Exercise 16.6

Question

Show that the countable collection
\begin{gather*}
\left\{(a,b) 	imes (c,d) \mid 	ext{$a < b$ and $c < d$ and $a,b,c,d$ are rational}ight\}
\end{gather*}
is a basis for ${\mathbb{R}}^2$.

Dan Whitman · Accepted Answer

\begin{proof}
It was proven in Exercise~13.8 part~(a) that the set
\begin{gather*}
\mathcal{B} = \left\{(a,b) \mid 	ext{$a < b$, $a$ and $b$ rational}ight\}
\end{gather*}
is a basis for the standard topology on ${\mathbb{R}}$.
It then follows that
\begin{gather*}
\mathcal{D} = \left\{B 	imes C \mid B,C \in \mathcal{B}ight\}
\end{gather*}
is a basis for the standard topology on ${\mathbb{R}}^2$ by Theorem~15.1.
Clearly, we have
\begin{gather*}
\mathcal{D} = \left\{(a,b) 	imes (c,d) \mid 	ext{$a < b$ and $c < d$ and $a,b,c,d$ are rational}ight\} \,,
\end{gather*}
which shows the desired result.
\end{proof}

Exercise 16.6

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Exercise 16.6

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