Notes on the od-Lindel\"of property
Abstract
A space is od-compact (resp. od-Lindel\"of) 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 Lindel\"ofness, od-Lindel\"ofness and linear Lindel\"ofness (and similar relations with `compact'). We prove in particular that if a space is od-compact, then the subset of its non-isolated points is compact. If a space is od-Lindel\"of, we only get that the subset of its non-isolated points is linearly Lindel\"of. Though, Lindel\"ofness follows if the space is moreover locally openly Lindel\"of (i.e. each point has an open Lindel\"of neighborhood).
Cite
@article{arxiv.1206.0722,
title = {Notes on the od-Lindel\"of property},
author = {Mathieu Baillif},
journal= {arXiv preprint arXiv:1206.0722},
year = {2015}
}
Comments
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