For a pair of metrizable spaces
X
X
and
Y
Y
, we investigate conditions under which there is a dense embedding
h
:
X
→
Z
h:X \to Z
, where
Z
Z
is completely metrizable and
Z
∖
h
(
X
)
Z\backslash h(X)
is homeomorphic to
Y
Y
. In such a case,
Z
Z
is called a topological completion of
X
X
and
Y
Y
is called a completion remainder of
X
X
. In case
X
X
and
Y
Y
are completely metrizable, we give necessary and sufficient conditions that
Y
Y
be a completion remainder of
X
X
. We characterize the completion remainders of
R
{\mathbf {R}}
and those of the rationals,
Q
{\mathbf {Q}}
. We also characterize the remainders of
Q
(
κ
)
{\mathbf {Q}}(\kappa )
, a nonseparable analogue of
Q
{\mathbf {Q}}
.