We construct an
n
n
-dimensional completely metrizable
A
E
(
n
)
AE(n)
-space
P
(
n
,
τ
)
P(n,\tau )
of weight
τ
≥
ω
\tau \geq \omega
with the following property: for any at most
n
n
-dimensional completely metrizable space
Y
Y
of weight
≤
τ
\leq \tau
the set of closed embeddings
Y
→
P
(
n
,
τ
)
Y \to P\left ( {n,\tau } \right )
is dense in the space
C
(
Y
,
P
(
n
,
τ
)
)
C\left ( {Y,P\left ( {n,\tau } \right )} \right )
. It is also shown that completely metrizable
L
C
n
L{C^n}
-spaces of weight
τ
≥
ω
\tau \geq \omega
are precisely the
n
n
-invertible images of the Hilbert space
ℓ
2
(
τ
)
{\ell _2}(\tau )
.