  We present a simple approach to questions of topological orbit equivalence
for actions of countable groups on topological and smooth manifolds. For
example, for any action of a countable group $\Gamma$ on a topological manifold
where the fixed sets for any element are contained in codimension two
submanifolds, every orbit equivalence is equivariant. Even in the presence of
larger fixed sets, for actions preserving rigid geometric structures our
results force sufficiently smooth orbit equivalences to be equivariant. For
instance, if a countable group $\Gamma$ acts on ${\mathbb T}^n$ and the action
is $C^1$ orbit equivalent to the standard action of $SL_n({\mathbb Z})$ on
${\mathbb T}^n$, then $\Gamma$ is isomorphic to $SL_n({\mathbb Z})$ and the
actions are isomorphic. (The same result holds if we replace $SL_n(\mathbb Z)$
by a finite index subgroup.) We also show that preserving a geometric structure
is an invariant of smooth orbit equivalence and give an application of our
ideas to the theory of hyperbolic groups.
  In the course of proving our theorems, we generalize a theorem of Sierpinski
which says that a connected Hausdorff compact topological space is not the
disjoint union of countably many closed sets. We prove a stronger statement
that allows "small" intersections provided the space is locally connected. This
implies that for any continuous action of a countable group $\G$ on a
connected, locally connected, locally compact, Hausdorff topological space,
where the fixed set of every element is "small", every orbit equivalence is
equivariant.
