, then

ind X = ind A

= ind A

.

2.1. Theorem. Let α = λ + n, where λ is 0 or a countable limit ordinal and n ≥ 1 is a natural number. For every partition K in H

between any pair of distinct points a, b ∈ B

, we have

ind K ≥ ind H

− 1 and Ind K ≥ λ + (n − 1) . P r o o f. As in the proof of Theorem 2.1 of [10] we can see that

(2.3) for every x ∈ B

and each closed set F ⊆ B

not containing x, there exists a partition L in H

between x and F such that ind L ≤ ind K.

We first prove that

(2.4) ind

H

≤ ind K + 1 for every x ∈ B

.

Let x ∈ B

and let F ⊆ H

be a closed set not containing x. Since H

= B

for α < ω

, we can assume that α ≥ ω

. Let E = F ∩ B

; by (2.3), there exists a partition M in H

between x and E such that ind M ≤ ind K. Let U, V ⊆ H

be disjoint open sets such that x ∈ U , E ⊆ V , and M = H

− (U ∪ V ).

By construction, we have H

= B

∪ [

{(K

− {p

}) × I

: β < λ} ;

from the definition of H

it also follows that each closed subset of H

disjoint from B

meets only a finite number of the sets (K

− {p

}) × I

. Thus

[F ∩ (U ∪ M )] ∩ [(K

− {p

}) × I

] 6= ∅

only for finitely many β, say for β = β

, . . . , β

.

Fig. 2.2

Let L

⊆ A

× I

, for i = 1, . . . , k, be a partition in K

between B

and [F ∩ (U ∪ M )] ∩ [(K

− {p

}) × I

] (see Fig. 2.2, where k = 2); since ind(A

× I

) = n + 1, we have ind L

≤ n + 1.

It is easily seen that M ∪ S

i=1

L

contains a partition L in H

between

x and E (see Fig. 2.2); by Theorem 2.A and monotonicity of ind, we have

ind L ≤ ind K.

Thus the proof of (2.4) is concluded. Applying (2.4) we prove the first inequality of our theorem by induction on α. For every α < ω

, the hypoth- esis of the theorem is equivalent to (2.4). Assume, therefore, that α ≥ ω

; that is, α = λ + n, where λ is a limit ordinal and n is a natural number.

Obviously, ind K ≥ ω

. Assume the inequality holds for each β < α. By (2.4) it suffices to show that ind

H

≤ ind K + 1 for each x ∈ H

− B

.

Observe that K contains a partition in H

× I

= H

between a pair of distinct points of the base B

for all but a finite number of β < λ.

Thus, by the inductive assumption, ind H

≤ ind K + 1 for those β; since ind H

≤ ind H

whenever ν ≤ µ, we have ind H

≤ ind K + 1 for all β < λ. By Theorem 2.A, every x ∈ H

− B

has a neighbourhood U in H

with ind U ≤ ind K + 1, which completes the proof of the inequality ind H

≤ ind K + 1.

Just as the base B

of H

, one can define the base B

of S

(see [10]).

The inequality Ind K ≥ λ+(n−1) follows from Theorem 2.1 of [10], because there exists an embedding of S

in H

mapping B

onto B

.

2.2. Lemma. Let β > 0 be a countable ordinal. For every x ∈ H

and each closed set E ⊆ H

not containing x, there exists a partition Y in H

between x and E such that

(2.5) for every cube C ∈ C

, the set Y ∩ C is the union of a finite number of cubes of dimension less than that of C, each parallel to a proper face of C; furthermore, if β = β(α) for some α, then ind Y < α.

P r o o f. For β < ω

the lemma is obvious. Thus assume that β ≥ ω

. Represent β as the sum λ + n of a limit ordinal λ and a natural number n.

Let H

, k = 0, 1, . . . , n, be the space obtained by sticking the (k + 1)- dimensional cube C

= I

to a k-dimensional face D of the base B

⊆ H

along its k-dimensional face (see Fig. 2.3, where β = ω

+ 1 and k = 1).

Precisely, define H

to be the subspace of H

× I consisting of all (y, z) such that either z = 0 or y ∈ D; let C

= C

∪ {C

}. Observe that from Theorem 2.A it follows that ind H

= ind H

.

We apply induction on β. Since H

⊆ H

and C

⊆ C

, it is sufficient to prove the counterpart of the lemma for each H

, k = 0, 1, . . . , n, and its covering C

.

Assume that λ = ω

or λ > ω

and the modified lemma holds for every ordinal β

= λ

+ n

such that λ

< λ. Fix k ∈ {1, . . . , n}. Then

H

= H

∪ C

= (H

× I

) ∪ C

=  [

{(A

∪ H

) × I

: γ < λ}



∪ C

, C

∩ [

{(A

∪ H

) × I

: γ < λ} = D ,

Fig. 2.3

and

[(A

∪ H

) × I

] ∩ [(A

∪ H

) × I

] = B

for distinct γ, δ < λ (see Fig. 2.3).

Let x ∈ H

and let E ⊆ H

be a closed set not containing x. If x 6∈ B

, then x ∈ C

−B

or x ∈ (A

∪H

)×I

−B

for some γ < λ. Since C

−B

and (A

∪H

)×I

−B

are open subsets of H

, the existence of a partition Y with the suitably modified property (2.5) is obvious whenever x ∈ C

− B

or x ∈ (A

∪ H

) × I

− B

and γ < ω

, and it follows from the inductive assumption if x ∈ (A

∪ H

) × I

− B

and γ ≥ ω

. Obviously, we can assume that Y is contained either in C

− B

or in (A

∪ H

) × I

− B

; thus ind Y < α for β = β(α).

Suppose now that x ∈ B

. Assume that β is a non-limit ordinal; for limit

β the proof is straightforward. Let Q ⊆ B

be an n-dimensional cube with

faces parallel to the faces of B

= I

such that x ∈ int Q, where int Q stands

for the interior of Q in B

, and E ∩ Q = ∅ (see Fig. 2.4, where β = ω

+ 1

Fig. 2.4

and k = 1). It follows that E ∩ [(A

∪ H

) × Q] 6= ∅ only for finitely many

γ < λ, say for γ = γ

, . . . , γ

(in Fig. 2.4, m = 2). For i = 1, . . . , m, take

r

∈ A

such that (p

r

× Q) ∩ E = ∅; recall that p

is an endpoint of

A

, and p

r

is the arc with endpoints p

and r

contained in A

.

Let

Y

= {r

} × Q ∪ p

r

× bd Q

where bd Q is the boundary of Q in B

(see Fig. 2.4). Next, take r ∈ I such that {(y, z) ∈ D × I : y ∈ Q and z ≤ r} ⊆ C

does not meet E, and set

Y

= (Q ∩ D) × {r} ∪ (bd Q ∩ D) × [0, r]

(see Fig. 2.4). Let Y

=  [

{A

∪ H

: γ < λ and γ 6= γ

, . . . , γ

}



× bd Q , and Y = S

i=0

Y

(see Fig. 2.4).

It is easily seen that Y is a partition in H

between x and E with the modified property (2.5).

It remains to show that if β = β(α) for some α, then ind Y < α. Let ν stand for the predecessor of β = β(α). Since ind( S

i=1

Y

) < ω

, it remains to verify that ind Y

< α (see Theorem 2.A).

The set bd Q is homeomorphic either to the (n − 1)-dimensional sphere or to the (n −1)-dimensional cube, and so it can be represented as the union of subspaces B

and B

homeomorphic to the (n − 1)-dimensional cube such that there exists a homeomorphism f of B

onto B

with f (x) = x for every x ∈ B

∩ B

. For i = 1, 2, let

A

=  [

{A

∪ H

: γ < λ and γ 6= γ

, . . . , γ

}



× B

.

Then Y

= A

∪ A

and there exists a homeomorphism h : A

→ A

such that h(x) = x for every x ∈ A

∩ A

; since A

and A

are homeomorphic to a subspace of H

, by Theorem 2.A, we have

ind Y

= ind A

= ind A

= ind H

< β (see the definition of β(α)).

2.3. Lemma. Let J be a segment contained in an edge of the base B

, and E ⊆ H

a closed set such that E ∩ J = ∅; let b

and b

be the endpoints of J. Then there exist a closed set Y ⊆ H

with the property (2.5) and open sets U, V ⊆ H

such that Y = H

− (U ∪ V ), J − {b

, b

} ⊆ U , E ⊆ V , and for i = 1, 2, we have

b

∈ U if b

is a vertex of the cube B

, b

∈ Y otherwise;

furthermore, if β = β(α) for some α, then ind Y < α.

P r o o f. Let Q ⊆ B

be an n-dimensional cube with faces parallel to the faces of B

and with the property that J is an edge of Q and E ∩ Q = ∅;

let bd Q stand for the boundary of Q in B

. A reasoning similar to that in

the proof of Lemma 2.2 shows that E ∩ [(A

∪ H

) × Q] 6= ∅ only for a finite

number of γ < λ, say for γ

, . . . , γ

. For i = 1, . . . , m, take r

∈ A

such that (p

r

× Q) ∩ E = ∅. Let

Y

= {r

} × Q ∪ p

r

× bd Q for i = 1, . . . , m, Y

=  [

{A

∪ H

: γ < λ and γ 6= γ

, . . . , γ

}



× bd Q, and

Y = [

i=0

Y

(see Fig. 2.5). Just as in the proof of Lemma 2.2 one can show that Y has the required properties.

Fig. 2.5

3. Examples of Cantor ind-manifolds. For each non-limit ordinal α = β + 1 such that ω

≤ α < ω

, we describe an α-dimensional Cantor ind-manifold Z

. For the convenience of the reader some technical reasonings showing that the construction is feasible are deferred to the Appendix.

First, we define an inverse sequence {Z

, r

} consisting of compact metrizable spaces Z

and retractions r

; simultaneously, we define count- able coverings D

of Z

by cubes with dimension greater than 1.

Let Z

= H

and D

= C

(see Section 2); recall that the covering C

consists of cubes of positive dimension for every ordinal β, and so it consists of cubes of dimension greater than 1 whenever β is a non-limit ordinal. Suppose that we have already defined the space Z

and its covering D

. Let %

be any metric on Z

compatible with its topology. Assume additionally that for k = 1, 2, . . . , there exists an arc L

⊆ Z

with the following properties:

(3.1) %

(x, L

) ≤ 1/k for every x ∈ Z

,

Fig. 3.1

(3.2) L

is contained in the union of a finite number of cubes belonging to D

,

(3.3) if J is a cube contained in a cube D ∈ D

and parallel to a proper face of D, then J ∩ L

is finite.

For k = 1, 2, . . . , denote by H

a copy of Henderson’s space H

and by I

an arbitrary edge of B

(see Section 2), and set D

= C

. Loosely speaking, in order to obtain Z

we stick a copy H

of Henderson’s space to each arc L

along the edge I

in such a way that the sets H

− I

are pairwise disjoint, and the space so obtained is compact, i.e., H

is contained in an arbitrarily small neighbourhood of L

for sufficiently large k’s (see Fig. 3.1). Strictly speaking, the space Z

can be defined as follows.

Let γ stand for the predecessor of β(α) (see (2.2)); then H

= H

× I. Set Z

= Z

× {(p

, p

, . . .)} ⊂ Z

× (H

)

, where p

denotes the distinguished point of H

(see Section 2). Next, let H

consist of all (x, (y

)

) ∈ Z

× (H

)

such that x ∈ L

and y

= p

for m 6= k. Put

Z

= Z

∪ [

k=1

H

.

Since L

is an arc, H

and H

are homeomorphic; obviously, so are Z

and Z

. In the sequel, we identify H

and H

as well as Z

and Z

.

Let D

= D

∪ S

k=1

D

and r

be the retraction of Z

onto Z

determined by the “orthogonal projections” of the spaces H

onto the edges I

of their bases, i.e.,

r

((x, (y

)

)) = x for (x, (y

)

) ∈ Z

⊆ Z

× (H

)

.

It is easy to see that Z

is a closed subspace of Z

× (H

)

, and so it is a compact metrizable space, and D

is a countable covering of Z

consisting of cubes with dimension greater than 1.

To complete our construction, we should check that for n = 1, 2, . . . and any metric %

on Z

, there exist arcs L

⊆ Z

with properties (3.1)–(3.3);

in the Appendix we show that there exist arcs L

⊆ Z

which have, apart from (3.1)–(3.3), some additional properties, and if we assume that there exist arcs L

with these additional properties, then there exist arcs L

⊆ Z

with these properties.

Now, assume that the inverse sequence {Z

, r

} is defined.

Let Z

= lim ←−{Z

, r

}; denote by r

the projection of Z

onto Z

. Obviously, Z

is a compact metrizable space. Since each bonding mapping r

is a retraction, we can assume that Z

⊆ Z

and r

is a retraction for every n = 1, 2, . . .

We now show that

(3.4) if K is a partition in Z

between any pair of distinct points, then ind K is not less than the predecessor of α.

Let U, V ⊆ Z

be disjoint open sets with K = Z

− (U ∪ V ). Take an n such that

Z

∩ U 6= ∅ 6= Z

∩ V ;

then, by (3.1), L

∩ U 6= ∅ 6= L

∩ V for a k ∈ N, and thus K ∩ H

is a partition in H

between a pair of distinct points from I

. By Theorem 2.1, ind(K ∩ H

) is not less than the predecessor of α and so is ind K.

It remains to prove that

(3.5) ind Z

≤ α .

To this end, we need the following technical lemma; the situation con- cerned by the lemma is illustrated in Fig. 3.2.

3.1. Lemma. Let {Z

, r

} be a sequence of compact spaces such that Z

⊆ Z

and r

is a retraction for every n ∈ N. Suppose Y

⊆ Z

, n = 1, 2, . . . , are closed subspaces with

(3.6) Y

= Y

∪ [

{A

: s ∈ S

} , where

(3.7) A

is closed and A

− Y

is open in Y

, (3.8) A

∩ A

⊆ Y

for distinct s, t ∈ S

,

and there is a natural number m such that:

(3.9) |A

∩ Y

| < ℵ

for every s ∈ S

, and |A

∩ Y

| > 1 only for a finite

number of s ∈ S

provided n < m,

Fig. 3.2

(3.10) |A

∩ Y

| = 1 for every s ∈ S

provided n ≥ m,

(3.11) r

(A

) = A

∩Y

for any n ∈ N and s ∈ S

such that |A

∩Y

| = 1.

Let Y = lim ←−{Y

, r

|Y

, n ≥ m}. If ind Y

≤ γ and ind A

≤ γ for every s ∈ S

n=1

S

, then ind Y ≤ γ.

P r o o f. We first show by induction that

(3.12) ind Y

≤ γ for every n ∈ N .

For n = 1, this is one of our assumptions. Assume (3.12) holds for an n; we will prove it for n + 1. Let y ∈ Y

, and let F ⊆ Y

be any closed set not containing y.

If y ∈ A

− Y

for some s ∈ S

, then the existence of a partition between y and F with small transfinite dimension less than γ follows from (3.7) and the inequality ind A

≤ γ. Assume therefore that y ∈ Y

(see (3.6)).

Set Z = S

{A

∩ Y

: s ∈ S

and |A

∩ Y

| > 1} − {y} (see Fig. 3.3); then Z is finite by (3.9) and (3.10) (Z = ∅ whenever n ≥ m). By the inductive assumption, there exists a partition K

in Y

between y and (F ∩ Y

) ∪ Z such that ind K

< γ (see Fig. 3.3); let U, V ⊆ Y

be disjoint open sets with y ∈ U , (F ∩ Y

) ∪ Z ⊆ V , and K

= Y

− (U ∪ V ).

Let s

, . . . , s

be all s ∈ S

such that y ∈ A

and |A

∩ Y

| > 1 (see (3.9)). For i = 1, . . . , j, ind A

≤ γ, and so there exists a partition K

in A

between y and (F ∩ A

) ∪ (A

∩ Y

− {y}) such that ind K

< γ (see Fig. 3.3, where j = 2).

We now show that

T = {s ∈ S

: |A

∩ Y

| = 1, A

∩ Y

⊆ U, and F ∩ A

6= ∅}

is finite.

Indeed, suppose that T is infinite. For every s ∈ T , choose x

∈ F ∩ A

;

since F ∩ U = ∅, we have x

∈ A

− Y

. Let x be an accumulation point of

Fig. 3.3

{x

: s ∈ T }. From (3.6)–(3.8) it follows that x ∈ Y

∩ F ⊆ V ; on the other hand, since r

(x

) ∈ U for every s ∈ T (see (3.11)) and r

is a retraction onto Y

, we have r

(x) = x ∈ U ∪ K

, contrary to V ∩ (U ∪ K

) = ∅.

Let s

, . . . , s

be all elements of T . For every i = j + 1, . . . , p, we have ind A

≤ γ, and so there exists a partition K

in A

between A

∩ Y

and F ∩ A

such that ind K

< γ (see Fig. 3.3, where p = 3).

It is easy to check that K = S

i=0

K

is a partition in Y

between y and F . The sets K

, i = 0, 1, . . . , p, are compact; since K

⊆ Y

and K

⊆ A

− Y

for i = 1, . . . , p, they are pairwise disjoint (see (3.8)). Thus

ind K ≤ max{ind K

: i = 0, 1, . . . , p} < γ .

Therefore the proof of (3.12) is concluded. We are now in a position to show that ind Y ≤ γ. Denote by r

the projection of Y onto Y

. Let y ∈ Y , and F ⊆ Y a closed set not containing y. Take n ≥ m such that r

(y) 6∈ r

(F ).

Since ind Y

≤ γ (see (3.12)), there exists a partition K

in Y

between r

(y) and r

(F ) with ind K

< γ; consider disjoint open sets U

, V

⊆ Y

such that r

(y) ∈ U

, r

(F ) ⊆ V

, and K

= Y

− (U

∪ V

). Set

K = K

, U = r

(U

), V = r

(V

∪ K

) − K

.

We prove that K is a partition in Y between y and F . Obviously, y ∈ U , F ⊆ V , U is open and K is closed in Y , and U ∩ K = ∅ = V ∩ K, U ∩ V = ∅.

We only need to show that V is open in Y .

Take z ∈ V . If r

(z) ∈ V

, then r

(V

) is a neighbourhood of z con- taining in V . Thus assume that r

(z) ∈ K

. Since z 6∈ K

, we have z 6∈ Y

. If r

(z) belonged to Y

for every k ≥ n, then z would belong to Y

.

Indeed, suppose that r

(z) ∈ Y

for k ≥ n. Then, in particular, r

(z) ∈ Y

. Assuming that r

(z) ∈ Y

for some k ≥ n, we obtain (recall that r

is a retraction) r

(z) = r

(r

(z)) = r

(z) ∈ Y

. Hence, by induction, r

(z) is in Y

for every k ≥ n, and so is z.

Thus r

(z) 6∈ Y

for some k ≥ n. Then by (3.6), r

(z) ∈ A

− Y

for some s ∈ S

, and by (3.7), r

(A

− Y

) is open. We now show that r

(A

− Y

) ⊆ V .

Indeed, since k ≥ n ≥ m, r

(A

) is a one-point set (see (3.10) and (3.11)); thus

r

(A

) = {r

(r

(z))} = {r

(z)} ⊆ K

. Obviously, r

(A

− Y

) ∩ K

= ∅, and so r

(A

− Y

) ⊆ V .

Having proved the lemma, we can turn to the proof of inequality (3.5);

recall that β stands for the predecessor of α. Let z ∈ Z

, and F ⊆ Z

a closed set not containing z. We prove that there exists a partition in Z

between z and F of dimension not greater than β.

Take m such that r

(z) 6∈ r

(F ), and p ≤ m such that r

(z) ∈ Z

− Z

; we assume that Z

= ∅, that is, if r

(z) ∈ Z

, then p = 1.

We shall define by induction for n = p, p + 1, . . . , m a partition Y

in Z

between r

(z) and r

(F ) ∩ Z

with the following property:

(3.13) for every cube D ∈ D

the set D ∩ Y

is the union of a finite number of cubes of dimension less than that of D, each parallel to a proper face of D;

moreover, we will require ind Y

≤ β. Simultaneously, we shall define sets S

and A

for s ∈ S

and n = p, p + 1, . . . , m − 1 satisfying (3.6)–(3.9), (3.11) and

(3.14) ind A

≤ β for every s ∈ S

.

Since Z

− Z

is a neighbourhood of r

(z) homeomorphic to an open subset of H

, the existence of a partition Y

with the required properties follows from Lemma 2.2. Assume that we have defined a partition Y

with the required properties for an n < m.

Let U

, V

⊆ Z

be open sets such that r

(z) ∈ U

, r

(F ) ∩ Z

⊆ V

and Y

= Z

− (U

∪ V

). Set

Y

= (r

)

(Y

), U

= (r

)

(U

), V

= (r

)

(V

) .

Then Y

is a partition in Z

between r

(z) and r

(F ) ∩ Z

. Since L

has properties (3.2)–(3.3) and Y

satisfies (3.13), (3.15) Y

∩ L

is finite for every k = 1, 2, . . .

Hence Y

∩ H

is the union of a finite number of pairwise disjoint sets homeomorphic to H

, where β(α) = ν + 1, for every k = 1, 2, . . . Denote these sets by A

, s ∈ T

(see Fig. 3.4). Observe that A

∩ Y

is a one-point set for every s ∈ T

and k = 1, 2, . . .

Since r

(F ) ∩ (U

∪ Y

) = ∅, it follows that r

(F ) ∩ (U

∪ Y

) ⊆ S {H

− L

: k ∈ N}; furthermore, since the sets H

− L

are pairwise disjoint and r

(F ) ∩ (U

∪ Y

) is compact, there exists l ∈ N such that r

(F ) ∩ (U

∪ Y

) ⊆ S

{H

− L

: k = 1, . . . , l}.

Fix k ≤ l, and an orientation of L

. Then L

= S

i=1

a

a

, where a

, a

, . . . , a

are ordered consistently with the orientation, and either

a

a

⊆ U

∪ Y

, whereas a

a

⊆ V

∪ Y

, or

a

a

⊆ V

∪ Y

, whereas a

a

⊆ U

∪ Y

for i = 1, . . . , j − 1, that is, {a

, . . . , a

} is the set of all points at which L

goes across Y

; of course, a

and a

are the endpoints of L

(see Fig. 3.4).

Let T

= {(i, k) : i = 1, . . . , j and a

a

⊆ U

∪ Y

}. For every s = (i, k) ∈ T

, the arc a

a

is identified with a segment contained in the edge I

of the base of H

= H

. Let A

, U

, V

stand for sets Y, U, V with the properties described in Lemma 2.3 for J = a

a

and E = r

(F ) ∩ H

(see Fig. 3.4).

The set Y

=



Y

− [

{H

− L

: k ≤ l}



∪ [

{A

: k ≤ l and s ∈ T

}

= Y

∪ [

{A

: k > l and s ∈ T

} ∪ [

{A

: k ≤ l and s ∈ T

} is a partition in Z

between r

(z) and r

(F ) ∩ Z

; indeed,

U

=



U

− [

{H

− L

: k ≤ l}



∪ [

{U

: k ≤ l and s ∈ T

} and

V

=



V

− [

{H

− L

: k ≤ l}



∪ [

{V

: k ≤ l and s ∈ T

} are open sets in Z

such that r

(z) ∈ U

, r

(F ) ∩ Z

⊆ V

, U

∩ V

= ∅ and Y

= Z

− (U

∪ V

).

Let S

= S

{T

: k ≤ l}∪ S

{T

: k > l}. It is easy to check that our sets

have the required properties. Thus we have constructed inductively the sets

Y

, Y

, . . . , Y

and the sets S

and A

, s ∈ S

, for n = p, p + 1, . . . , m − 1.

Fig. 3.4

We define Y

for n = m + 1, m + 2, . . . by induction setting Y

= (r

)

(Y

) (see Fig. 3.5).

Let S

= (L

∩ Y

) × {k} for n = m, m + 1, . . . and k = 1, 2, . . . , and

Fig. 3.5

next S

= S

k=1

S

; let

A

= (r

)

(x) ∩ H

for s = (x, s) ∈ S

(see Fig. 3.5).

One can check by induction that Y

satisfies (3.13) for n = m, m + 1, . . . , and hence Y

∩ L

is finite for k = 1, 2, . . . (see (3.2) and (3.3)). By construction and the above observation, it follows that (3.6)–

(3.8), (3.10)–(3.11) are also satisfied for n ≥ m; since each A

, s ∈ S

, is homeomorphic to Henderson’s space H

, where ν is the predecessor of β(α), condition (3.14) is also satisfied (recall that ind H

< α for every µ < β(α), see Section 2).

Since Y

is a partition in Z

between r

(z) and r

(F ), it follows that r

(Y

) is a partition in Z

between z and F . It is easily seen that r

(Y

) is homeomorphic to lim ←−{Y

, r

|Y

, n ≥ m}; thus ind r

(Y

) ≤ β by Lemma 3.1.

4. Appendix. We complete the description of the construction of {Z

, r

}. To wit, we show that there exist arcs L

in Z

satisfying (3.1)–

(3.3), and having some additional properties: each L

is a D

-broken line (see Definition 4.1). We also show that if each L

⊆ Z

is a D

-broken line, then there exist D

-broken lines L

⊆ Z

with properties (3.1)–(3.3).

First, we have to prepare an auxiliary apparatus.

4.1. Definition. Let D be a countable covering of a topological space

X by cubes. An arc L is said to be a D-broken line in X if it is contained in

the union of a finite number of cubes belonging to D, and for every D ∈ D, L ∩ D is the union of a finite number of segments and one-point sets.

4.2. Definition. Let D be a countable covering of a topological space X by cubes of dimension greater than 1. We say that D has property (∗) if the following conditions are satisfied:

(4.1) for every pair of distinct cubes C, D ∈ D, C ∩ D is either a proper face of C and a proper face of D, or is the union of a finite number of segments and one-point sets contained either in a proper face of C or in a proper face of D,

(4.2) for every pair of cubes C, D ∈ D, there exists a sequence of cubes D

, . . . , D

∈ D such that C = D

, D = D

, and |D

∩ D

| ≥ ℵ

for i = 1, . . . , n − 1.

Note that (4.1) does not exclude that C ∩ D = ∅ for some C, D ∈ D, and it implies that if |C ∩ D| ≥ ℵ

, then C ∩ D contains a segment.

4.3. Lemma. For every countable non-limit ordinal α > 1, the covering C

of H

has property (∗).

The proof is by induction on α.

4.4. Lemma. Let Y be a topological space. For k = 0, 1, . . . , let X

be a subspace of Y , E

a covering of X

by cubes with property (∗), and (L

)

a sequence of E

-broken lines in X

. Furthermore, suppose that

(4.3) X

∩ X

= L

, and X

∩ X

= L

∩ L

for distinct k, m = 1, 2, . . . , (4.4) for every cube D ∈ E

, L

∩ D is the union of a finite number of

segments and one-point sets contained in a proper face of D.

Then E = S

k=0

E

is a covering of X = S

k=0

X

by cubes with property (∗).

P r o o f. Obviously, E is a countable covering of X by cubes of dimension greater than 1. It is a simple matter to check that (4.1) is satisfied. We now show that (4.2) is also satisfied.

Let C, D ∈ E. If C, D ∈ E

for some k = 0, 1, . . . , then the existence of a sequence D

, . . . , D

with the required properties follows from the assump- tion that E

has property (∗); thus suppose that C ∈ E

and D ∈ E

, where k 6= m. We only consider the case when k, m > 0; if k = 0 or m = 0, the reasoning is similar.

Since X

∩ X

= L

, and E

and E

are countable, |C

∩ C

| ≥ ℵ

for some C

∈ E

and C

∈ E

; by a similar argument, there exist D

∈ E

and D

∈ E

such that |D

∩ D

| ≥ ℵ

. Let

• D

, . . . , D

∈ E

be such that D

= C, D

= C

, and |D

∩ D

| ≥ ℵ

for i = 1, . . . , j − 1,

• D

, . . . , D

∈ E

be such that D

= C

, D

= D

, and |D

∩D

| ≥ ℵ

for i = j + 1, . . . , l − 1, and

• D

, . . . , D

∈ E

be such that D

= D

, D

= D, and |D

∩ D

| ≥ ℵ

for i = l + 1, . . . , n.

Then the sequence D

, . . . , D

has the required properties.

4.5. Lemma. Let (X, %) be a totally bounded metric space, and D its covering by cubes with property (∗). Then for every ε > 0, there exists a D-broken line L in X with the following properties:

(4.5) for every x ∈ X, the distance between x and L is not greater than ε, (4.6) if J is a cube contained in a cube D ∈ D, the dimension of J is less than that of D, and J is parallel to a proper face of D, then L ∩ D is finite.

The proof of Lemma 4.5 will be preceded by two preliminary lemmas, both concerning the situation described in Lemma 4.5.

4.6. Lemma. Let C ∈ D; suppose T ⊆ C is finite and x, y ∈ C − T . Then for every δ > 0, there exists a broken line K ⊆ C − T satisfying (4.6) with endpoints x and y and such that

(4.7) %(z, K) ≤ δ for every z ∈ C .

P r o o f. Since the dimension of C is not less than 2, it is a simple matter to find a broken line K ⊆ C − T satisfying (4.7) with endpoints x and y (see Fig. 4.1).

Fig. 4.1

Since D is countable and satisfies (4.1), C ∩ [ S

(D − {C})] is the union

of a number of faces of C, a countable number of segments (say F

, F

, . . .),