Universal metric spaces and extension dimension

Abstract

For any countable $CW$-complex $K$ and a cardinal number $τ\geqω$ we construct a completely metrizable space $X(K,τ)$ of weight $τ$ with the following properties: $\e X(K,τ)\leq K$, $X(K,τ)$ is an absolute extensor for all normal spaces $Y$ with $\e Y\leq K$, and for any completely metrizable space $Z$ of weight $\leqτ$ and $\e Z\leq K$ the set of closed embeddings $Z\to X(K,τ)$ 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.