Not long after Penrose's proof, Dennis Gannon(1975) and C. W. Lee (1976) independently rederived Penrose's result, not using the "trapped surface" conditions of Penrose, but rather using purely topological assumptions. This work culminated in a theorem by Friedman, Schleich, & Witt (1995)called "the topological singularity theorem," which has a striking interpretation. In their own words: "general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it." Gannon's and FSW's papers will be the subject of this lecture. As before, I will distribute lecture notes those who are interested.