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.