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Ok, so I know open sets of the real line are the

**countable**union of disjoint open intervals (or open balls). Does this in any way extent to R^n? Say, any open set in R^n is the countable union of open rectangles or balls? I ask because I was reading some proof, and at a crucial step they use the fact that open sets in R^n can be expressed as the

**countable**union of open

*rectangles*, and I have no idea where this comes from! It doesn't even seem plausible to me, if one considers the open ball in the plane-- how would you describe

*that*as the union of countably many open rectangles?

I know every open set is the union of open balls, and maybe the Heine-Borel theorem comes in somewhere...but I'm just lost.

Any help?

Thanks.