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 X^{1}.

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 left^{2} as a base. **L** is Lindelöf, but **L x L** is not.

^{1}That is, every cover of `X` consisting of open sets has a subset that is countable, and also covers `X`.

^{2}That 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.