Notes on linearly H-closed spaces and od-selection principles

Abstract

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 Fréchet-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 $♢$) 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.

Corrected version including many remarks by the referee. In particular, some results due to him/her are included

Cited by in corpus (1)

• Three small results on normal first countable linearly H-closed spaces