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. (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,

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

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.

References[]

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

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