Notes on the od-Lindel\"of property

TL;DR

This paper explores the properties of od-compact and od-Lindel"of spaces, examining their behavior under unions and their relations to other Lindel"ofness concepts, with results linking these properties to the structure of non-isolated points.

Contribution

It provides new insights into od-Lindel"of spaces, showing how these properties relate to classical Lindel"ofness and the structure of non-isolated points, including conditions for Lindel"ofness.

Findings

od-compact spaces have compact non-isolated points

od-Lindel"of spaces have linearly Lindel"of non-isolated points

Lindel"ofness follows in locally openly Lindel"of spaces

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 $T_1$ space is od-compact, then the subset of its non-isolated points is compact. If a $T_1$ 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).