A perfect map $f$ is a closed continuous surjective function such that the preimage of every point is compact. One property of perfect maps is that if $f \, \colon \, X \to Y$ is perfect, and $Y$ is compact, then $X$ is compact too.
My question (rephrased): if $f$ is a continuous surjective function such that the preimage of every point is compact, and $Y$ is compact, does it follow that $X$ is compact?