Past exam of the Probability, statistics and options II course of the University of Oxford / 2007 / E / 3 / c / Solution

True. The Hausdorff assumption makes $\{y\}$ closed in $Y$. Its inverse image under the continuous map $f$ is closed in the compact space $X$, and a closed subset of a compact space is compact. This is the compactness of the fiber with its subspace topology.