Show that the countable collection
\[ \ts{(a, b) \times (c, d) \mid a < b \text{ and } c < d, \text{ and } a, b, c, d \text{ rational}} \]
is a basis for \(\mathbb{R}^2\).
Solution
The collection is countable because it is indexed by a subset of \(\mathbb Q^4\).
Let \(U\subseteq\mathbb R^2\) be open and \((x,y)\in U\). Choose an ordinary open rectangle \[ (x-\varepsilon,x+\varepsilon)\times(y-\delta,y+\delta)\subseteq U. \] By density of \(\mathbb Q\), choose rationals \[ a<x<b,\qquad c<y<d \] with \[ (a,b)\subseteq(x-\varepsilon,x+\varepsilon),\qquad (c,d)\subseteq(y-\delta,y+\delta). \] Then \[ (x,y)\in(a,b)\times(c,d)\subseteq U. \] By the basis criterion, the stated countable collection is a basis for the standard topology on \(\mathbb R^2\).