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\begin{document}
\parindent=0cm
\section*{V.1 Simplicial complex in $\mathbb{R}^N$}

\begin{defn}\textit{
$\{a_0,\cdot \cdot \cdot ,a_n\}\subset \mathbb{R}^N$ is
geometrically independent(or affinely independent) if
$a_1-a_0,\cdot \cdot \cdot, a_n-a_0$ are linearly independent.}
\end{defn}

{\bf Note.} geometrically independent$\Leftrightarrow$
$\displaystyle{\sum_{i=0}^{n}t_i a_i}=0\,\,\,\,$ with
$\,\,\displaystyle{\sum_{i=0}^{n}t_i}=0$ $\Rightarrow t_0=\cdot
\cdot \cdot = t_n=0$.

Affine independence is a notion in affine space, i.e., invariant
under affine transformations.

\begin{defn}\textit{(n-simplex)\\
$\{a_0,\cdot \cdot \cdot a_n\}\subset \mathbb{R}^N$ 가
geometrically
independent하다고 하자. 이 때,\\
$\sigma=<a_0,\cdot \cdot \cdot ,a_n>=\{x\in\mathbb{R}^n |
x=\displaystyle{\sum_{i=0}^{n}t_i a_i\,\,,\,\, t_i\geq 0
\,\,\,and\,\,\, \sum_{i=0}^{n}t_i=1 }\}$ \\
=convex hull of $\{a_0,\cdot \cdot \cdot a_n\}$ \\
=n-simplex spanned by $\{a_0,\cdot \cdot \cdot,a_n\}$}라 정의한다.\\
\end{defn}

{\bf Remark}\\
{\bf (1)} $t_i=t_i(x)$ for $x\in \sigma$ is uniquely
determined and called a \textit{barycentric coordinate} of $x$,
and $t_i$ is a continuous function of $x\in \sigma$.

{\bf (2)} $a_i$=vertex of $\sigma$, n=dim$\sigma$일 때 a simplex
spanned by a subset of $\{a_0,\cdot \cdot \cdot a_n\}$ is called a
\textit{face} of $\sigma$.

$\overset{\circ}{\sigma}:=int(
\sigma)=\{x\in\sigma\,|\,\,t_i(x)>0\,,\,t_i=0 ,\cdot \cdot \cdot,n
\}$.

$\partial \sigma$:=boundary of
$\sigma=\sigma-\overset{\circ}{\sigma}=\{x\in\sigma\,|\,\,t_i(x)=0,\,\,for\,\,
some \,\,i\}$

{\bf (3)} $\forall x\in \sigma$, $\exists!$ face $\tau$ of
$\sigma$(denoted by $\,\tau<\sigma$) such that $x\in
\overset{\circ}{\tau}$.Indeed,

$\tau=<a_{i_0},\cdot \cdot \cdot,a_{i_k}\,|\,\,t_{i_j}(x)>0\,,\,\,
\,j=0,\cdot\cdot\cdot,k>$. Therefore $\sigma = \underset{\tau <
\sigma}{\coprod}\overset{\circ}{\tau}$


\begin{defn}
\textit{(Simplicial complex)\\ A simplicial complex $K$ in
$\mathbb{R}^N$ } is a collection of simplices in $\mathbb{R}^N$
such that

(1)$\tau<\sigma,\,\sigma\in K\Rightarrow \tau\in K$, and


(2)$\sigma,\tau \in K \Rightarrow \sigma \cap \tau <\sigma$ and
$\sigma\cap\tau < \tau$.
\end{defn}

\begin{defn}\textit{(Subcomplex, Dimension, p-skeleton)\\
(1) $L\subset K$ is a subcomplex (denoted by $L<K$) if $L$ is a
simplicial complex in its own right.}

(2) $dimK:=max\{dim\,\sigma | \sigma \in K\}$.

(3) $p-skeleton$ of $K:=K^p=$the subcomplex consisting of all simplices of $K$ of dim$\leq p$.\\

\end{defn}

주어진 simplicial complex $K$에 대해
$|K|=\displaystyle{\bigcup_{\sigma\in K}} \sigma\subset
\mathbb{R}^N$ 를 생각해보자. $|K|$에 topology를 다음과 같이 준다.

Topology of $|K|$ :

(1)each of $\sigma$ has the usual induced subspace topology in
$\mathbb{R}^N$.

(2)$A\subset |K|$ is closed (open, respectively) if $A\cap \sigma$
is closed(open, respectively) in $\sigma$, $\forall \,\sigma\in
K.$

$|K|$의 closed set을 (2)와 같이 정의하면 이는 $|K|$에 topology
구조를 주고 이를 weak (or coherent) topology 라고 부른다. 또한
$|K|$ with a weak topology 를 $K$의
underlying space(or a polytope) 라고 한다.\\


{\bf 숙제 14.} 일반적으로 어떤 집합 $X$에서 $S_{\alp}\subset
X,\forall\,\alp$ 이고 각 $S_{\alp}$는 topological spaces일 때,
다음 조건을 만족한다고 하자.

1.$S_{\alp}\cap S_{\beta}$ is open(closed, respectively) in
$S_{\alp}$ and $S_{\beta}\,$, $\,\forall\,\,$ $\alp,\beta$

2.topology on $S_{\alp}\cap S_{\beta}$ induced from $S_{\alp}$=
topology on $S_{\alp}\cap S_{\beta}$ induced from $S_{\beta}$

이 때 $X=\displaystyle{\bigcup_{\alp}}S_{\alp}$ 에 다음과 같이
topology를 정의할 수 있다.

$A \subset X $ is open(closed, respectively) if $A\cap S_{\alp}$
is open(closed, respectively) in each $S_{\alp}$.

그러면 이런 $A$들은 $X$상에 topology를 잘 정의하게 되고 다음을
만족한다.

the subspace topology of $S_{\alp}$ as a subset of $X$=the
original topology of
$S_{\alp}$.\\

이 topology를 $\{S_{\alp}\}$에 의해 induced된 weak or coherent
topology라고 부른다. 그리고 $X$가 polytope 과 homeomorphic 할 때
$X$를 polyhedron이라고
한다.\\

{\bf Note.} The weak topology of $|K|$ is finer than the subspace
topology of $|K|\subset \mathbb{R}^N$, i.e., $id:|K|_w \rightarrow
|K|_s$ is continuous.

(증명) $A\subset |K|$ 가 closed in $|K|_s$ 이면 $A$는 subspace
topology 로 closed이고 $A\cap \sigma$는 $\sigma$에서 closed이다. 따라서 $A$는 weak topology로 closed이다.\\

{\bf Examples.}

1 . $[0,1]$\\

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$(0,1]$은 subspace topology로 보면 K에서 open이지만, weak
topology로 보면 이는 closed이다. 왜냐하면 $K$의 각 simplex들과
$(0,1]$과의 교집합은 $\emptyset$ 혹은 simplex 자신으로 나오므로
이는 closed이다.

2 .

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$\sigma$에 대해 $\overset{\circ}{\sigma}$는 $|K|_w$에서 open
이다. 왜냐하면 $\overset{\circ}{\sigma}$와, $\sigma$를 제외한
나머지 $\tau$ 와의 교집합은 모두 $\emptyset$ 이고 이는 $\tau$에서
open이다. 또한 $\sigma$와의 교집합은 $\overset{\circ}{\sigma}$
인데 이 역시 $\sigma$에서 open이므로 $\overset{\circ}{\sigma}$는
$|K|_w$에서 open 이다.

하지만 $\overset{\circ}{\sigma}$는 $|K|_s$에서 open 이 아니다.
subspace topology 로 봤을 때, $\overset{\circ}{\sigma}$상의 한
점에서 어떤 근방을 잡아도 다른 simplex $\tau\in K $ 와 만나므로
interior point가 될 수 없다. 따라서 $\overset{\circ}{\sigma}$는
open일 수가 없다.

3 . If $K$ is a \textit{finite} simplicial complex in
$\mathbb{R}^N$ , then

$\hspace{4em}$ weak topology of $|K|$=subspace topology of $|K|$.

\begin{proof}($\supseteq$)는 이미 앞에서 보였고,
$(\subseteq)$를 보이면 된다. F를 $|K|_w$ 에서 closed인 subset
이라고 하자. 그러면 모든 $\sigma$에 대해 $F\cap \sigma$ 는
$\sigma$에서 closed이고, $\sigma$ 는 $\mathbb{R}^N$에서 closed
이므로 $F\cap \sigma$ 는 $\mathbb{R}^N$에서 closed이다. 그러면,
$\displaystyle{F=\bigcup_{\sigma}(F\cap \sigma)}$ 이므로 closed
subset의 finite union은 역시 closed하다는 성질에 따라 $F$는
$\mathbb{R}^N$에서 closed이다.\end{proof}\\

{\bf Simplicial complex in $\mathbb{R}^J$}\\
Let $J$ be an arbitrary index set and $\mathbb{R}^J = \{f : J \rightarrow \mathbb{R} \}$.\\
Write $f$ as $(x_{\alp})_{\alp \in J} \hspace{1.0em},i.e., f(\alp)=x_{\alp}$.\\

$\mathbb{R}^{J}$ is a vector space with the usual addition and scalar multiplication.\\
$\mathbb{E}^{J} := \{x=(x_{\alp})_{\alp \in J} \in \mathbb{R}^{J} | x_{\alp}=0 $ for all but finitely many $\alp$'s \}\\

Topology of $\mathbb{E}^{J}$:\\
Define a metric on $\mathbb{E}^{J}$ by $|x-y|=max\{|x_{\alp}-y_{\alp}| | \alp \in J\}$.\\
Then $\mathbb{E}^{J}$ with this topology is called a generalized Euclidean space.\\

{\bf Note} $span\{e_{\alp_1}, \cdots ,e_{\alp_N}\} \cong \mathbb{R}^{N}$(as a topological vector space) and a simplex $\sigma=<a_{0}, \cdots , a_{n}>$ in $\mathbb{E}^{J}$ can be viewed as a simplex in $\mathbb{R}^{N}$.\\

All the previous definitions go through for a simplicial complex in $\mathbb{E}^{J}$.\\
$\mathbb{R}^{\infty} := \mathbb{E}^{N}$.\\


\end{document}