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corrections from piazza

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Zheyuan Wu
2025-05-05 00:48:03 -05:00
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10
pages/Math4121/Exam_reviews/Math4121_Final.md
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@@ -42,7 +42,15 @@ $$
c_e(S)=c_i(S)+c_e(\partial S)
$$
Note the boundary of $S$ is defined as $\partial S=\overline{S}\setminus S$.
Note the boundary of $S$ is defined as $\partial S=\overline{S}\setminus S^\circ$ (corrected by Nathan Zhou).
> Some common notations for sets:
>
> $S^\circ$ is the interior of $S$. $S^\circ=\{x\in S| \exists \epsilon>0, B(x,\epsilon)\subset S\}$ (largest open set contained in $S$)
>
> $S'$ is the set of limit points of $S$ (derived set of $S$). $S'=\{x\in \mathbb{R}^n|\forall \epsilon>0, B(x,\epsilon)\setminus \{x\}\cap S\neq \emptyset\}$ (Topological definition of limit point).
>
> $\overline{S}$ is the closure of $S$. $\overline{S}=S\cup S'$ (smallest closed set containing $S$)
Equivalently, $\forall x\in \partial S$, $\forall \epsilon>0$, $\exists p\notin S$ and $q\notin S$ s.t. $d(x,p)<\epsilon$ and $d(x,q)<\epsilon$.