Chapter 2.13, Problem 8E

 (a) Apply Lemma 13.2 to show that the countable collection 

 
 $B=\left\{\left(a,b\right)|a<b,a\text{and}b\text{rational}\right\}$ 

 is a basis that generates the standard topology on 
 $ℝ$. 

(b) Show that the collection

$C=\left\{\left(a,b\right)|a<b,a\text{and}b\text{rational}\right\}$

is a basis that generates a topology different from the lower limit topology on $ℝ$.