Let f be a transcendental entire function of disjoint type. The Julia set of f consists of an uncountable union of disjoint curves each of which joins a finite endpoint to infinity. Following recent results on the topology of the set of non-escaping endpoints for functions in the exponential family, we show that the union of non-escaping endpoints of f with infinity is a totally separated set. Combined with a result of Alhabib and Rempe-Gillen this gives a strong dichotomy on the topological properties of the set of endpoints which escape and those which do not escape for disjoint-type functions. This is joint work with D. Sixsmith.