  A locally compact stable plane of positive topological dimension will be
called semiaffine if for every line $L$ and every point $p$ not in $L$ there is
at most one line passing through $p$ and disjoint from $L$. We show that then
the plane is either an affine or projective plane or a punctured projective
plane (i.e., a projective plane with one point deleted). We also compare this
with the situation in general linear spaces (without topology), where P.
Dembowski showed that the analogue of our main result is true for finite spaces
but fails in general.
