Lindelöf's lemma

Every second-countable topological space has the Lindelöf property: every open cover contains a countable subcover.

Second-countable space: A topological space with a countable base: every open set is a union of base elements. Its countable base supplies the sets used to extract a subcover.
Topology: The study of open sets and the structures they define on spaces. The lemma is a theorem about open sets and topological spaces.
Separable space: A topological space containing a countable dense subset. Every second-countable space is separable, giving another consequence of a countable base.
Compactness (topology): The property that every open cover has a finite subcover. Compactness guarantees a finite subcover, while this lemma guarantees a countable one.
Countable base: A countable collection of open sets such that every open set is a union of its members. For each base element lying inside a cover member, choose one such member.
Basis (topology): A collection of open sets from which every open set in a topology can be formed by unions. Second-countability means that some basis for the topology is countable.
Second-countable implies Lindelöf: The theorem that every second-countable topological space is Lindelöf. This implication is the lemma’s direct conclusion.
Paracompact space: A topological space in which every open cover has a locally finite open refinement. Paracompactness controls refinements rather than merely selecting countably many cover members.
Open cover: A collection of open sets whose union contains the entire space. The lemma starts with an arbitrary open cover.
Open set: A member of a topology, describing a set treated as open within a space. Both basis elements and cover members must be open.

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