Lemma 6.21.1 (008D)—The Stacks project

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Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.21: Continuous maps and sheaves
Lemma 6.21.1 (cite)

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Lemma 6.21.1. Let $f : X \to Y$ be a continuous map. Let $\mathcal{F}$ be a sheaf of sets on $X$. Then $f_*\mathcal{F}$ is a sheaf on $Y$.

Proof. This immediately follows from the fact that if $V = \bigcup V_ j$ is an open covering in $Y$, then $f^{-1}(V) = \bigcup f^{-1}(V_ j)$ is an open covering in $X$. $\square$

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