Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. This is a generalization of the more commonly used notion of compactness, which requires that the subcover be finite.

In general, no implications hold (in either direction) between the Lindelöf notion and other compactness notions, such as paracompact, which are discussed in the compactness page. Any second countable space is Lindelöf, but not conversely.

However, the matter is simpler for metric spaces. A metric space is Lindelöf if and only if it is separable if and only if it is second countable.

An open subspace of a Lindelöf space is not necessarily Lindelöf. However, a closed subspace must be Lindelöf.

Lindelöf is preserved by continuous maps. However, it is not necessarily preserved by products, not even by finite products.

References

• Michael Gemignani, Elementary Topology (ISBN 0-486-66522-4) (see especially section 7.2)
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology (ISBN 0-486-68735-X)