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How to prove that the Sorgenfrey line is hereditarily separable?

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Hint: Let $S$ be a subset of the Sorgenfrey line. From each interval of the form $[q,r)$ in the original line, where $q,r \in \mathbb{Q}$, pick one point from $S$ if possible. Then characterize the points in $S$ that are not limits of the points you just chose.

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This has a proof that for any ordered space separable implies hereditarily separable. And the Sorgenfrey line is a subspace of a separable ordered space (e.g. the double arrow).

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