Notes on the od-Lindelöf property

Abstract

A space is od-compact (resp. od-Lindelöf) provided any cover by open dense sets has a finite (resp. countable) subcover. We first show with simple examples that these properties behave quite poorly under finite or countable unions. We then investigate the relations between Lindelöfness, od-Lindelöfness and linear Lindelöfness (and similar relations with `compact'). We prove in particular that if a $T_1$ space is od-compact, then the subset of its non-isolated points is compact. If a $T_1$ space is od-Lindelöf, we only get that the subset of its non-isolated points is linearly Lindelöf. Though, Lindelöfness follows if the space is moreover locally openly Lindelöf (i.e. each point has an open Lindelöf neighborhood).

13 pages, one figure. V2: We added a note concerning the fact that Mills and Wattel had proved a more general result in 1979. The author was unaware of it at the time of publication

References in corpus (2)

• Various topologies on trees
• Metrisability of Manifolds

Cited by in corpus (1)

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