Distinguishing endpoint sets from Erd\H{o}s space

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.