A pairwise disjoint family of intervals with nonempty interior in $\RR$ is countable

Solution

1. A family $\mathcal I$ of pairwise disjoint intervals in $\RR$, each with nonempty interior, is at most countable.

1.1. Each $I \in \mathcal I$ contains a rational number $q_I$.

$I$ contains a nonempty open interval, and $\QQ$ is dense in $\RR$.

1.2. The map $I \mapsto q_I$ is injective.

If $I \neq J$, then $I \cap J = \emptyset$, so $q_I \neq q_J$.

1.3. Q.E.D.

Steps 1 and 2 give an injection of $\mathcal I$ into the countable set $\QQ$.

2. The family of singletons $\theset{\theset{x} : x \in \RR}$ is an uncountable family of pairwise disjoint degenerate intervals $[x,x]$.

Distinct singletons are disjoint, and $\RR$ is uncountable.

3. Q.E.D.

Step 1 is the statement for intervals with nonempty interior, in particular for open intervals; step 2 shows that it fails for degenerate intervals.

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