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\begin{document}
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\section*{Abstract Simplicial complex}


\begin{defn}
\textit{An (abstract) simplicial complex} consists of a set $V$ of
vertices and a collection $K$ of finite non-empty subsets of $V$
called simplices such that

1 . $v\in V \Rightarrow \{v\}\in K$,

2 . $\sigma \in K\,,\,\,\emptyset\neq\tau\subset \sigma
\,\,\Rightarrow \,\,\tau \in K$ .( �� �� $\tau$�� $\sigma$��
face��� �ϰ�,
$\tau<\sigma$�� ����.)\\
\end{defn}

�տ��� �����ߴ� �Ͱ� ���������� dimension, subcomplex,
p-skeleton�� ��������.

$dim\,K:=sup\{dim\, \sigma \,|\,\sigma\in K\}$.

$L\subset K$ is a \textit{subcomplex}($L<K$) if $L$ is a
simplicial complex in its own right.

$K^p=p-skeleton$ of $K$= collection of all simplices of $ K$ of
dim$\leq\,p$

{\bf Examples.}

1 . ������ �־��� ���� A�� ���� $F(A)$�� A�� ���
���Ѻκ�����(������ ����)���� �����̶�� ����. �׷��� $K=F(A)$��
$V=A$�� �ϴ� simplicial complex�� �ȴ�.

2. K, L�� simplicial complex �� �� $K$�� $L$�� join$K*L$�� ������
���� ��������.

$K*L=K\coprod L \coprod \{\sigma \coprod \tau \,\,|\,\, \sigma \in
K, \tau \in L \}$.

�̰� ���� simplicial complex�� �ȴ�. ���� ��� �� 1-dim simplex
$K,L$ �� join�ϸ� 3����¥�� ���ü�� ������ �ǰ�, $K=\{point\}$ ��
��� 2���� �ﰢ�� L�� join�ϸ� ���� �� ���ü�� ������
�ȴ�. ($K$�� �� ���� �� $K$�� $L$�� join�� $L$���� cone�̶�� �Ѵ�.)\\

{\bf ���� 13.} $\sigma^n *\sigma^m=\sigma^{n+m+1}$.

$\hspace{4em}S^n*S^m=S^{n+m+1}$.

($\sigma^n$�� n���� simplex�̰� $S^n$�� n���� sphere�� ���Ѵ�.)\\

���� abstract simplicial complex�� underlying space�� ��������.\\



$K$�� simplicial complex, $\sigma=\{v_0,\cdot\cdot\cdot,v_n\} \in
K$��� ����. �� ��, ������ ���� $|K|$�� �����Ѵ�.

Let $|\,\sigma|=\{\displaystyle{\sum_{i=0}^n
t_iv_i\,\,|\,\,\sum_{i=0}^n }t_i=1,\,\,t_i\geq 0\}$ and,
$|K|=\displaystyle{\bigcup_{\sigma \in K}}|\,\sigma|$

with identification($1\cdot v=v$ and) $\,0\cdot v=0$.

�̸� formal�ϰ� ����,

$|K|=\{x:V\rightarrow [0,1]\,\,|\,\,\{v\in K\,\,|\,\,x(v)\neq
0\}\in K,\displaystyle{\sum_{v\in K}}x(v)=1 \}$ and

$|\,\sigma|=\{x\in |K|\,\,|\,\,x(v)=0 \,\,if\,\, v\notin \sigma\}$
and $x(v_i)=t_i$.

$x(v)=t_v(x)$�� ���ǵ� $t_v:|K|\rightarrow [0,1]\,\,$ �� $x$��
$v$��° barycentric coordinate�̶�� �Ѵ�. ���� ��� $v$�� ����
$t_v(x)=t_v(y)$ �̸� $x=y$ �̴�.\\

���� $|K|$�� topology�� ����.

topology of $|\,\sigma|$ : $x,y \in \sigma$, $x=\sum
t_iv_i\,,\,\,\, y=\sum s_iv_i$ �� ����

$d(x,y):=\sqrt{\sum{(t_i-s_i)^2}}$ �� �ָ�

$|\,\sigma|\cong \,\,standard \,\,simplex
<e_1,\cdot\cdot\cdot,e_n>\subset \mathbf{R}^{n+1}$.(isometric
�ϴ�.)

$\hspace{1.7em}\cong \,\,any\,\, affine\,\, simplex
\,\,<a_0,\cdot\cdot\cdot ,a_n>\subset \mathbf{R}^{N} $ with the
subspace topology.

( �� affine simplex $<a_0,\cdot\cdot\cdot ,a_n>$�� a geometric
realization of $|\,\sigma|$��� �Ѵ�.)

topology of $|K|$�� weak topology generated by $\{|\,\sigma|
\,\,|\,\,\sigma\in K\}$�� �����Ѵ�.

\begin{thm}
$f:|K|\rightarrow X$ is continuous $\Leftrightarrow$
$f|_{|\,\sigma|}$ is continuous $\,\forall \,\,|\,\sigma|\in K$.
\end{thm}

\begin{proof}
$\Rightarrow$ �� �翬�ϰ� $\Leftarrow$ �� ������.

$X$���� closed�� $C$�� ���� $f^{-1}(C)$�� closed in $|K|$����
���̸� �ȴ�. �׷��� ������ $|\,\sigma|$�� ���� $f^{-1}(C)\cap
|\,\sigma|=f|_{|\,\sigma|}^{-1}(C)$ �̰� �̴� $|\,\sigma|$ ����
closed �̹Ƿ� ���� $f^{-1}(C)$�� $|K|$���� closed�̴�.
\end{proof}\\

\begin{cor}
$\hspace{1em}$

(1) $t_v:|K|\rightarrow [0,1]$ is continuous.

(2) $|K|$ is a Hausdorff space.

(3) $A\subset |K|$ is compact $\Leftrightarrow$ $A$ is closed
subset of $|L|$ for some finite subcomplex $L$ of $K$. In
particular, $|K|$ is compact if and only if K is a finite
simplicial complex.

\end{cor}

\begin{proof}
(1)�� �� simplex �󿡼� coordinate function $t_v$�� �����̹Ƿ�
���� $|K|$������ �����̴�.

(2)�� ���� $x\neq y $ in $|K|$ �̶�� $t_v(x)\neq t_v(y)$ ��
�����ϴ� $v$�� �����ϹǷ� ���� (1)�� ���� $ x,y$�� separate ��ų
�� �ִ�.

(3)���� $\Leftarrow$�� ������. $|L|=\displaystyle{\bigcup_{\sigma
\in L}}\sigma$�̰� �� $\sigma$�� compact�̴�. $L$ �� finite �̸�
$|L|$�� compact set���� finite union�̹Ƿ� $|L|$ ���� compact
�̴�. ���� $L$�� closed subset $A$ ���� compact�̴�.

$\Rightarrow$�� ���̱� ���� compact�� $A\subset |K|$ �� $\forall
\, \sigma \in K$ �� ���� $A\cap \overset{\circ}{\sigma}$ ��
��������. $A\cap \overset{\circ}{\sigma}$�� non-empty��
$\sigma$���� $A\cap \overset{\circ}{\sigma}$ �� ���ϴ� ���Ҹ�
�ϳ��� �̾� $x_{\sigma}$�� �ΰ� �̵��� ���� ���� $A'$�� ����. ��
�� $|K|=\displaystyle{\coprod_{\sigma\in
K}}\overset{\circ}{\sigma}$�̹Ƿ� $A'$�� finite���� ���̸�
����ϴ�. $A'\subset A$ �̰� $A'$�� ��� subset B�� closed��
�ǹǷ� ($B\cap \sigma $�� finite set�̹Ƿ� $\sigma$�� closed
subset�� �ǰ� weak topology�� ���ǿ� ���� $B$�� closed�̴�.)
$A'$�� discrete�ϴ�. ���� $A'$�� compact�� $A$�� closed
subset�̹Ƿ� $A'$���� compact�̴�. �� $A'$�� compact, discrete
set�̹Ƿ� finite set�� �Ǿ�� �Ѵ�.


\end{proof}












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