  We study topological properties of the escaping endpoints and fast escaping
endpoints of the Julia set of complex exponential $\exp(z)+a$ when $a\in
(-\infty,-1)$. We show neither space is homeomorphic to the whole set of
endpoints. This follows from a general result stating that for every
transcendental entire function $f$, the escaping Julia set $I(f)\cap J(f)$ is
first category.
