----- > [!proposition] Proposition. ([[sheaf triviality can be tested on stalks]]) > A [[sheaf]] $\mathcal{F}$ on a [[topological space]] $X$ is the zero sheaf iff $\mathcal{F}_{p}=(0)$ for all $p \in X$. ^proposition > [!proof]+ Proof. ([[sheaf triviality can be tested on stalks]]) > One direction is immediate. For the other, suppose $\mathcal{F}_{p}=0$ for all $p \in X$. Let $U$ be an open subset of $X$, let $s \in \mathcal{F}(U)$. By hypothesis, for each $p \in U$, there is an open [[neighborhood]] $V_{p} \ni p$ for which $s |_{V_{p}}=0$. Since these $V_{p}$ form an [[cover|open cover]] of $U$, locality implies $s=0$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```