Consider $\mathscr{B}=\{\{q\}:q\in\mathbb{Q}\cap[0,1]\}\cup\{]a,b[:0\leqslant a<b\leqslant 1\}\}$ Prove that $\mathscr{B}$ is a base for the topology $\tau$ in $[0,1]$ and show $([0,1],\tau)$ is compact.
Proving it is basis of topology I wrote it down on my notebook but it took me a while to deal with all the cases of intersection. I proved the union of all the open sets of the basis equals $[0,1]$ and I intersect two arbitray open sets of the following kind ${q_i}\cup]a_i,b_i[$ and $q_j\cup ]a_j,b_j[$ which gave me 8 cases. When I tried to prove $[0,1]$ to be a compact I thought the basis at [0,1] ,since it would contain the interval and the rationals are dense in $\mathbb{R}$, I thoughtit would be the same basis as the Euclidian basis. That would allow me to use Heine-Borel Theorem and finally prove $[0,1]$ is compact.
Question:
I am aware my answer is imprecise however it is imprecise as my ideas regarding the problem. I guess it is wrong. How should I then solve the exercise?
The showing a base part is not very hard: they clearly cover $[0,1]$ and any two basic subsets that intersect have their intersection in the base again (open intervals or singleton rationals). That makes the check pretty easy.
It seems to me that a set like $C=\{\frac{1}{n} \mid n \ge 1\}$ is actually closed and discrete in this topology, so that would make the topology non-compact.
It certainly is not the Euclidean topology on $[0,1]$, as it has plenty of isolated points; a dense set of them. It's also not connected for that reason.