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

Abstract

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