A theorem of real analysis states that any open set in [itex]\Re^{n}[/itex] can be written as the countable union of nonoverlapping intervals, where "interval" means a parallelopiped in [itex]\Re^{n}[/itex], and nonoverlapping means the interiors of the intervals are disjoint. Well, what about something as simple as an open ball in [itex]\Re^{2}[/itex] or [itex]\Re^{3}[/itex]? Intuitively, I can't visualize how non-overlapping rectangles, even a countably infinite number of them, could ever make a circle. If you could write it as such a union, then just pick a point on the circumference of the circle: it is either a corner of a rectangle or on the side of a rectangle. Either way, it would not look like a circle near that point.(adsbygoogle = window.adsbygoogle || []).push({});

Any help would be greatly appreciated.

Thank You in Advance.

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# Can open sets be written as unions of intervals?

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