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}
].