Monotonically monolithic spaces, Corson compacts, and D-spaces

Monotonically monolithic spaces were recently introduced by V.V. Tkachuk, and monotonically κ-monolithic spaces by O. Alas, V.V. Tkachuk, and R. Wilson. In this note we answer some of their questions by showing that monotonically ω-monolithic compact spaces must be Corson compact, yet there is a Corson compact space which is not monotonically ω-monolithic. We obtain a characterization of monotonic monolithity that shows its close relationship to condition (G) of P. Collins and R. Roscoe. We also give an easy proof of Tkachukʼs result that monotonically monolithic spaces are hereditarily D-spaces by applying a result involving nearly good relations, and finally, we generalize nearly good to nearly OK to similarly obtain L.-X. Pengʼs result that weakly monotonically monolithic spaces are D-spaces.