Several conditions are given which together imply that a 2-manifold M in a 3-manifold is locally tame from one of its complementary domains, U, at all except possibly one point. One of these conditions is that certain arbitrarily small simple closed curves on M can be collared from U. Another condition is that there exists a certain sequence
M
1
,
M
2
,
…
{M_1},{M_2}, \ldots
of 2-manifolds in U converging to M with the property that each unknotted, sufficiently small simple closed curve on each
M
i
{M_i}
is nullhomologous on
M
i
{M_i}
. Moreover, if each of these simple closed curves bounds a disk on a member of the sequence, then it is shown that M is tame from
U
(
M
≠
S
2
)
U(M \ne {S^2})
. As a result, if U is the complementary domain of a torus in
S
3
{S^3}
that is wild from U at just one point, then U is not homeomorphic to the complement of a tame knot in
S
3
{S^3}
.