Lemma 5.21.6 (03HR)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 5: Topology
Section 5.21: Nowhere dense sets
Lemma 5.21.6 (cite)

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Lemma 5.21.6. Let $f : X \to Y$ be a continuous map of topological spaces. Let $T \subset Y$ be a subset. If $f$ is open and $T$ is a closed nowhere dense subset of $Y$ , then also $f^{-1}(T)$ is a closed nowhere dense subset of $X$ . If $f$ is surjective and open, then $T$ is closed nowhere dense if and only if $f^{-1}(T)$ is closed nowhere dense.

Proof. Omitted. (Hint: In the first case the interior of $f^{-1}(T)$ maps into the interior of $T$ , and in the second case the interior of $f^{-1}(T)$ maps onto the interior of $T$ .) $\square$

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