Could someone please show that an open ball is open where the definition of "open" is: A set is open if for each $x$ in $U$ there is an open rectangle $A$ such that $x$ in $A$ is contained in $U$. Where an open rectangle is $(a_1,b_1)\times\ldots\times(a_n,b_n)$. I also realize that one can use rectangles or balls, but I would like to see the proof using rectangles, as this is the definition used in Spivak's calculus on manifolds. So for example, the solution located at An open ball is an open set is unacceptable.
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