Open Sets with respect to certain intervals

Question

So I am trying to generally see results for open sets, for example:

What form do open sets take when open in the interval of the form $[a,b),$ $[a,b]$, $(a,b)$, $(a,b]$ in the usual topology?

For instance, in $[0,1],$ I now know that sets in the form $[0,a)$ are open in $[0,1]$ in the usual topology. But how would the open sets change when they are open in say $[0,1)$ in the usual topology?

Answer 1

The open sets in $[0,1)$ (besides the empty set) are the unions of $2$ types of intervals:

• those of the type $(a,b)$, with $0\leqslant a<b<1$;
• those of the type $[0,a)$, with $0<a<1$.