# path connectedness

Try the map $p:[0,1]\to X,$ $p(t)=a$ if $t\ne1$ and $p(1)=b.$
Then $p^{-1}(\O)=\O,$ $p^{-1}(\{a\})=[0,\,1),$ $p^{-1}(X)=[0,\,1]$ which are all open in $[0,\,1].$