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 is a topological space such that every subspace of it is Lindelöf. Such a space is sometimes called strongly Lindelöf, but confusingly that terminology is sometimes used with an altogether different meaning. The term hereditarily Lindelöf is more common and unambiguous. Lindelöf spaces are named after the Finnish mathematician Ernst Leonard Lindelöf. Every compact space, and more generally every σ-compact space, is Lindelöf. In particular, every countable space is Lindelöf. A Lindelöf space is compact if and only if it is countably compact. Every second-countable space is Lindelöf, but not conversely. For example, there are many compact spaces that are not second countable. A metric space is Lindelöf if and only if it is separable, and if and only if it is second-countable. Every regular Lindelöf space is normal. Every regular Lindelöf space is paracompact. A countable union of Lindelöf subspaces of a topological space is Lindelöf. Every closed subspace of a Lindelöf space is Lindelöf. Consequently, every Fσ set in a Lindelöf space is Lindelöf. Arbitrary subspaces of a Lindelöf space need not be Lindelöf. The continuous image of a Lindelöf space is Lindelöf. The product of a Lindelöf space and a compact space is Lindelöf. The product of a Lindelöf space and a σ-compact space is Lindelöf. This is a corollary to the previous property. The product of two Lindelöf spaces need not be Lindelöf. For example, the Sorgenfrey line is Lindelöf, but the Sorgenfrey plane is not Lindelöf. In a Lindelöf space, every locally finite family of nonempty subsets is at most countable. A space is hereditarily Lindelöf if and only if every open subspace of it is Lindelöf. Hereditarily Lindelöf spaces are closed under taking countable unions, subspaces, and continuous images.

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