# 族生成的筛　`prop.sieve-generated-by-family`
命题：$R = \coprod_{i} \operatorname{im}(h_{f_{i}})$
layer 11 · 命题 · 层与拓扑 · 范畴论
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族 $(f_{i} : X_{i} \to X)_{i \in I}$ 生成的筛是

$$R \;=\; \coprod_{i \in I} \operatorname{im}\bigl(h_{f_{i}}\bigr),$$
> 陈述续见 [nodes/prop.sieve-generated-by-family.2.md](./prop.sieve-generated-by-family.2.md)


## 为什么成立（入边，证明在 proofs/）
- [`def.sieve`](../nodes/def.sieve.md) 筛：用到了定义 筛　[proofs/def-link.sievegen-sieve.md](../proofs/def-link.sievegen-sieve.md)

refs: Le Stum, An introduction to condensed mathematics, Exercise 3.10

> 说明见 `notes/prop.sieve-generated-by-family.md`

引用：`prop.sieve-generated-by-family`（内容 v0.9.0）· 条款见 [LICENSE.txt](../LICENSE.txt)