Universal maps and surjective characterizations of completely metrizable $\textrm {LC}^ n$-spaces

Abstract:

We construct an $n$-dimensional completely metrizable $AE(n)$-space $P(n,\tau )$ of weight $\tau \geq \omega$ with the following property: for any at most $n$-dimensional completely metrizable space $Y$ of weight $\leq \tau$ the set of closed embeddings $Y \to P\left ( {n,\tau } \right )$ is dense in the space $C\left ( {Y,P\left ( {n,\tau } \right )} \right )$. It is also shown that completely metrizable $L{C^n}$-spaces of weight $\tau \geq \omega$ are precisely the $n$-invertible images of the Hilbert space ${\ell _2}(\tau )$.