Countably compact inverse semigroups and Nyikos' problem

A regular separable first-countable countably compact space is called a Nyikos space. In this paper, we give a partial solution to an old problem of Nyikos by showing that each locally compact Nyikos inverse topological semigroup is compact. Also, we show that a topological semigroup $S$ that contains a dense inverse subsemigroup is a topological inverse semigroup, provided (i) $S$ is compact, or (ii) $S$ is countably compact and sequential. The latter result solves a problem of Banakh and Pastukhova and provides the automatic continuity of inversion in certain compact-like inverse semigroups.