  $t$-spanners are used to approximate the pairwise distances between a set of
points in a metric space. They have only a few edges compared to the total
number of pairs and they provide a $t$-approximation on the distance of any two
arbitrary points. There are many ways to construct such graphs and one of the
most efficient ones, in terms of weight and the number of edges of the
resulting graph, is the greedy spanner. In this paper, we study the edge
crossings of the greedy spanner for points in the Euclidean plane. We prove a
constant upper bound for the number of intersections with larger edges that
only depends on the stretch factor of the spanner, $t$, and we show there can
be more than a bounded number of intersections with smaller edges. Our results
imply that greedy spanners for points in the plane have separators of size
$\mathcal{O}(\sqrt n)$, that their planarizations have linear size, and that a
separator hierarchy for these graphs can be constructed from their
planarizations in linear time.
