What exactly is the topology generated by the union of a family of topologies? I recall reading somewhere that the supremum of a family of topologies on a set is simply the topology generated by the union of all topologies in the family. My question is, what does "generated" mean in this case? For instance, I know when dealing with filter bases, the generated filter is the set of all subsets containing some set in the filter base. But for topologies, how do we know what subsets to include in the generated topology? I would like to be able to visualize what kind of subsets are contained in this supremum. Thanks for any insight.

The topology generated by any family of subsets $F$ of a set $X$ is the intersection of all topologies on $X$ containing $F$; $F$ then forms a subbase for the resulting topology.  In practice this means that you take all finite intersections of the elements of $F$ to get a base $F'$, then take arbitrary unions of the elements of $F'$.  
