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The wikipedia article for the real line says that the order topology and the metric topology of the reals are the same.

What is the explanation that these two topologies are in fact identical?

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up vote 4 down vote accepted

They both have the same basis: the family of all open intervals $(a,b)$, where $a$ and $b$ are real numbers with $a < b$.

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Whoops, duh, thanks. – James Pond Dec 10 '11 at 17:06
A bit more should be added: the open balls for the usual metric are intervals of that kind. – GEdgar Dec 10 '11 at 19:06

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