Show that this countable collection is a basis for $\mathbb R^2$ Show that the countable collection of rectangles 

$\{ (a,b)\times (c,d) \mid a<b \text{ and } c<d, \text{ and } a,b,c,d \text{ are rational} \}$ 

is a topological basis for $\mathbb{R^2}$.
This a question from the book Topology by James Munkres. I am not able figure out. How to go about this question?
 A: Just for the sake of having an answer. (See http://meta.math.stackexchange.com/questions/1148/what-should-one-do-when-ones-question-has-been-answered-in-the-comments
or http://meta.math.stackexchange.com/questions/1559/dealing-with-answers-in-comments
or http://meta.math.stackexchange.com/questions/3138/unanswered-questions
)
If we already know that
$$\{(a,b)\times(c,d); a,b,c,d\in\mathbb R,a<b.c<d\}$$
is a basis, then it suffices to show that if $[x,y]\in(a,b)\times(c,d)$ then $x$ is contained in a rectangle with rational endpoints which is a subset of $(a,b)\times (c,d)$. (I am using $[x,y]$ for an ordered pair and $(a,b)$ for an open interval in this answer.)
We have $x\in(a,b)$, $y\in(c,d)$. Then there are rational numbers $a',b',c',d'$ such that $a'\in(a,x)$, $b'\in(b,x)$, $c'\in(c,y)$, $d'\in(y,d)$. Obviously
$$[x,y]\in (a',b')\times(c',d')\subset(a,b)\times(c,d).$$ 

Note that the proof is an easy generalization of the proof of an analogous result for the real line. I've copied here Henno Brandsma's answer from Topology Q+A
board.
In reply to "Countable basis of open intervals in R", posted by math
layman on Feb 3, 2005:

R is $(-\infty, +\infty)$, how to find a countable basis of
  open intervals so that for any open point x inside of an open set B,
  there is an open interval within the basis which contains this point.

The set of intervals with rational endpoints does the trick.
If $x$ is inside an open set $B$, then there is an open interval
$x \in (a,b) \subset B$.
But the interval $(a,x)$ contains a rational number $r_1$
and the interval $(x,b)$ contains a rational number $r_2$
and so $x \in (r_1, r_2) \in (a,b) \subset B$ and so
there is an interval with rational endpoints inside every open set.
Henno 
