Munkres provides the following definition of subbasis. A few threads seem to have touched on this definition but I haven't reached clarification yet.
A subbasis $S$ for a topology $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis $S$ is defined to be the collection $\mathcal{T}$ of all unions of finite intersections of elements of $S$
My understanding is that the first sentence implies that $S$ is closed under finite intersection, otherwise the union of its sets would not necessarily be a topology.
But then the second sentence adds that a topology generated by a subbasis is generated by finite intersections of elements of $S$, suggesting that $S$ may not be closed under intersection.
I would just like to clarify that a subbasis is indeed, always, closed under finite intersections, and that the 'finite intersections' stated in the second sentence of the definition is superfluous.
EDIT Sorry, it was a silly misunderstanding For some reason I was thinking of the first sentence as discussing the construction of the 'topology' that is the set of subsets. But I understand now it is just saying that $S$ covers the underlying set. Indeed if the union of elements of $S$ were a 'topology', the elements of $S$ would seem to need to be sets of subsets of $X$.
No, a subbasis need not be closed under finite intersections. He does not say in the first sentence that the union of the subbasis elements is a topology; he says only that the union of the subbasis elements is the entire space.
Look at the simplest example. In $\Bbb R$, the collection of open intervals $(-\infty,b)$ and $(a,\infty)$, as $a,b$ range over $\Bbb R$, is a subbasis for the usual topology. This collection is obviously not closed under finite intersections!