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I could get the proof for characterization of open sets in $\mathbb{R}$ from the book by NL Carothers (Theorem 4.6). However, I could not extend it to higher dimensions. Could any point me to a reference (a text book would be great) or answer this question?

Any help is much appreciated.

Thanks, Phanindra

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    I cannot access math.stackexchange.com (this website) $f$rom my work place. I am not sure where to post this problem. Any help?2011-08-06

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Yes; $\mathbb R^n$ is second-countable, meaning that any open set in it is the countable union of many intervals (although you cannot guarantee that these sets will be disjoint), where here intervals are boxes $a_i; for $i=1,2,..,n $. This is true because $\mathbb R^n$ is second-countable, and has a basis of open intervals centered at points of $\mathbb Q^n$, with rational "length". So any proof of the second countability should do.

By definition of basis, every open subset of $\mathbb R^n$ is then the union of countably-many open basic open sets ; open boxes.

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    I apologize if the question was not clear. I wanted to know if every open set in \mathbb{R}^m, m>1 can be written as a union of countable disjoint open balls. Thanks all for replies.2011-08-04