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\begin{document}
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\section*{IV.4 Urysohn Embedding and Metrization theorem}

%\subsection* {1. Urysohn Embedding and Metrization theorem }

\begin{thm}
\textbf{(Urysohn Embedding Theorem) }\\
 $X$:normal and 2nd countable $\Rightarrow\exists$an embedding $f:X\hookrightarrow I^\infty\hookrightarrow
 H$.\\
\end{thm}

\begin{thm}
\textbf{(Urysohn Metrization Theorem) }\\
 $X$:normal and 2nd countable $\Rightarrow$ X is metrizable.\\
\end{thm}

\begin{proof}
Urysohn Metrization Theorem follows from Urysohn Embedding Theorem
immediately, and let's show the embedding theorem.\\
Let $\mathfrak{U}=\{O_i\}$ be a countable basis. 
$\forall O_i \in \mathfrak{U}$ and $\forall x \in O_i$, $\exists O_j
\in \mathfrak{U}$ such that $x \in O_j\subset \overline{O_j} \subset O_i$.\\
All such pairs $(O_i, O_j)$ form a countable collection $\{P_1, P_2, \cdots\}$.\\
$\forall P_n=(O_i,O_j)$, by Urysohn lemma, $\exists f_n:X\rightarrow [0,1]$  such that $f_n(\overline O_j)=0, f_n(O_i^c)=1$.\\
Define $f:X\rightarrow I^\infty$ by $f(x)=(f_1(x), f_2(x),\cdots)$. Then\\
1. $f$ is 1-1:\\
$x\neq y\\
\Rightarrow \exists O_k, O_l$: disjoint basic open neighborhoods of x,y respectively in
$\mathfrak{U}$\\
$\Rightarrow f_m(x)\neq f_m(y)$.\\
i.e., \emph{"$\{f_n\}$ is a family of enough functions separating points of X"} in the
sense that $x\neq y\Rightarrow\exists f_m$  s.t. $f_m(x)\neq
f_m(y)$.\\
2. $f$ is continuous: obvious.\\
3.$f^{-1}:f(X)\rightarrow X$is continuous:\\
Show $\forall x\in X$ and $O_i$ containing x, $\exists
\delta$ s.t. $d(f(x),f(y))<\delta \Rightarrow y \in O_i$:\\
Let $P_{n_0}=(O_i,O_j)$ s.t. $x\in O_j\subset\overline{O_j}\subset
O_i$.\\
If $d(f(x),f(y))<\delta<1/n_0$\\
$\Rightarrow \frac{|f_n(x)-f_n(y)|}{n}<\delta$, $ \forall n$\\
$\Rightarrow \frac{|f_{n_0}(x)-f_{n_0}(y)|}{n_0}<\delta\Rightarrow
\frac{f_{n_0}(y)}{n_0}<\delta\Rightarrow f_{n_0}(y)<n_0\delta<1$.\\
Since $f_{n_0}(y)=1$ if $y\in O_i^c$, $f_{n_0}(y)<1\Rightarrow y \in O_i$
\end{proof}

\begin{rem}
We can change normality to regularity in Urysohn
Embedding and Metrization Theorem since "regular+2nd countable $\Rightarrow$ normal".
(See Munkres Theorem 32.1.)
 \end{rem}
\textbf{Homework} 1. Is uncountable product of $I=[0,1]$ normal? metrizable?\\
2. Read or prove by yourself Munkres' Theorem 32.1.
 \end{document}
