Lindelöf's lemma

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

Alternate version
Lindelöf's lemma is also known as the statement that every open cover in a second-countable space has a countable subcover. This means that every second countable space is also Lindelöf.