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Lindelöf's lemma

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In mathematics, Lindelöf's lemma is a simple but useful lemma in topology on the real line, named for the Finnish mathematician Ernst Leonard Lindelöf.

Statement of the lemma

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Let the real line have its standard topology. Then every open subset of the real line is a countable union of open intervals.

Generalized Statement

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Lindelöf's lemma is also known as the statement that every open cover in a second-countable space has a countable subcover (Kelley 1955:49). This means that every second-countable space is also a Lindelöf space.

Proof of the generalized statement

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Let be a countable basis of . Consider an open cover, . To get prepared for the following deduction, we define two sets for convenience, , .

A straight-forward but essential observation is that, which is from the definition of base.[1] Therefore, we can get that,

where , and is therefore at most countable. Next, by construction, for each there is some such that . We can therefore write

completing the proof.

References

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  1. ^ Here, we use the definition of "base" in M.A.Armstrong, Basic Topology, chapter 2, §1, i.e. a collection of open sets such that every open set is a union of members of this collection.
  1. J.L. Kelley (1955), General Topology, van Nostrand.
  2. M.A. Armstrong (1983), Basic Topology, Springer.