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.
Let the real line have its standard topology.
Then every open subset of the real line is a countable union of open intervals.
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.
be a countable basis of
Consider an open cover,
To get prepared for the following deduction, we define two sets for convenience,
β ∈
: β ⊂
A straight-forward but essential observation is that,
β ∈
β
which is from the definition of base.
β ∈
completing the proof.
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