Need explanation: Subbase of a Topology example in Royden I'm currently using Royden's Real Analysis for a course in real variables, and I was reading Chapter 11, which is on Topological Spaces.
It gives the following definition for a subbase:

Definition: For a topological space $(X, \mathcal{T})$, a subcollection $\mathcal{S}$ of $\mathcal{T}$ that covers $X$ is a subbase for the topology $\mathcal{T}$ provided intersections of finite subcollections of $\mathcal{S}$ are a base of $\mathcal{T}$.

Now, it gives the following example of a subbase, which I'm having a bit of trouble wrapping my head around:

Consider a closed, bounded interval $[a,b]$ as a topological space with the topology it inherits from $\mathbb{R}$. This space has a subbase consisting of intervals of the type $[a,c)$ or $(c,b]$ for $a<c<b$.

From what I understand, the "topology it inherits from $\mathbb{R}$" is the standard topology on $\mathbb{R}$, which has as its base elements the open intervals $(a,b)$. Intersections of half-open intervals of the same type (i.e., two intervals of the type $[a,c)$) result in more half-open intervals, not open ones. So, I don't understand how the half-open intervals serve as a subbase for the standard topology on $[a,b]$. Could somebody please explain this to me? 
Thanks.
 A: The open intervals are a base of $\mathbb{R}$. The intersections of open intervals with $[a,b]$ are a base of $[a,b]$.  These have three forms, $[a,c)$, $(c,b]$, and $(c,d)$ with $a<c<d<b$.  
Intersections of the first and second type are already in our candidate subbase. Furthermore we have $(c,d)=[a,d)\cap(c,b]$ whenever $a<c<d<b$, i.e., $(c,d)$ is the intersection of a finite number of elements (to be precise, two) of our candidate subbase.  Thus finite intersections of elements of our candidate subbase give all intersections of open intervals with $[a,b]$, and so the candidate subbase is in fact a subbase.
A: Standard topology of $[a, b]$ is defined as follows:
$$\mathcal{T}_{[a, b]} = \left\{[a, b] \cap \mathcal{O} : \mathcal{O}\textrm{ open in }\mathbb{R}\right\},$$
This contains more than open intervals. In particular, $[a, b]$ itself is contained here (this is in fact a requirement when you think about it: topology on a set $X$ always contains the set itself), along with half open intervals $[a, c)$ and $(c, b]$.
