Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 2, Problem 29

Problem

Prove that every open set in $\mathbb{R}^1$ is the union of an at most countable collection of disjoint segments.

Answer

Let $K_l:=\{[m,m+1/2^l)\;|\; m \in \mathbb{Z}/2^l\}$. Then $K_l$ is a collection of coutable disjoint segments with length $1/2^l$. For open set $G \subset \mathbb{R}^1$, define

$$S_0:=\{Q \in K_0\;|\; Q \subset G\}.$$

$$S_n:=\{Q \in K_n\;|\; Q \subset G, \text{$Q \not\subset Q'$ for some $Q' \in \bigcup_{i=1}^{n-1}K_i$}  \}.$$

$$S:=\bigcup_{n=1}^\infty S_n.$$

Then $S$ is at most countable collection of disjoint segments and $\bigcup_{Q \in S}Q \subset G$.

For $x \in G$, there exists $N \in \mathbb{N}$ such that $N_{1/2^N}(x) \subset G$. Then, we can find $m \in \mathbb{Z}/2^N$ such that $m \leq x$ and $m \in N_{1/2^N}(x)$. This means $x \in Q$ for some $Q \in \bigcup_{n=1}^N S_n$. Hence $G \subset \bigcup_{Q \in S}Q$.

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