Lindelöf's Lemma

Theorem
Let $C$ be a set of open real sets.

Let $S$ be a set that is covered by $C$.

Then there is a countable subset of $C$ that covers $S$.

Proof
Let $U = \displaystyle \bigcup_{O \mathop \in C} O$.

Let $x$ be an arbitrary point in $U$.

Since $U$ is the union of the sets in $C$, $x$ belongs to a set in $C$.

Name such a set $O_x$.

Since $O_x$ is open, $O_x$ contains an open interval $I_x$ that contains $x$.

By Between two Real Numbers exists Rational Number, two rational numbers exist between $x$ and each endpoint of $I_x$.

Form an open interval $R_x$ that has two such rational numbers as endpoints.

By Rational Numbers are Countably Infinite, we have that the rationals are countable.

By Cartesian Product of Countable Sets is Countable, a set consisting of the pair of two rational numbers is countable as the rationals are countable.

$R_x$ is defined from its two endpoints, which are rational numbers.

Therefore, the set $\left\{ {R_y: y \in U} \right\}$ is countable.

This allows us to define a different index $i$ for $R_x$ like this:


 * $R^i = R_x$

where


 * $i \in N$


 * $N \subseteq \N$

Note that:
 * $\left\{ {R^j: j \in N} \right\} = \left\{ {R_y: y \in U} \right\}$

Starting with $R^i$, we know that a $y$ in $U$ exists so that $R_y = R^i$.

Also, $R_y \subset I_y$ and $I_y \subset O_y$.

Define $O^i = O_y$.

We observe that $R^i \subset O^i$.

We define a mapping from $\left\{ {R^j: j \in N} \right\}$ to $\left\{ {O^j: j \in N} \right\}$ that sends $R^i$ to $O^i$.

By Image of Countable Set under Mapping is Countable, $\left\{ {O^j: j \in N} \right\}$ is countable since $\left\{ {R^j: j \in N} \right\}$ is countable.

We observe that $\left\{ {O^j: j \in N} \right\}$ is a countable subset of $C$.

Putting all this together:

Since $S$ is covered by $C$, we have $S \subset U$.

Also, $S \subset \displaystyle \bigcup_{i \mathop \in N} \left\{ {O^i} \right\}$ since:
 * $U = \displaystyle \bigcup_{i \mathop \in N} \left\{ {O^i} \right\}$

In other words, the collection $\left\{ {O^i: i \in N} \right\}$ covers $S$.