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\begin{document}
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\section*{Simplicial map}

$f:K\rightarrow L$�� simplicial map �̶�� ����. ��
$f:V(K)\rightarrow V(L)$, $f(\sigma)\in L$ if $\sigma\in K$. �̶�
������ ���� map�� ��������.

$"f":|K|\rightarrow|L|$ defined as :
$\displaystyle{\sum_{i=0}^nt_i v_i=x\in|K|\Rightarrow
f(x)=\sum_{i=0}^n t_i f(v_i)}$

�� $"f"(|\sigma|)=|f(\sigma)|$ �̴�. �� $"f"$�� $f$�� ����
induced�� simplicial map�̶�� �Ѵ�.

{\bf Note.} 1. simplicial map $\circ$ simplicial map = simplicial
map.

$\hspace{3.2em}$2. $"f"$ is continuous.\\



$f:K\rightarrow L$ �� simplicial isomorphism �̸�
$"f":|K|\rightarrow |L|$ is a homeomorphism �̰�, �̸� simplicial
homeomorphism�̶�� �θ���.\\

\begin{prop}
$K$ �� finite simplicial complex �̸� $|K|$�� $\mathbf{R}^N$��
embedd�ȴ�.
\end{prop}
\begin{proof}
$K$�� finite�̹Ƿ� $V(K)=\{v_0,\cdot\cdot\cdot v_N\}$ ��� ����,
$\mathbf{R}^N$�ȿ��� geometrically independent�ϰ�
$a_0,\cdot\cdot\cdot,a_N$�� ��´�. �׸���
$<a_0,\cdot\cdot\cdot,a_N>=\sigma^N$���� ���� �� $\sigma^N$��
�װ��� face ��� �̷���� simplicial complex �� $\triangle^N$ ��
����. �� �� $K$�� $\triangle^N$���̿� ������ ���� �Լ��� ��������.

Define $f:K\rightarrow \triangle^N \hspace{1em} $ by $f(v_i)=a_i$,
$\,\,\,i=0,\cdot\cdot\cdot,N$ .

�׷��� $\{v_0,\cdot\cdot\cdot,v_N\}$�� subset���� image����
$\{a_0,\cdot\cdot\cdot,a_N\}$���� subset���� �Ǵµ� �̵��� ���
$\triangle^N$�󿡼� simplex �� �ǹǷ� $f$�� simplicial map�� �ȴ�.

���� $f$�� $K$�� $\triangle^N$�� subcomplex $L=f(K)$������
isomorphism�� �ֹǷ� $"f":|K|\rightarrow|L|_w$ �� homeomorphism ��
�ȴ�.

�� �� L�� finite �ϹǷ� $\mathbf{R}^n$���� $|L|_s=|L|_w$ �̰�
����

$"f":|K|\rightarrow|L|_s\subseteq \mathbf{R}^n$ �� homeomorphism��
�Ǿ� $|K|$�� $\mathbf{R}^n$�� embedding�� �ȴ�.



\end{proof}

\begin{defn}\textit{star(v)}

$v\in V(K)$�� ����
$st(v):=\displaystyle{\bigcup_{v\in\sigma}}int(|\sigma|)=\{x\in|K|\,\,|\,\,t_v(x)\neq
0 \}$.

\end{defn}



�� ��,
$\overline{st(v)}=\displaystyle{\bigcup_{v\in\sigma}}|\sigma|$
���� ������.

$(\subseteq)\,\,\,$ $ st(v)=\bigcup int(|\sigma|)\subseteq \bigcup
|\sigma|$ �̰�, $\bigcup |\sigma|$�� closed�̹Ƿ�
$\overline{st(v)}\subseteq
\displaystyle{\bigcup_{v\in\sigma}}|\sigma|$�̴�.

$(\supseteq)\,\,\,$ $v\in\sigma$�� �� $\sigma$����
$\sigma\in\overline{st(v)}$ �̹Ƿ�
$\displaystyle{\bigcup_{v\in\sigma}}|\sigma|\subseteq\overline{st(v)}$
�̴�.


���� $\overline{st(v)}$�� $|K|$���� closed�̰�, $lk(v):=\overline{sk(v)}-st(v)$ �� �����Ѵ�.\\


 \begin{figure}[htb]
    \centerline{\includegraphics*[scale=1,clip=true]{graph8.eps}}

    \end{figure}

$\hspace{15em}$ �׸� 8\\


{\bf ���� 14.} ������ ������.

$K$ is locally finite.

$\Leftrightarrow |K|$ is locally compact.

$\Leftrightarrow |K|$ is metrizable with respect to $d$, $d(x,y)=\sqrt{\displaystyle{\sum_{v\in V}(t_v(x)-t_v(y))^2}}$.\\

{\bf ���� 15.} If $K$ is countable and locally finite and
$dim\,K\leq n$, then $|K|$ can be embedded as a closed subset in
$\mathbf{R}^{2n+1}$.


(\textit{Hint.}) Use a curve $C=(t,t^2,\cdot\cdot\cdot,t^{2n+1})$.

�� $C$���� � 2n+2���� ���� �̾Ƶ� �̵��� geometrically
independent�ϴ�.




\end{document}
