Lindelöf's Lemma
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Theorem
Let $C$ be a set of open real sets.
Let $S \subseteq \R$ be a subset of the real numbers that is covered by $C$.
Then there exists a countable subset of $C$ that covers $S$.
Proof
Lemma
Let $C$ be a set of open real sets.
Then there is a countable subset $D$ of $C$ such that:
- $\ds \bigcup_{O \mathop \in D} O = \bigcup_{O \mathop \in C} O$
$\Box$
We have that $S$ is covered by $C$.
This means that $S$ is a subset of $\ds \bigcup_{O \mathop \in C} O$.
From the lemma:
- $\ds \bigcup_{O \mathop \in D} O = \bigcup_{O \mathop \in C} O$
where $D$ is a countable subset of $C$.
Hence $S$ is also a subset of $\ds \bigcup_{O \mathop \in D} O$.
In other words, $S$ is covered by $D$.
That is, $S$ is covered by a countable subset of $C$.
Hence the result.
$\blacksquare$
Source of Name
This entry was named for Ernst Leonard Lindelöf.