If $\mathbb{R}$ has the topology consisting of sets $A$ such that $\mathbb{R}-A$ is either countable or all of $\mathbb{R}$, is $[0,1]$ a compact subspace?
I've found the example $A_n = [0,1]-\{\frac{1}{n},\frac{1}{n+1},\cdots\}$ but I don't know how it answers the question. Why this set cannot be covered by finitely many open subspaces?