I realise this is a very easy question. But it seems to me that from the standard (open sets) definition of a topology ($X$ and $\varnothing$ open, closed under arbitrary unions and finite intersections) that the collection $\{\Bbb R, [0, 1], \varnothing\}$ forms a topology on $\mathbb R$. Why is this not the case?
Also, can someone point me in the direction of a good proof for why an open set in a topology does not contain it's boundary points (a proof from the axioms of a topology not from the concept of an open set in a metric space).