vii, 34 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We study the Bousfield localization functors known as [Special characters omitted], as described in [MahS]. In particular we would like to understand how they interact with suspension and how they stabilize. We prove that suitably connected [Special characters omitted]-acyclic spaces have suspensions which are built out of a particular type n space, which is an unstable analog of the fact that [Special characters omitted]-acyclic spectra are built out of a particular type n spectrum. This theorem follows Dror-Farjoun's proof in the case n = 1 with suitable alterations. We also show that [Special characters omitted] applied to a space stabilizes in a suitable way to [Special characters omitted] applied to the corresponding suspension spectrum. / Committee in charge: Hal Sadofsky, Chairperson, Mathematics; Arkady Berenstein, Member, Mathematics; Daniel Dugger, Member, Mathematics; Dev Sinha, Member, Mathematics; William Rossi, Outside Member, English