Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover.

A strongly Lindelöf space is a topological space such that every open subspace is Lindelöf. Such spaces are also known as hereditarily Lindelöf spaces, because all subspaces of such a space are Lindelöf.

Lindelöf spaces are named for the Finnish mathematician Ernst Leonard Lindelöf.

In general, no implications hold (in either direction) between the Lindelöf property and other compactness properties, such as paracompactness. But by the Morita theorem, every regular Lindelöf space is paracompact.

Any second-countable space is a Lindelöf space, but not conversely. However, the matter is simpler for metric spaces. A metric space is Lindelöf if and only if it is separable, and 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.

A Lindelöf space is compact if and only if it is countably compact.

Any σ-compact space is Lindelöf.

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