Lindelöf

Lindelöf refers to a concept in topology, specifically a property of topological spaces. A space is called Lindelöf if every open cover of the space has a countable subcover. The term honors a mathematician associated with the study of this property.

Basic relations and examples: Every compact space is Lindelöf, since every open cover admits a finite subcover, which is in particular a countable one. The property is preserved under taking continuous images: if X is Lindelöf and f: X → Y is continuous, then f(X) is Lindelöf. Closed subspaces of a Lindelöf space are Lindelöf as well; a closed subspace of a Lindelöf space inherits the Lindelöf property. By contrast, open subspaces need not be Lindelöf.

Products and limitations: The Lindelöf property is not in general preserved under taking products. A classical counterexample is provided by the Sorgenfrey line: it is Lindelöf, but its square is not Lindelöf. Nevertheless, the product of a Lindelöf space with a compact space remains Lindelöf.

Special cases and terminology: In familiar metric spaces, the Lindelöf condition is often easy to verify, and spaces such as the real line with its usual topology are Lindelöf without being compact. The concept has various related notions, including local Lindelöf properties and its role in covering properties and dimension theory.