 Failure of nested compact image intersection without closed fibers

Take $X=[0,1]$ with its usual topology and $Y$ the Sierpiński space with open singleton $\{1\}$. The map $f(0)=0$, $f(x)=1$ for $x>0$ is continuous, since $f^{-1}(\{1\})=(0,1]$ is relatively open in $X$. The sets $K_n=[0,1/n]$ are nonempty, nested and compact, butThe fiber of $1$ is not closed, which is exactly where the compact-intersection proof fails. This distinguishes the valid Hausdorff-codomain theorem from a false extension to arbitrary codomains.