subbasis for a topology Munkres 
I have a question about the following definition: 
A subbasis $S$ for a topology on a set $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis $S$ is defined to be the collection $T$ of all unions of finite intersections of elements of $S$. 
If you have a subbasis $S$ for a topology $A$, then is the topology generated by $S$ necessarily also $A$? It seems like you could have many different subbases for $A$, but my intuition is that they might not all generate the same topology $A$. Is there something i'm missing? Thanks for any help/clarification.
Sincerely,
Vien
 A: Other definitions that are equivalent:
Let $X$ be a topological space with topology $\tau$. A subbase of $\tau$ is usually defined as a subcollection $\mathcal{B}$ of $\tau$ satisfying one of the two following equivalent conditions:


*

*The subcollection $\mathcal{B}$ generates the topology $\tau$.
[This means that T is the smallest topology containing B: any topology U on X containing B must also contain T.]

*The collection of open sets consisting of all finite intersections of elements of $\mathcal{B}$, together with the set $X$ and the empty set, forms a basis for $\tau$.
[This means that every non-empty proper open set in T can be written as a union of finite intersections of elements of $\mathcal{B}$. Explicitly, given a point x in a proper open set $U$, there are finitely many sets $S_1, \dots, S_n \in \mathcal{B}$, such that the intersection of these sets contains $x$ and is contained in $U$. (wiki)]
A collection of subsets of a topological space that is contained in a basis of the topology and can be completed to a basis when adding all finite intersections of the subsets. (wolfram mathworld)
