Serhii Bardyla, Around Nyikos problem

A Nyikos problem asks whether ZFC implies the existence of a regular separable first-countable countably compact non-compact space? Due to its profound relation to Set Theory, this problem is listed among 20 central problems in Set-theoretic Topology by Hrušak and Moore. In this talk, we shall discuss questions closely related to Nyikos problem, and present countably compact non-compact spaces, which still possess some nice properties. The existence of some of our examples requires additional axioms. Moreover, some of our topological results are equiconsistent with the equalities between certain cardinal characteristics of the continuum. Also, we consider Nyikos problem in the class of topological inverse semigroups and find a ZFC solution in a locally compact case. The proof of this solution is partly based on the generalization of a result of Gutik, Pagon and Repovš. (For more background information, see Problem 1 in https://people.math.sc.edu/nyikos/1000.pdf)