Monotone covering properties and properties they imply

We study properties of spaces that were proven in an earlier paper of Chase and Gruenhage (2013) [5] to follow from monotonic metacompactness. We show that all of the results of that earlier paper that follow from the monotonic covering property follow just from these weaker properties. The results we obtain are either strengthenings of earlier results or are new even for the monotonic covering property. In particular, some corollaries are that monotonically countably metacompact spaces are hereditarily metacompact, and separable monotonically countably metacompact spaces are metrizable. It follows that the well-known examples of stratifiable spaces given by McAuley and Ceder are not monotonically countably metacompact; we show that they are not monotonically meta-Lindelöf either. Finally, we answer a question of Gartside and Moody by exhibiting a stratifiable space which is monotonically paracompact in the locally finite sense, but not monotonically paracompact in the sense of Gartside and Moody.