The upward separation technique was developed by Hartmanis, who used it to show that E=NE iff there is no sparse set in NP-P [Ha-83a]. This paper shows some inherent limitations of the technique. The main result of this paper is the construction of an oracle relative to which there are extremely sparse sets in NP-P, but NEE = EE; this is in contradiction to a result claimed in [Ha-83, HIS-85]. Thus, although the upward separation technique is useful in relating the existence of sets of polynomial (and greater) density in NP-P to the NTIME(T(n)) = DTIME(T(n)) problem, the existence of sets of very low density in NP-P can not be shown to have any bearing on this problem until proof techniques are developed which do not relativize. The oracle construction is also of interest since it is the first example of an oracle relative to which EE = NEE and E ≠ NE. (The techniques of [BWX-82], [Ku-85], and [Li-87] do not suffice to construct such an oracle.) The construction is novel and the techniques may be useful in other settings. In addition, this paper also presents a number of new applications of the upward separation technique, including some new generalizations of the original result of [Ha-83a].