Universal metric spaces and extension dimension

For any countable $CW$-complex $K$ and a cardinal number $\tau\geq\omega$ we construct a completely metrizable space $X(K,\tau)$ of weight $\tau$ with the following properties: $\e X(K,\tau)\leq K$, $X(K,\tau)$ is an absolute extensor for all normal spaces $Y$ with $\e Y\leq K$, and for any completely metrizable space $Z$ of weight $\leq\tau$ and $\e Z\leq K$ the set of closed embeddings $Z\to X(K,\tau)$ is dense in the space $C(Z,X(K,\tau))$ of all continuous maps from $Z$ into $X(K,\tau)$ 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.