Solution
= Solution
\b[False.] Take $X=[-1,1]$, $Y=[0,1]$ and $f(x)=x^2$, with their usual topologies. The domain is <compact> and <connected>, the codomain is <Hausdorff>, and the map is continuous and onto. But the fibre over one is $\{-1,1\}$, a disconnected two-point subspace.