A topological space that possesses a weaker form of compactness. A space X is Lindelöf iff every open cover of the
space contains a countable subcover of X1.
This property is important because the Lindelöf Theorem (proved by Finnish mathematician Ernst Leonard Lindelöf in 1903)
Every second countable space is a Lindelöf space.
provides a connection between the separability properties of topological spaces and compactness properties. For example, a sequentially compact space
is compact iff it is also Lindelöf.
Since the Euclidean topology on the real numbers is second countable, it is also Lindelöf as a result of the Lindelöf Theorem.
The Lindelöf property is not as well-behaved as the other separability and compactness properties for topological spaces:
- Given a Lindelöf space, only closed subspaces are necesarily Lindelöf.
- Product spaces of a Lindelöf space are not necessarly Lindelof. For example, consider the topology L with all real intervals closed on the left2 as a base. L is Lindelöf, but L x L is not.
1That is, every cover of X consisting of open sets has a subset that is countable, and also covers X.
2That is, for every a, b ∈ R, { r | a <= r < b } is open in
L.
Source: John Greever, Theory and Examples of Point-set Topology, Brooks/Cole Publishers, Belmont, California, 1969.