Let $B = \{ \mathbb{R} \} \cup \{ (a,b) \cap\mathbb {Q} \ ,\ a\lt b \ ,\ a,b \in\mathbb{Q}\}$
Thus, a set $V \in B$ if it is either equal to $\mathbb{R}$ or if it is in the intersection of $\mathbb{Q}$ with an open interval with rational endpoints.
a) Show that $B$ forms a base for a topology $T$ on $\mathbb{R}$.
b) Show that this topology does not contain nor be contained in the usual topology of $\mathbb{R}$.
c) Is $(\mathbb{R}, T)$ compact? Is it separable? Is it connected?
I think have solved a)
All irrational numbers are contained in the set $\mathbb{R}$ and all the other numbers are contained in some intersection of an open set with $\mathbb{Q}$. So each point $x$, is contained in some set $v \in B$. Also if a point $x$ belongs to the intersection of two base sets then I can find some set containing $x$ that is contained in that intersection.