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\begin{document}
 \parindent=0cm
  \section*{Baire Space and Category}

 \begin{defn}
$A \subset X$ is nowhere dense if $\overline{A}^i = \emptyset$
    , or equivalently $\overline{A}^c$ is dense.
\end{defn}

\begin{prop}
Suppose $X$ is a space, then the followings are equivalent.
    \begin{enumerate}
    \item A countable intersection of open dense sets in $X$ is dense.
    \item A countable union of closed sets with empty interior has empty interior.
    \item A countable union of nowhere dense sets has empty interior.
    \end{enumerate}
\end{prop}

\begin{proof}

(1 $\LRa$ 2) It is clear since $U_n$ is open dense iff $(U_n)^c$ is closed with an empty interior, and
    \[
    (\bigcap_{n=1}^{\infty} U_n)^c = (\bigcup_{n=1}^{\infty} U_n^c).
    \]

(2 $\Ra$ 3) Suppose $\{U_n\}$ is a countable collection of nowhere dense sets.
Since $\overline{U_n}$ is a closed set with an empty interior, by the assumption, $\bigcup_{n=1}^{\infty} \overline{U_n}$ has an empty interior.
Thus $\bigcup_{n=1}^{\infty} U_n$ has an empty interior.

(3 $\Ra$ 2) By definition a closed set with an empty interior is nowhere dense.
\end{proof}

    \begin{defn}

      A space $X$ is a Baire space if it satisfies any of the equivalent statements
       in the above proposition.

    \end{defn}

    Note: $A \subset X$ �� dense �ϴٴ� ���� $A^c$�� interior��
    empty��� �Ͱ� ��ġ�̴�.
    �� �ٸ������ϸ� $X$�� ������ open set�� $A$�� �����ٴ�
    ���̴�.\\


    {\bf ��1}. $\mathbb{Q}$ �� Baire space�� �ƴϴ�.

    $\mathbb{Q} = \bigcup_{r \in \mathbb{Q}}\{r\}$����
    ��$\{r\}$���� closed�̰� interior�� empty������.
    �̰͵��� countable union�� $\mathbb{Q}$ ��ü�� �ǰ�
    ���� interior�� empty�� �ƴϴ�.\\

    {\bf ��2}. $\mathbb{Z}$ �� Baire space�̴�.

    $\mathbb{Z}$�� dense subset�� �ڱ��ڽŻ��̴�. �ֳ��ϸ�
    $\mathbb{Z}$�� topology�� discrete topology�̰� ����
    �������� �� ���� $\{n\}$�� open set�� �ȴ�. �׷��Ƿ�
    �����̶� ������ dense�� �ȵǱ� �����̴�.

    ���� ���Ƿκ��� �ڸ��ϰ� $\mathbb{Z}$�� Baire space��
    �ȴ�.\\

    �׷��� �ڿ������� $\mathbb{R}$, $\mathbb{R}^2$,
    $\mathbb{R}^n$�� Baire space���� �ƴ����� ���ο� ���� ������
    �����. �� ������ ���� ����� �������ε� ������ ������ �����ϸ� �ٷ�
    �� �� �ִ�.

    \begin{thm}

    (Baire) A locally compact Hausdorff or a complete metric space is a Baire space.

    \end{thm}

    \begin{proof}

    $X$�� locally compact Hausdorff �̸� $\forall x \in X$�� $x$��
    neighborhood $U$�� ���� $x \in V \subset \overline{V} \subset
    U$�̰� $\overline{V}$�� compact�� �Ǵ� $V$�� �����Ѵ�.

    ���� $D_n$, $n=1, 2, 3, \cdots$�� dense open set�̶� �ΰ�
    $\forall U$�� ���� ������ ���̷� �Ѵ�.
    $${\bigcap_{n=1}^{\infty}D_{n} \cap U \neq \emptyset}:$$

    $D_n$�� dense��� ��Ƿκ���

    $D_1 \cap U \neq \emptyset \Rightarrow \exists x \in D_1 \cap U
    \Rightarrow \exists V_1$ such that $x \in V_1 \subset
    \overline{V_1} \subset D_1 \cap U$

    $D_2 \cap V_1 \neq \emptyset \Rightarrow \exists x \in D_2 \cap V_1
    \Rightarrow \exists V_2$ such that $x \in V_2 \subset
    \overline{V_2} \subset D_2 \cap V_1 \subset D_1 \cap U$

    \hspace{5em}  \vdots\\

    $\exists V_n$ such that $V_n \subset \overline{V_n} \subset D_n \cap V_{n-1}.$

    ���� $\cdots \subset \overline{V_n} \subset V_{n-1} \subset \overline{V_{n-1}} \subset \overline{V_1}$ �̰� $\overline{V_1}$�� compact�̴�.

    $\{\overline{V_n}\}$ has F.I.P.
    $\Rightarrow$ $\bigcap\overline{V_n} \neq \emptyset.$ �׸���
    $\bigcap \overline{V_n} \subset \bigcap_{m=1}^{\infty}D_m \cap
    U$.

    ���� locally compact Hausdorff ������ ���ؼ��� ������
    ������.\\\\

    �������� complete metric ������ ���ؼ��� �̿� ����� �����
    ����ϴµ� $V_n$ ��ſ� open ball $B_n$�� ����Ѵ�.

    ���������� $D_n$, $n=1, 2, 3, \cdots$�� dense open set�̶� �ΰ�
    $\forall U$�� ���Ͽ� ������ ���δ�.
    $$ \bigcap_{n=1}^{\infty} D_{n} \cap U \neq \emptyset:$$

    $D_1 \cap U \neq \emptyset \Rightarrow \exists$ an open ball $B_1(x_1, r_1)$ of
    radius $r_1 < 1$ such that $\overline{B_1} \subset
    D_1 \cap U$

    $D_2 \cap B_1 \neq \emptyset \Rightarrow \exists$ an open ball $B_2(x_2, r_2)$ of
    radius $r_2 < \frac 1 2$ such that $\overline{B_2} \subset
    D_2 \cap B_1$

    \hspace{5em}  \vdots\\

    $\exists B_n$ of radius $r_n < \frac 1 n$ such that
    $\overline{B_n} \subset D_n \cap B_{n-1}$


    $\{x_n\}$�� Cauchy ���� $\Rightarrow$ $x_n \rightarrow \exists x \in X$

    $\forall m, \{x_n\}_{n \geq m} \subset B_m \Rightarrow x \in \overline {B_m} \subset D_m \cap B_{m-1} \subset D_m \cap U.$

    $\therefore x \in \bigcap_{m=1}^{\infty} D_m \cap U$


    \end{proof}\\

    {\bf Interesting applications}

    1. $\mathbb{Q} \subset \mathbb{R}$�� $G_\del$ -set(i.e., a countable intersection of open sets)�� �ƴϴ�:

    ���� $G_\del$ -set�̶�� �����ϸ� ������ open set $O_n$���� �����ؼ�
    $\displaystyle{\mathbb{Q}=\bigcap_{n=1}^{\infty}O_n}$����
    ��Ÿ������ �� �ִ�. �׷��� ������ $O_n$��
    $\mathbb{R}$�� open dense subset�� �Ǿ�� �Ѵ�.
    ($\because \mathbb{Q}\subset O_n$)

    ���� $U_r=\mathbb{R}\setminus\{r\}$, $r\in \mathbb{Q}$��
    ������ $U_r$�� dense �ΰ��� �ڸ��ϴ�.

    �׷��� ������ $O_n$, $U_r$���� open dense subset�̹Ƿ�

    $$\emptyset = \mathbb{Q}\cap \mathbb{Q}^c =
    \bigcap_{n=1}^{\infty}O_n \cap \bigcap_{r\in\mathbb{Q}}U_r$$
    �� $\mathbb{R}$�� dense subset�̴�.

    �������� dense subset �� �� �����Ƿ� �̰��� ����̰� ����
    ������ �Ǿ���.


    2. $\nexists$ $f:\mathbb{R} \rightarrow \mathbb{R}$ is
    continuous precisely at $\mathbb{Q}\subset \mathbb{R}$:

    �ݸ鿡 $\mathbb{R}\setminus\mathbb{Q}$���� precise�ϰ� ������
    �Լ��� �����ϴµ� �׷��� ���� ������ ���� ��ü������ ����� ����.

    ���� $\mathbb{Q}$�� $\mathbb{N}$���̿� 1-1 ������ �ִ� �Լ�
    $\varphi$�� ���� $f$�� ����ó�� ���´�.\\

    $f= \left\{ \begin{array}{ll}\frac{1}{\varphi(x)} & x \in \mathbb{Q}
    \\ \\0 & x \in \mathbb{Q}^c \end{array}\right.$\\

    �׷��� �� �Լ��� ��Ȯ�� ���������鿡�� �����̴�.

    (�ֳ��ϸ� ������ $\epsilon$�� ���Ͽ�
    $\frac{1}{\varphi(x)}>\epsilon$ �� $x$�� ���Ѱ� ���̴�.)\\

    �׷��� �ٽ� ó������ ���ư��� �ݴ�� ��Ȯ�� ���������鿡��
    ������ �Լ��� �������� �ʴ°��� ������.

    �׷��� �Լ� $f$�� �����Ѵٰ� ��������.
    $O_n=\bigcup \{O^{open}\subset \mathbb{R} \mid diam f(O)<\frac 1 n
    \}$���� �θ�
    $$\mathcal{C}(f):=\bigcap_{n=1}^{\infty}O_n$$
    �� �ٷ� $f$�� ������ ������ �����̵ȴ�.\\

    �������κ��� $\mathcal{C}(f)=\mathbb{Q}$�̴�.

    ���Ƿκ��� $\mathcal{C}(f)$�� $G_\del$ -set�̴�.
    (open set���� countable intersection�̹Ƿ�)

    �׷��� 1������ $\mathbb{Q}$�� $G_\del$ -set�� �ƴ��� ������
    ���� ����̴�.

    �׷��Ƿ� ��Ȯ�� ���������鿡�� ������ �Լ��� �������� �ʴ´�.


    {\bf ��}. point, line, curve�� nowhere dense�̴�.

    $\mathbb{Q}$�� $\mathbb{R}$������ nowhere dense�� �ƴϴ�.

    cantor set C�� nowhere dense�̴�.

    \begin{defn}

    $X$�� nowhere dense set���� countable union��
    1st category set�̶��ϰ� �ٸ� ��� set�� 2nd category
    set�̶� �Ѵ�.

    \end{defn}

    {\bf ��}. $\mathbb{Q}$�� 1st category
    ($\because \mathbb{Q}=\bigcup_{r \in \mathbb{Q}}\{r\}$)

    \begin{thm}

    (Baire) $X$ is a Baire space

    $\Leftrightarrow$ the interior of a 1st category set is empty

    $\Leftrightarrow$ a set containing open set is of 2nd category

    $\Leftrightarrow$ a non-empty open set is of 2nd category.

    \end{thm}

    \begin{proof}
    A subset of a nowhere dense set is nowhere dense and hence a subset of 1st category set is of 1st category.
    \end{proof}\\

    ������ $X$ �ڽŵ� �и��� open�̹Ƿ� ������ ���뿡�� �ٷ�
    ������ ���� �� �ִ�.

    \begin{cor}

    1. Baire space �ڽ��� 2nd category.\\
    2. $X$�� Baire space�� �� $X$�� subset $A$�� 1st category �̸�
    $A^c$�� 2nd category set�̴�.
    \end{cor}    

    (���� �׷��� �ʴٰ� �����ϸ� $X = A \cup A^c$�̹Ƿ� $X$�� 1st
    category�ε� �̰��� ���� ���������� ���� ����̴�.)\\

    {\bf ��}. $\mathbb{R}\setminus \mathbb{Q}$�� 2nd category\\

    {\bf Homework}

    1. Baire space�� open subset�� Baire space.

    2. Baire space�� $G_{\del}$ -subset�� Baire space.

\end{document}
