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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)\\
$\{a_0,\cdot \cdot \cdot a_n\}\subset \mathbb{R}^N$ ��
geometrically
independent�ϴٰ� ����. �� ��,\\
$\sigma=<a_0,\cdot \cdot \cdot ,a_n>=\{x\in\mathbb{R}^n |
x=\displaystyle{\sum_{i=0}^{n}t_i a_i\,\,,\,\, t_i\geq 0
\,\,\,and\,\,\, \sum_{i=0}^{n}t_i=1 }\}$ \\
=convex hull of $\{a_0,\cdot \cdot \cdot a_n\}$ \\
=n-simplex spanned by $\{a_0,\cdot \cdot \cdot,a_n\}$}�� �����Ѵ�.\\
\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$\sigma$�� �� 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}

�־��� simplicial complex $K$�� ����
$|K|=\displaystyle{\bigcup_{\sigma\in K}} \sigma\subset
\mathbb{R}^N$ �� �����غ���. $|K|$�� topology�� ������ ���� �ش�.

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 $\sigma$, $\forall \,\sigma\in
K.$

$|K|$�� closed set�� (2)�� ���� �����ϸ� �̴� $|K|$�� topology
������ �ְ� �̸� weak (or coherent) topology ��� �θ���. ����
$|K|$ with a weak topology �� $K$��
underlying space(or a polytope) ��� �Ѵ�.\\


{\bf ���� 14.} �Ϲ������� � ���� $X$���� $S_{\alp}\subset
X,\forall\,\alp$ �̰� �� $S_{\alp}$�� topological spaces�� ��,
���� ������ �����Ѵٰ� ����.

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 $S_{\beta}$

�� �� $X=\displaystyle{\bigcup_{\alp}}S_{\alp}$ �� ������ ����
topology�� ������ �� �ִ�.

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

�׷��� �̷� $A$���� $X$�� topology�� �� �����ϰ� �ǰ� ������
�����Ѵ�.

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

�� topology�� $\{S_{\alp}\}$�� ���� induced�� weak or coherent
topology��� �θ���. �׸��� $X$�� polytope �� homeomorphic �� ��
$X$�� polyhedron�̶��
�Ѵ�.\\

{\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.

(����) $A\subset |K|$ �� closed in $|K|_s$ �̸� $A$�� subspace
topology �� closed�̰� $A\cap \sigma$�� $\sigma$���� closed�̴�. ���� $A$�� weak topology�� closed�̴�.\\

{\bf Examples.}

1 . $[0,1]$\\

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$(0,1]$�� subspace topology�� ���� K���� open������, weak
topology�� ���� �̴� closed�̴�. �ֳ��ϸ� $K$�� �� simplex���
$(0,1]$���� �������� $\emptyset$ Ȥ�� simplex �ڽ����� �����Ƿ�
�̴� closed�̴�.

2 .

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$\sigma$�� ���� $\overset{\circ}{\sigma}$�� $|K|_w$���� open
�̴�. �ֳ��ϸ� $\overset{\circ}{\sigma}$��, $\sigma$�� ������
������ $\tau$ ���� �������� ��� $\emptyset$ �̰� �̴� $\tau$����
open�̴�. ���� $\sigma$���� �������� $\overset{\circ}{\sigma}$
�ε� �� ���� $\sigma$���� open�̹Ƿ� $\overset{\circ}{\sigma}$��
$|K|_w$���� open �̴�.

������ $\overset{\circ}{\sigma}$�� $|K|_s$���� open �� �ƴϴ�.
subspace topology �� ���� ��, $\overset{\circ}{\sigma}$���� ��
������ � �ٹ��� ��Ƶ� �ٸ� simplex $\tau\in K $ �� �����Ƿ�
interior point�� �� �� ����. ���� $\overset{\circ}{\sigma}$��
open�� ���� ����.

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}($\supseteq$)�� �̹� �տ��� ������,
$(\subseteq)$�� ���̸� �ȴ�. F�� $|K|_w$ ���� closed�� subset
�̶�� ����. �׷��� ��� $\sigma$�� ���� $F\cap \sigma$ ��
$\sigma$���� closed�̰�, $\sigma$ �� $\mathbb{R}^N$���� closed
�̹Ƿ� $F\cap \sigma$ �� $\mathbb{R}^N$���� closed�̴�. �׷���,
$\displaystyle{F=\bigcup_{\sigma}(F\cap \sigma)}$ �̹Ƿ� closed
subset�� finite union�� ���� closed�ϴٴ� ������ ���� $F$��
$\mathbb{R}^N$���� closed�̴�.\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}
