Infinite-dimensional manifolds related to C-spaces

Haver’s property C turns out to be related to Borst’s transfinite extension of the covering dimension. We prove that, for a uncountably many countable ordinals β there exists a strongly universal kω-space for the class of spaces of transfinite covering dimension ă β. In some sense, our result is a kω-counterpart of Radul’s theorem on existence of absorbing sets of given transfinite covering dimension. Анотація. Як з’ясувалося, властивість C Гейвера споріднена з трансфінітним розширенням Борста покриттєвого виміру. Ми доводимо, що для нескінченного числа зліченних ординалів β існує сильно універсальний kω-простір для класу просторів трансфінітного покриттєвого виміру ă β. В деякому сенсі наш результат є kω-аналогом теореми Радула про існування поглинаючих множин заданого трансфінітного покриттєвого виміру.


INTRODUCTION
The notion of metric C-space was introduced by W. Haver [9]. He applied these spaces to the theory of retracts. In [1] the property C was defined for all topological spaces. The C-spaces play an important role in the dimension theory. P. Borst [7] introduced a transfinite extension of the covering dimension dim which characterizes property C.
T. Radul [13] proved that there exists an uncountable set of countable ordinals β such that there exist noncountable-dimensional pre-Hilbert spaces 50 O. Polivoda, M. Zarichnyi D β which are absorbing spaces (in the sense of Bestvina and Mogilski [6]) for the class of compacta with dim C less than β.
In some sense, our main result is a counterpart of Radul's theorem in the category of k ω -spaces. We prove that, for an uncountable set of countable ordinals β, there exists a k ω -space which is strongly universal for the class of compacta with dim C less than β.

PRELIMINARIES
A family V of subsets of a space X is said to refine a family U of subsets of X if for each element V P V there exists U P U for which V Ă U . A family V is said to star-refine a family U if for every U P U there exists V P V such that U 1 Ă V for any U 1 P U such that U X U 1 ‰ H. If U is a family of subsets in a metric space, then we define

Definition 2.1.
A space X has property C (briefly is a C-space) if for each sequence tα n | n P Nu of open coverings of X there exists a sequence tβ n | n P Nu of open disjoint families such that each family β n refines α n and Y 8 n=1 β n P cov(X). We will need the following properties of compact metrizable C-spaces (see [1]), where they are proved for more general class of spaces).

Proposition 2.2. Every closed subspace of a
Proof. Let tU n | n P Nu be a countable family of neighborhoods of A such that A X 8 n=1 U n . Then X/A is the sum of the sets homeomorphic to XzU n and a singleton. □ The following statement is proved in [8] for the paracompact spaces.

Proposition 2.5. Let f be a closed map from a compact metrizable space
X onto a C-space Y . If f´1(y) has property C for each y P Y , then X is a C-space.
We will need the following result by T. Radul [13].
Theorem 2.6. For each α ă ω 1 there exists a compact metrizable Cspace L α which contains topologically each compact metrizable space K with dim C K ď α.
2.7. Dimension dim C . P. Borst [7] introduced the transfinite extension dim C of the covering dimension. We recall some necessary definition. Let us start with the ordinal number Ord. Given a set L, we denote by Fin L the collection of all finite, nonempty subsets of L. Let M be a subset of Fin L. For σ P tHu Y Fin L, put Define the ordinal number Ord M inductively as follows: Let X be a topological space and K(X) denote the set of the all locally finite coverings of X. Put .
Remark that the dimension dim C coincides with classical covering dimension dim for finite-dimensional spaces [7] and for any compact metric space K dim C K exists if and only if K has property C.
Also, it is an easy consequence of the definition that if A is a closed subset of X, then dim C A ď dim C X.
Let us denote by D(β) the class of compact metric spaces with dim C less than β. We say that a topological space Y is D(β)-universal if Y contains topologically all compacta from D(β).
T. Radul [13,Theorem 3] proved that for each ordinal α ă ω 1 there exists an ordinal β, α ď β ă ω 1 , and a C-compact metric space X such that dim C X = β and X is D(β)-universal. Lemma 2.9. There exists a function h : ω 1 Ñ ω 1 such that, for any compact metric C-space X and any closed subset A of X, Proof. Let K be a universal space for compact metric spaces X with dim C X ď α. Let exp K denote the hyperspace of K, i.e., the space of all nonempty closed subsets of X endowed with the Vietoris topology (see, e.g., [11]). There exists a continuous map of the Cantor discontinuum C onto exp X. Let , and therefore is homeomorphic to a subset of Z/B. We conclude that □ Remark 2.10. Actually, no example is known witnessing that h is not the identity map.

RESULTS
Recall that an absolute retract (AR-space) is a space X which is a retract of every metric space containing X as a closed subset. Proposition 3.1. Let X be a compact metrizable C-space. Then there exists a compact metrizable C-spaceX that contains a topological copy of X and is an AR-space.
Proof. We assume that X is a metric space. Define inductively a sequence (U i ) of open covers of X as follows. Let U 1 = tXu. If U j is already defined for j ă i, let U i be a cover of X which star-refines U i´1 and such that mesh(U i´1 ) ď 2´i.
Assume that X is isometrically embedded into a Banach space L which, in turn, is identified with the subset Lˆt0u of Lˆ[0, 1]. Let N (U i ) be the nerve of the cover U i . Assume also that N (U i ) is a subpolyhedron of Lˆt2 1´i u with the following property: for any U P U i , the vertex of N (U i ) Infinite-dimensional manifolds related to C-spaces 53 corresponding to U is an element of the set Uˆt2 1´i u. For every natural i, let K i be the mapping cylinder of a natural map f i : N (U i+1 ) Ñ N (U i ). The latter is a simplicial map sending the vertex corresponding to U P N (U i+1 ) to the vertex corresponding to any V P N (U i ) such that U Ă V . We assume as well that the mapping cylinder consists of all linear segments in Using standard arguments we show thatL is a compact metrizable AR-space. The projection ofL onto [0, 1] is a closed map such that every its preimage is a C-space (either L or a polyhedron). Therefore,L is a C-space. □ Proposition 3.2. For any α ă ω 1 there exists a pointed compact metrizable C-space (L,˚) that contains a topological copy of each pointed compact metrizable C-space (K,˚) with dim C K ď α.
Proof. Let L α be a universal space for compact metrizable C-spaces K with dim C K ď α. Denote byL the quotient space The set ∆ is regarded as the base point ofL. Denote by q : L αˆLα ÑL the quotient map. Suppose that (K, x 0 ) is a compact metrizable C-space with dim C K ď α. Then, clearly, the map f : K ÑL defined by the formula f (x) = q(x, x 0 ), x P K, is a pointed embedding.
Finally, remark thatL is a C-space. Indeed, one can representL as the countable union of the singleton tq(∆)u and the spaces q((LˆL)zU i ), where tU i | i P Nu is a countable base neighborhoods of ∆ in the product LˆL, and then apply Propositions 2.2 and 2.3. □ Recall that a topological space X is said to be a k ω -space if X = lim Ý Ñ X i , where (X i ) is an increasing sequence of its compact subspaces.
We say that a space X is strongly D(β)-universal (resp. locally strongly D(β)-universal) if for every compact metric space A with dim C (A) ă β and every embedding f : B Ñ X of its closed subset B into X there exists an embeddingf : A Ñ X (resp. an embeddingf : U Ñ X, where U is a neighborhood of B in A) that extends f .

Theorem 3.3.
There is an uncountable set Φ Ă ω 1 such that, for every β P Φ there exists a strongly D(β)-universal k ω -space K β which is the countable direct limit of an increasing sequence of compact spaces from the class D(β).
Proof. Let α ă ω 1 . Let X 1 = t˚u be any compact metrizable C space with dim C X 1 = α. Suppose that compact metrizable C-spaces X i are already constructed for all i ă n.
By Proposition 3.1, there exists a compact metric C-spaceX n´1 which contains X n´1 and is an absolute retract.
By Proposition 3.2, there exists a pointed compact C-space (Y n´1 ,˚) which is universal for pointed compact metric spaces of dim C not exceeding h(α). Take X n =X n´1ˆYn´1 .
We are going to show that X is strongly D(β)-universal. Suppose that (A, B) is a pair of compact metric spaces and dim C A ă β. Suppose also that f : B Ñ X is an embedding. Since X is a k ω -space, there exists n P N such that f (B) Ă X n . SinceX n is an AR-space, there is a continuous extension f 1 : A ÑX n .
Let q : A Ñ A/B be the quotient map. By Proposition 3.2, there exists an embedding g : A/B Ñ Y n such that g(B) =˚.
Finally, definef : A Ñ X n+1 by the formulaf (x) = (f 1 (x), gq(x)). It is easy to see thatf is an embedding that extends f . □ The following is a characterization theorem for the spaces K β .
Theorem 3.4. Let X be a k ω -space and X is countable direct limit of compact metric spaces from the class D(β). Then the following properties are equivalent: (1) X is strongly D(β)-universal; (2) X is homeomorphic to K β .
Proof. We apply the back and forth argument used in [14] as well as in another publications. For the sake of reader's convenience, we provide some details.
. is a sequence of compact spaces such that Y n P D(β) for every n P N.
Let m 1 = 1. There exists an embedding f 1 : Continuing in this way we obtain a commutative diagram Hausdorff space which is locally homeomorphic to open subsets in K β . We will assume that the K β -manifolds are k ω -spaces.
The following is a characterization theorem for the K β -manifolds.
Theorem 3.5. Let X be a k ω -space and X is countable direct limit of compact metric spaces from the class D(β). Then the following properties are equivalent: (1) X is locally strongly D(β)-universal; (2) X is a K β -manifold.
Actually, the proof of this result can be performed along the line of the proof of [14,Theorem 1.3]. And, similarly as in [14], we obtain the following Open Embedding Theorem (see also [10]).

UNIVERSAL MAPS
The following notion is introduced in [15]. Let K fd denote the class of metrizable finite-dimensional compacta. A map f : X Ñ Y is called strongly K fd -universal if, for any if for every embedding α : B Ñ X of a closed subset B of a space A P K fd and any map γ : A Ñ Y with f α = γ|B there is an embeddingᾱ : A Ñ X such that fᾱ = γ andᾱ|B = α.
Let R 8 be the direct limit of the sequence Proof. Without loss of generality we may assume that K β is embedded in Q 8 as a closed subset. Let f : R 8 Ñ Q 8 be an K fd -universal map. Define X = f´1(K β ) and let f 1 denote the restriction f 1 = f |X : X Ñ K β . Since 56 O. Polivoda, M. Zarichnyi K β is closed in Q 8 , we see that X is a k ω -space. Clearly, X = lim Ý Ñ X i , where X i P K fd for all i. Since K β is an absolute extensor, the space X satisfies the conditions of the characterization theorem for R 8 (see [14]). Also, this implies the strong K fd -universality of f 1 . □ Remark also that a strongly K fd -universal map from Theorem 4.1 is unique up to homeomorphism.

REMARKS
In connection with Radul's results on existence of D(β)-absorbing sets in the sense of Bestvina and Mogilski [6] the following question arises.
Question 5.1. Are there ordinals β ă ω 1 for which both D β and K β exist? Is there a bitopological characterization of this pair in the spirit of Banakh and Sakai [4]?
Recall that in [4] a characterization of the bitopological space (R 8 , ℓ 2 f ) is given.
We expect the negative answer to the following question related to Theorem 4.1. Also, we conjecture that the universal map from Theorem 4.1 is not locally self-similar. (Here, a map π : X Ñ Y is said to be locally selfsimilar if for every point x P X and every neighborhood U Ă X of x there is a neighborhood V Ă U of x such that the map π|V : V Ñ π(V ) is homeomorphic to π. It is proved in [3] that the universal map R 8 Ñ Q 8 is not locally self-similar. ) Banakh and Repovš [3] proved that there exists a linear realization of the universal map R 8 Ñ Q 8 . It looks plausible that such a realization can be found for the universal map from Theorem 4.1. This would provide another construction of the spaces K β , namely as a linear topological space. See, e.g., [2,5,15] for various results concerning infinite-dimensional manifolds in topological algebra.
Finally, remark that some of the results of this note are announced in [12]; here they are given with new proofs.