in may's "concise course" treatment of groupoid van kampen (
math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf page 17 (or p25 of the pdf)), it seems that the proof doesn't use connectedness at all. am i missing something? (i'd like to apply this to a cover of $S^1$ by two connected opens, but their intersection won't be connected. of course, certainly the conclusion is still true.)