Here is a Proposition from my real analysis book.

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> **Proposition.** Suppose $G \subset \mathbb{R}$ is open. Then $G$ can be written as the countable union of disjoint open intervals.
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The proof begins with the following.

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> Let $G$ be an open subset of the real line and for each $x \in G$, let$$A\_x = \inf\{a : \text{there exists }b\text{ such that }x \in (a, b) \subset G\}$$and$$B\_x = \sup\{d : \text{there exists }c \text{ such that }x \in (c, d) \subset G\}.$$Let $I\_x = (A\_x, B\_x)$.
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I just realized I don't really get what this beginning part is saying. Can somebody explain it to me/add some more detail? Thanks in advance!