TITLE: A subbasis $S$ for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$.
QUESTION [7 upvotes]: Munkres definition says the following:

A subbasis $S$ for a topology on $X$ 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 $S$.

Now, a collection of subsets of $X$ whose union equals $X$ could be made up of a collection of disjoint subsets of $X$.  In that event, the collection $\mathcal T$ as defined above is an empty set?  How does it generate the topology then?

REPLY [6 votes]: One finite intersection of the sets is the original sets themselves...
Consider for instance the discrete topology $\tau$ on a set $X$.  Then the singletons $\{x\}$ would be a subbasis... Every subset is then open, as a union of these singletons,  which are intersections of a single set with itself (the trivial intersection,  so to speak).