I'm trying to prove that all continuous maps of pairs $f:([-1,1], \{-1,1\})\to (\{-1,1\},\{-1,1\})$ are constant, and I've almost got a working argument, but it reduces down to the following situation:
Since $\{-1\}$ and $\{1\}$ are both open in $\{-1,1\}$, so too must be $f^{-1}(\{-1\})$ and $f^{-1}(\{1\})$. By definition of $f$, their union must equal all of $[-1,1]$.
So now I have a pair of disjoint open subsets of $[-1,1]$ whose union is all of $[-1,1]$. This is impossible, and easy to prove if they are both basic open sets (but I can't assume that).
Any suggestions?