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.
 A: 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 $F'$.  
A: The following short piece of lecture notes is almost entirely devoted to an answer of your question:
http://alpha.math.uga.edu/~pete/TopSection4.pdf
Unfortunately, for reasons that now escape me, I decided to break the file here, so the important remark that the statements "$\tau$ is the topology generated by the family of sets $\mathcal{F}$" and "The family of sets $\mathcal{F}$ is a subbase for the topology $\tau$" are equivalent does not appear until the beginning of the next little piece of notes:
http://alpha.math.uga.edu/~pete/TopSection5.pdf
A: It might also be useful to view your object as the ("projective") limit of the set topologized variously. The indexing is by the topologies, with the finer ones mapping to the coarser ones. A minor virtue of this is that it characterizes the result without necessarily giving a construction or description of the opens.
A: You can also do this.  The intersection of any family of topologies on a set is itself a topology.  The discrete topology (all subsets open) is a topology for any set.  So, take the set of all topologies containing the family of topologies (remember: the discrete one is here) and you will get a topology.  It will be the smallest topology containing all topologies in the family.
This is integral to the notion of weak topology.  The first answer has a second correct characterization of this topology.  See the book {\it Topology} by Wilansky.  
