The Nyikos problem asks whether ZFC implies the existence of a regular, separable, first-countable, countably compact, non-compact space.
In this talk, we discuss the Nyikos problem and several related questions. We present examples of countably compact, non-compact spaces with interesting additional properties, and we find sufficient conditions that guarantee compactness of Nyikos topological semigroups.
This is the first part of a series of 3 talks; the next 2 talks will take place on December 11 and 18.