A note on monotone covering properties

In this note, we show that a monotonically normal space that is monotonically countably metacompact (monotonically meta-Lindelöf) must be hereditarily paracompact. This answers a question of H.R. Bennett, K.P. Hart and D.J. Lutzer. We also show that any compact monotonically meta-Lindelöf T 2 -space is first countable. In the last part of the note, we point out that there is a gap in Proposition 3.8 which appears in [H.R. Bennett, K.P. Hart, D.J. Lutzer, A note on monotonically metacompact spaces, Topology Appl. 157 (2) (2010) 456–465]. We finally give a detailed proof of how to overcome the gap.