fix typos
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Trance-0
2025-12-11 14:14:01 -06:00
parent c42e7e6489
commit 7abc8d7e80
8 changed files with 751 additions and 9 deletions

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@@ -98,7 +98,7 @@ A $T_0$ space is regular if for any $x\in X$ and any close set $A\subseteq X$ su
A $T_0$ space is normal if for any disjoint closed sets, $A,B\subseteq X$, there are **disjoint open sets** $U,V$ such that $A\subseteq U$ and $B\subseteq V$.
<details>
<summary></summary>
<summary>Finer topology may not be normal</summary>
Let $\mathbb{R}_K$ be the topology on $\mathbb{R}$ generated by the basis:
@@ -106,7 +106,7 @@ $$
\mathcal{B}=\{(a,b)\mid a,b\in \mathbb{R},a<b\}\cup \{(a,b)-K\mid a,b\in \mathbb{R},a<b\}
$$
where $K=\coloneqq \{\frac{1}{n}\mid n\in \mathbb{N}\}$.
where $K\coloneqq \{\frac{1}{n}\mid n\in \mathbb{N}\}$.
**This is finer than the standard topology** on $\mathbb{R}$.