= Continuous images of nested compact intersections
{title2=$f(\bigcap_nK_n)=\bigcap_nf(K_n)$}
Let $X$ be a <Hausdorff space>, let $K_n$ be a nested sequence of nonempty compact subsets of $X$, and let $f:X\to Y$ be a <continuous map> into a <T1 space>. Then the displayed equality holds. The forward inclusion is true for every map. For the reverse, fix $y\in\bigcap_nf(K_n)$. Each $F_n=K_n\cap f^{-1}(\{y\})$ is nonempty and closed in the compact set $K_1$: the $K_n$ are closed by the Hausdorff assumption, while the fiber is closed because $Y$ is T1 and $f$ is continuous. The nested sets $F_n$ have the <finite intersection property>, so compactness gives a point in their intersection. Its image is $y$.
The proof only needs the relevant fibers to be closed in $K_1$, rather than all of $Y$ to be T1. A <Hausdorff space> as codomain is a familiar sufficient special case. Mere continuity into an arbitrary space does not suffice.
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