I am reading a proof of The Van Kampen Theorem from "Topology: J. R. Munkres, second edition", section - 70, page no - 426. In the hypothesis of the theorem, we assume that the space $X=U \cup V$ where $U$ and $V$ are open in $X$; and $U, V,$ and $ U\cap V$ are path connected. I have gone through the the proof (as given in the text) but I am not able to figure out where the assumption that $U$, $V$ are being open is being used.
Thanks in advance!