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\begin{document}
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  \section*{Homeomorphism}

    \begin{defn}
    $f: X \rightarrow Y$ is an open map if $f$(open set) is
    open.
Also, $f: X \rightarrow Y$ is a closed map if $f$(closed set) is
closed.\\
    \end{defn}

    Note: $f$ is open $\; \Longleftrightarrow \;$ $f$(basic open set) is open

     \hspace{3em}  (������ $f(\bigcup B_\alpha )=\bigcup f(B_\alpha)$ �� ����
    �и��ϴ�.)\\


     {\bf{��1}}. The projection map $p_\alpha: \prod_\alpha X_\alpha \rightarrow X_\alpha$ is open and continuous but not closed.

     {\bf{��2}}. $\exists f$ which is open and closed but not
    continuous.(exercise)

     {\bf{��3}}. $\exists f$ which is closed and continuous but not
    open.(exercise)\\


    ���� �� �������� �˼� �ֵ��� �Լ��� open, closed Ȥ�� �����̶�� ���� ���� ������
    �����̴�.

\begin{prop}
  $f: X \rightarrow Y$ is open
    $\; \Longleftrightarrow \;$ $f(A^i) \subset f(A)^i, \;\; \forall A \subset X.$

    \hspace{3em} $f: X \rightarrow Y$ is closed $\; \Longleftrightarrow \;$
    $f(\bar{A}) \supset \overline{f(A)}, \;\; \forall A \subset X.$
\end{prop}
   \begin{proof}
   
   ù��° ����:
   
     $\Rightarrow $) ����:\\
     $f$ is open $\Rightarrow $ $f(A^i)$ is open $\Rightarrow $ $f(A^i) = f(A^i)^i \subset f(A)^i$

    $\Leftarrow$) ����:\\
     $A$ is open $\Rightarrow $ $A= A^i$. ���� �������� ���� $f(A)=f(A^i) \subset
     f(A)^i$�̵ǹǷ� $f(A) = f(A)^i$�� �ǰ� $f(A)$�� open �̴�. 
	 \\




�ι�° ����:

    $\Rightarrow $) ����:\\
     $f$ is closed $\Rightarrow $ $f(\bar{A})$ is closed $\Rightarrow $ $f(\bar{A}) =\overline{f(\bar{A})} \supset \overline{f(A)}$

    $\Leftarrow$) ����:\\
     $A$ is closed $\Rightarrow $ $A= \bar{A}$. ���� �������� ���� $f(A)=f(\bar{A}) \supset
    \overline{f(A)}$�� �ǹǷ� $f(A) = \overline{f(A)}$�� �ǰ� $f(A)$�� closed �̴�.
\end{proof}
\begin{defn}
 $f: X \rightarrow Y$ is a homeomorphism if $f$ is a
    continuous bijection whose inverse $f^{-1}$ is also
    continuous.
    \end{defn}
    ���� homeomorphism�� 1-1, onto �Լ��ν� open set�� open
    set���� ��Ȯ�� �������� �ִ�, �ٽø��ϸ� topology�� ��Ȯ��
    �����ϴ� map�̴�.

    �ΰ��� ������� X�� Y���̿� homeomorphism �� �����ϸ�
    �������� ������ X�� Y�� ������ �����ϸ� �̰�� �츮�� X��
    Y�� homeomorphic�ϴٰ� �θ���. �̰��� �ΰ��� group���̿�
    isomorphism�� �ΰ��� ��������� �����ϴ� �Ͱ� ���� ���������
    ���̿� equivalence�� �ش�.
\begin{prop}
   $f:X \rightarrow Y$ �� bijection�̸� ������ ���
    ��ġ�̴�.

    1. $f$ �� homeomorphism�̴�.

    2. $f$ �� �����̰� open�̴�.

    3. $f$ �� �����̰� closed�̴�.

    4. $f(\bar{A}) = \overline{f(A)}$ (i.e., $f$�� closure�� ����)

    5. $f(A^i) = f(A)^i$ (i.e., $f$�� interior�� ����)
\end{prop}
\begin{proof}
    1 $\Leftrightarrow $ 2:

    $f^{-1}$�� ���� $\Leftrightarrow $ $(f^{-1})^{-1}$(open set) is open. $\Leftrightarrow $ $f$(open set) is open $\Leftrightarrow $
    $f$�� open


    1 $\Leftrightarrow $ 3:

 $f^{-1}$�� ���� $\Leftrightarrow $ $(f^{-1})^{-1}$(closed set) is closed. $\Leftrightarrow $ $f$(closed set) is closed $\Leftrightarrow $
    $f$�� open


    3 $\Leftrightarrow $ 4:

    $f(\bar{A}) \subset \overline{f(A)}$ ; $f$�� ������ �ʿ��������

    $f(\bar{A}) \supset \overline{f(A)}$ ; $f$�� closed�� �ʿ��������
    �̹Ƿ� �и��ϴ�.

    1 $\Rightarrow $ 5:

    homeomorphism�� topology�� �����ϹǷ� topology�� ���� ���ǵǴ� ��� ������� �����ϰԵǰ� ���� interior�� �����Ѵ�.

    5 $\Rightarrow $ 1:

     $f(A^i) = f(A)^i$�̸� �� proposition���� $f$ �� open. ���� $B = f(A)$ �� ������ $A = f^{-1}(B)$ ($f$ �� bijection�̹Ƿ�)
     �̰� �����ϸ� $f(f^{-1}(B)^i)=B^i$, ���� ${f^{-1}(B)}^i =
     f^{-1}(B^i)$ �̹Ƿ� �� proposition���� $f^{-1}$ �� open �̸� ���� $f$�� ������ �ȴ�.
\end{proof}\\

    {\bf{��1}}. ������ �� �������� homeomorphic�ϴ�.

    (0,1) �� (a, b) �� ���ؼ��� �����ϸ� ����ϰ� �� �� ������
    homeomorphism�� $y = (b-a)x + a$�� �־�����.

    ������ �� �󱸰��� ���ؼ��� ���������� ������ �� �ִ�.


     {\bf{��2}}. ������ �������� $\mathbb{R}$�� homeomorphic�ϴ�.

    $y$ = tan $x$�� $(-\frac{\pi}{2},\frac{\pi}{2})$ �� $\mathbb{R}$������ homeomorphism�� �ǰ� ��1������ ����� ������ ���������� �Ȱ��� ������ �����Ѵ�.

     {\bf{��3}}. $(1, 3)-\{2\} \cong (1, 2)\cup(3, 4)$

     {\bf{��4}}. ������, �ݰ�����, �󱸰��� ���� homeomorphic �����ʴ�.(��?)

     {\bf{��5}}. ��, Ÿ��, �ﰢ���� ���� ���� homeomorphic�ϴ�.

     {\bf{��6}}. �����̿� ���ڰ��� ����  homeomorphic �����ʴ�.(��?)

     {\bf{��7}}. $\mathbb{Z}\cong\mathbb{N}\ncong\mathbb{Q}$

     {\bf{��8}}.
    $\mathbb{Z}\cong\mathbb{Z}\times\mathbb{Z}\cong\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}$

     {\bf{��9}}. ����band�� M\"{o}bius band�� ���� homeomorphic ����
    �ʴ�.(��?)

\end{document}
