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?

Hints:
(i) Show that $(-\infty,b)$ is in $\tau$ for every $b$.
(ii) Show that $[a,+\infty)$ is in $\tau$ for every $a$.
(iii) Deduce that $[a,b)$ is closed in $\tau$ for every $a\lt b$.
(iv) Show that $a$ is in the closure of $(a,b)$ with respect to $\tau$ for every $a\lt b$.
(v) Conclude that the closure of $(a,b)$ is $[a,b)$ for every $a\lt b$.
