Prove that the unit open ball in $\mathbb{R}^2$ cannot be expressed as a countable disjoint union of open rectangles.
Open rectangles in $\mathbb{R}^2$ are subsets of the form $(a,b)\times(c,d)$.
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[Math] Prove that the unit open ball in $\mathbb{R}^2$ cannot be expressed as a countable disjoint union of open rectangles.
general-topologyreal-analysis
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Best Answer
Suppose that it is possible to cover unit ball the by a sequence of disjoint open sets $U_1,U_2,U_3,U_4,U_5...$. Let U be $U_1$ and V be the union of $U_2,U_3,U_4,U_5...$. It follows easily that U,V are disjoint open sets and the unit open ball is contained in the union of U,V. This contradicts the fact that the unit open ball in $R^2$ is connected