---- > [!definition] Definition. ([[subpresheaf]]) > Let $X$ be a [[topological space]]. Let $\mathcal{F}$ be a [[presheaf]] on $X$. A **subpresheaf** $\mathcal{F}'\subset \mathcal{F}$ is a [[presheaf]] $\mathcal{F}'$ satisfying $\mathcal{F}'(U) \leq \mathcal{F}(U)$ for all open $U \subset X$, with restrictions $\rho'_{UV}$ induced by $\rho_{UV}$. (Here, $\leq$ represents structure-preserving inclusion in whatever the data [[category]] is.) ^definition > [!basicnonexample] Warning. > A [[subpresheaf]] of a [[sheaf]] need not be a [[sheaf]] (locality will be satisfied, by gluing may not be). The [[sheaf]] $\mathcal{C}$ of continuous real-valued functions on a [[topological space]] $X$ has a [[subpresheaf]] $\mathcal{C}_{\text{bounded}}$ given by $\mathcal{C}_{\text{bounded}}(U):=\{ f:U \to \mathbb{R} \ | \ f \text{ is continuous and bounded}\} \leq \mathcal{C}(U).$ This [[presheaf]] does not satisfy the gluing condition. ^nonexample ---- #### ---- #### 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 > ```