Notes on the od-Lindel"of property

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).

💡 Research Summary

The paper introduces two new topological notions: od‑compactness and od‑Lindelöfness. A space is called od‑compact (respectively od‑Lindelöf) if every cover consisting solely of open dense subsets admits a finite (respectively countable) subcover. These definitions restrict the usual compactness and Lindelöf properties to a special class of covers, which leads to markedly different behaviour.

First, the authors provide very simple examples showing that both od‑compactness and od‑Lindelöfness are poorly behaved under finite or countable unions. For instance, two od‑compact spaces X and Y may each satisfy the finite‑subcover condition for open‑dense covers, yet their union X∪Y can fail to be od‑compact because a cover by open dense sets of the union need not restrict to open dense covers of the components. Analogous constructions demonstrate that a countable union of od‑Lindelöf spaces need not be od‑Lindelöf. These examples underline that the od‑properties are not closed under the usual set‑theoretic operations that preserve ordinary compactness or Lindelöfness.

The core of the paper then investigates the relationship between od‑compactness/od‑Lindelöfness and the classical notions of compactness, Lindelöfness, and linear Lindelöfness (the latter meaning that every open cover has a countable subcover which can be ordered linearly by inclusion). All results are proved under the T₁ separation axiom.

The first main theorem states: if a T₁ space X is od‑compact, then the subspace D consisting of all non‑isolated points of X is compact. The proof proceeds by observing that any open cover of D can be extended to an open dense cover of X by adding the isolated points as singleton open sets (which are dense in the subspace of isolated points). Since X is od‑compact, a finite subcover exists, and after discarding the isolated‑point pieces we obtain a finite subcover of D, establishing its compactness.

The second theorem shows: if a T₁ space X is od‑Lindelöf, then the set D of non‑isolated points is linearly Lindelöf. Here the authors construct, for each point of D, an open dense set that contains it and is minimal with respect to inclusion. The family of such minimal dense opens is linearly ordered by inclusion, and the od‑Lindelöf hypothesis guarantees a countable subfamily covering D. Hence D enjoys linear Lindelöfness, a weaker form of the usual Lindelöf property.

A further significant result concerns the additional hypothesis of local open Lindelöfness: a space is locally openly Lindelöf if every point has an open neighbourhood that is Lindelöf in the subspace topology. Under this assumption, od‑Lindelöfness actually implies full Lindelöfness. The argument uses the local Lindelöf neighbourhoods to refine any open dense cover into an ordinary open cover; the od‑Lindelöf condition then yields a countable subcover, which by the local Lindelöf property can be enlarged to a countable ordinary open cover of the whole space. Consequently, the space becomes Lindelöf in the classical sense.

The paper concludes with a diagram summarising the implications among the various properties (compact ⇐ od‑compact ⇒ compactness of non‑isolated set, Lindelöf ⇐ od‑Lindelöf ⇒ linear Lindelöfness of non‑isolated set, and the strengthening to Lindelöf under local open Lindelöfness). It also discusses several illustrative classes of spaces—such as metric spaces, spaces with dense isolated points, and spaces where every non‑isolated point has a countable local base—to show where the new notions coincide with or diverge from the classical ones.

Finally, the authors point out open problems: whether od‑compactness or od‑Lindelöfness is preserved under continuous images, subspaces, or product constructions; and how these properties interact with other separation axioms beyond T₁. The paper thus establishes od‑compactness and od‑Lindelöfness as distinct, nuanced concepts that enrich the landscape of covering properties in topology.