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.

Not to be confused with Lindelöf's theorem in complex analysis.

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

Consider ${\displaystyle B=\{\beta$ :\beta {\text{ is an element of basis}}\}} . Since ${\displaystyle X}$ is of a countable base, we consider it as ${\displaystyle B=\left\{\beta _{1},\beta _{2},\cdots \right\}}$ to infinity. Consider an open cover, ${\displaystyle {\mathcal {F}}=\bigcup _{\alpha }U_{\alpha }}$. To get prepared for the following deduction, we define two sets for convenience, ${\displaystyle B_{\alpha }:=\left\{\beta \in B:\beta \subset U_{\alpha }\right\}}$, ${\displaystyle B':=\bigcup _{\alpha }B_{\alpha }}$.

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

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

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

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

completing the proof.

References

1. J.L. Kelley (1955), General Topology, van Nostrand.
2. M.A. Armstrong (1983), Basic Topology, Springer.