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{\bf Question}

Show that any non-empty open set in ${\bf R^2}$ (or ${\bf R^n}$)
can be expressed as the union of a countable collection of closed
rectangles.



\vspace{0.25in}

{\bf Answer}

Consider the collection ${\cal C}$ of all rectangles $R=\{{\bf
x}|a_r\leq x_r\leq b_r\}$ contained in $S$, an open set, where
$a_r$ and $b_r$ are rational.

Then ${\cal C}$ is countable and $\ds \bigcup_{\cal C}R\subseteq
S$.

If $x\epsilon S$ then there is a neighbourhood
$N_\epsilon(x)\subseteq S$

$N_\epsilon(x)$ contains a member of ${\cal C}$ containing $x$.

So $\ds x\epsilon\bigcup_{\cal C}R$, therefore $\ds
S=\bigcup_{\cal C}R$




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