Dense embeddings of sigma-compact, nowhere locally compact metric spaces

Abstract:

It is proved that a connected complete separable ANR $Z$ that satisfies the discrete $n$-cells property admits dense embeddings of every $n$-dimensional $\sigma$-compact, nowhere locally compact metric space $X(n \in N \cup \{ 0,\infty \} )$. More generally, the collection of dense embeddings forms a dense ${G_\delta }$-subset of the collection of dense maps of $X$ into $Z$. In particular, the collection of dense embeddings of an arbitrary $\sigma$-compact, nowhere locally compact metric space into Hilbert space forms such a dense ${G_\delta }$-subset. This generalizes and extends a result of Curtis [Cu$_{1}$].