Let
M
M
and
N
N
be smooth manifolds, where
M
⊂
N
M\subset N
and
dim
(
N
)
−
dim
(
M
)
≥
3
\dim (N)-\dim (M)\ge 3
. A disjunction lemma for embeddings proved recently by Goodwillie leads to a calculation up to extension problems of the base point component of the space of smooth embeddings of
M
M
in
N
N
. This is mostly in terms of
i
m
m
(
M
,
N
)
\mathbf {imm}(M,N)
, the space of smooth immersions, which is well understood, and embedding spaces
e
m
b
(
S
,
N
)
\mathbf {emb}(S,N)
for finite subsets
S
S
of
M
M
with few elements. The meaning of few depends on the precision desired.