  A space is called linearly H-closed iff any chain cover possesses a dense
member. This property lies strictly between feeble compactness and
H-closedness. While regular H-closed spaces are compact, there are linearly
H-closed spaces which are even collectionwise normal and Fr\'echet-Urysohn. We
give examples in other classes, and ask whether there is a first countable
normal linearly H-closed non-compact space in ZFC. We show that PFA implies a
negative answer if the space is moreover either locally separable or locally
compact and locally ccc. Ostaszewski space (built with $\diamondsuit$) is an
example which is even perfectly normal. We also investigate Menger-like
properties for the class of od-covers, that is, covers whose members are open
and dense.

General Topology