James Munkres defines a subbasis $\mathcal S$ for a topology on a set $X$ as a collection of subsets of $X$ whose union equals $X$. Then the topology generated by the subbasis $\mathcal S$ is defined to be the collection $\mathcal T$ of all unions of finite intersections of elements of $\mathcal S$. To prove that $\mathcal T$ is indeed a topology, one only needs to show that the collection $\mathcal B$ of all finite intersections of elements of $\mathcal S$ is a basis.
My question is what if $\mathcal S$ is a partition of $X$? Then any finite intersection of the elements of $\mathcal S$ is either $\emptyset$ or the elements of $\mathcal S$. It is clear that $\emptyset$ is not enough to generate a topology. Hence the topology is essentially generated by all the unions of the elements of $\mathcal S$, which is in fact the property of a basis rather than a subbasis. Hence, if my argument is correct, then I can conclude that if elements of $\mathcal S$ form a partition of $X$, then there is no difference between basis and subbasis. Is this right, please? Thank you!