1 6 A ug 1 99 9 UNIVERSAL METRIC SPACES AND EXTENSION DIMENSION

For any countable CW -complex K and a cardinal number τ ≥ ω we construct a completely metrizable space X(K, τ) of weight τ with the following properties: e-dimX(K, τ) ≤ K, X(K, τ) is an absolute extensor for all normal spaces Y with e-dimY ≤ K, and for any completely metrizable space Z of weight ≤ τ and e-dimZ ≤ K the set of closed embeddings Z → X(K, τ) is dense in the space C(Z,X(K, τ)) of all continuous maps from Z into X(K, τ) endowed with the limitation topology. This result is applied to prove the existence of universal spaces for all metrizable spaces of given weight and with a given cohomological dimension. The existence of universal separable metric spaces for extension dimension with respect to countable CW -complexes was proved by Olszewski in [12]. In the class of all metric spaces of a given weight this problem was recently solved by Levin [11]. In the present note we show the existence of universal metric spaces having some extra properties (see Theorem 1 below). The concept of extension dimension was introduced by Dranishnikov [5] (see also [2], [6]). For a normal space X and a CW -complex K we write e-dimX ≤ K (the extension dimension of X does not exceed K) if K is an absolute extensor for X. This means that any continuous map f : A → K, defined on a closed subset A of X, admits a continuous extension f̄ : X → K. Since not every CW complex is an absolute neighborhood extensor for normal spaces, we can enlarge the class of normal spaces X with e-dimX ≤ K (K is a CW -complex) by introducing the following notion (see [14, Definition 2.5]): A normal space X is in the class α(K) if every continuous map from a closed A ⊂ X to K which extends to a map of a neighborhood of A to K can be extended to a map of X to K. Obviously, if K ∈ ANE(X) (this, for example, holds for every X admitting a perfect map onto a first countable paracompact space [8]) then X ∈ α(K) if and only if e-dimX ≤ K. We also adopt the following definition: 1991 Mathematics Subject Classification. 5510; 54B35.