Lindelöf's lemma

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.

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.

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