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A space is od-compact (resp. od-Lindel\"of) provided any cover by open densesets has a finite (resp. countable) subcover. We first show with simpleexamples that these properties behave quite poorly under finite or countableunions. We then investigate the relations between Lindel\"ofness,od-Lindel\"ofness and linear Lindel\"ofness (and similar relations with`compact'). We prove in particular that if a $T_1$ space is od-compact, thenthe subset of its non-isolated points is compact. If a $T_1$ space isod-Lindel\"of, we only get that the subset of its non-isolated points islinearly Lindel\"of. Though, Lindel\"ofness follows if the space is moreoverlocally openly Lindel\"of (i.e. each point has an open Lindel\"ofneighborhood).

Notes on the od-Lindel\"of property