Lindelöf's lemma
# 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.

## Generalized Statement

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

Let $B$ be a countable basis of $X$. Consider an open cover, ${\mathcal {F}}=\bigcup _{\alpha }U_{\alpha }$. To get prepared for the following deduction, we define two sets for convenience, $B_{\alpha }:=\left\{\beta \in B:\beta \subset U_{\alpha }\right\}$, $B':=\bigcup _{\alpha }B_{\alpha }$.

A straight-forward but essential observation is that, $U_{\alpha }=\bigcup _{\beta \in B_{\alpha }}\beta$ which is from the definition of base. Therefore, we can get that,

${\mathcal {F}}=\bigcup _{\alpha }U_{\alpha }=\bigcup _{\alpha }\bigcup _{\beta \in B_{\alpha }}\beta =\bigcup _{\beta \in B'}\beta$

where $B'\subset B$, and is therefore at most countable. Next, by construction, for each $\beta \in B'$ there is some $\delta _{\beta }$ such that $\beta \in U_{\delta _{\beta }}$. We can therefore write

${\mathcal {F}}=\bigcup _{\beta \in B'}U_{\delta _{\beta }}$

completing the proof.

