The topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$. 
Let $\tau$ be to topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$.
How would you find the closure of $(0,1)$ in $\tau$?

I'm trying to find the smallest closed set containing $(0,1)$ in that topology but then I realised I don't fully understand what an 'open' interval is. Is an open interval in this topology one that is half open like in the sub-basis?

By definition of a sub-basis, a set is open iff it is a union of sets which are finite intersections of sets in the sub-basis. Here if you play around though, you can see that the intersection of two of these intervals is another such interval, or is empty.
You should look at basic open sets containing 0, here. Must they always intersect (0,1)? Draw a conclusion about 0 being in the closure or not. You should be able to spot the closure very quickly if you grasp this.
