TITLE: When is the union of topologies a topology?
QUESTION [13 upvotes]: The union of two topologies on some set may or may not be a topology. When is it a topology?

REPLY [4 votes]: Years late to answer this but:
There is actually a problem in Munkres related to this: prove for a family of topologies $\{\tau_\alpha \}$ on a set $X$ that there is a unique smallest topology containing all of the topologies in the family.
If you set $S = \{ U : U \in \tau_\alpha \text{ for some } \alpha \}$ and consider the topology generated by $S$ (as a subbasis) I think this does the trick.  It also provides a less wordy sufficient and necessary condition to your question.