  The family of exponential maps $f_a(z)= e^z+a$ is of fundamental importance
in the study of transcendental dynamics. Here we consider the topological
structure of certain subsets of the Julia set $J(f_a)$. When $a\in
(-\infty,-1)$, and more generally when $a$ belongs to the Fatou set of $f_a$,
it is known that $J(f_a)$ can be written as a union of "hairs" and "endpoints"
of these hairs. In 1990, Mayer proved for $a\in (-\infty,-1)$ that, while the
set of endpoints is totally separated, its union with infinity is a connected
set. Recently, Alhabib and the second author extended this result to the case
where $a \in F(f_a)$, and showed that it holds even for the smaller set of all
escaping endpoints.
  We show that, in contrast, the set of non-escaping endpoints together with
infinity is totally separated. It turns out that this property is closely
related to a topological structure known as a `spider's web'; in particular we
give a new topological characterisation of spiders' webs that may be of
independent interest. We also show how our results can be applied to Fatou's
function, $z\mapsto z + 1 + e^{-z}$.
