  We prove that the set of all endpoints of the Julia set of $f(z)=\exp(z)-1$
which escape to infinity under iteration of $f$ is not homeomorphic to the
rational Hilbert space $\mathfrak E$. As a corollary, we show that the set of
all points $z\in \mathbb C$ whose orbits either escape to $\infty$ or attract
to $0$ is path-connected. We extend these results to many other functions in
the exponential family.
