  In 1988, Mayer proved the remarkable fact that infinity is an explosion point
for the set of endpoints of the Julia set of an exponential map that has an
attracting fixed point. That is, the set is totally separated (in particular,
it does not have any nontrivial connected subsets), but its union with the
point at infinity is connected. Answering a question of Schleicher, we extend
this result to the set of "escaping endpoints" in the sense of Schleicher and
Zimmer, for any exponential map for which the singular value belongs to an
attracting or parabolic basin, has a finite orbit, or escapes to infinity under
iteration (as well as many other classes of parameters).
  Furthermore, we extend one direction of the theorem to much greater
generality, by proving that the set of escaping endpoints joined with infinity
is connected for any transcendental entire function of finite order with
bounded singular set. We also discuss corresponding results for *all* endpoints
in the case of exponential maps; in order to do so, we establish a version of
Thurston's "no wandering triangles" theorem.
