# 自由紧 Hausdorff 空间　`def.free-compact-hausdorff`
自由紧 Hausdorff 空间（Free Compact Hausdorff Space）
layer 16 · 定义 · 紧 Hausdorff 空间 · 拓扑学
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**自由紧 Hausdorff 空间**就是某个离散空间 $I$ 的 Stone–Čech 紧化 $F \cong \beta I$。称 $I$ 是 $F$ 的一组**基**：此时

$$\operatorname{Hom}_{\mathbf{CHaus}}(F, S) \;\cong\; \operatorname{Hom}_{\mathbf{Top}}(I, S) \;=\; S^{I}$$
> 陈述续见 [nodes/def.free-compact-hausdorff.2.md](./def.free-compact-hausdorff.2.md)


## 为什么成立（入边，证明在 proofs/）
- …另有入边，续页见 [nodes/def.free-compact-hausdorff.3.md](./def.free-compact-hausdorff.3.md)

## 它能推出什么 / 谁在用它
- …另有出边，续页见 [nodes/def.free-compact-hausdorff.4.md](./def.free-compact-hausdorff.4.md)

refs: Le Stum, An introduction to condensed mathematics, §2.1–2.2

> 说明见 `notes/def.free-compact-hausdorff.md`

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