# 点给出的拓扑斯态射　`prop.topos-morphism-point`
命题：$x : \mathbf{1} \to X$ 给出 $(x_{*}E)(U) = E$（$x \in U$）与 $x^{-1}\mathcal{F} = \mathcal{F}_{x}$
layer 18 · 命题 · 拓扑斯 · 范畴论
[在星图里打开](../#n=prop.topos-morphism-point)　·　[节点索引](../index.md)

设 $X$ 是拓扑空间、$x : \mathbf{1} \to X$ 是一个点。则它给出的拓扑斯态射满足

$$(x_{*}E)(U) = \begin{cases} E & x \in U,\\ \emptyset & x \notin U,\end{cases} \qquad x^{-1}\mathcal{F} = \mathcal{F}_{x}.$$
> 陈述续见 [nodes/prop.topos-morphism-point.2.md](./prop.topos-morphism-point.2.md)


## 为什么成立（入边，证明在 proofs/）
- …另有入边，续页见 [nodes/prop.topos-morphism-point.3.md](./prop.topos-morphism-point.3.md)

refs: Le Stum, An introduction to condensed mathematics, Exercise 3.56

> 说明见 `notes/prop.topos-morphism-point.md`

引用：`prop.topos-morphism-point`（内容 v0.9.0）· 条款见 [LICENSE.txt](../LICENSE.txt)