0.1  (§33) The Urysohn lemma

• $\mathrm{★}$ Thm. (Urysohn’s lemma.) $A,B$ disjoint closed subsets of a normal space $X$ $\Rightarrow$ there is a map $f:X\to [0,1]$ with $f\equiv 0$ on $A$ and $f\equiv 1$ on $B$.

• Proof, on the board in four moves.

  1. Dyadic rationals $r=a/2^n$ are dense in $ℝ$.

  2. Build a nested family. $U_1=X-B$; normality gives $U_0$ with $A\subset U_0\subset {\overline{U}}_0\subset U_1$, then $U_{1/2}$ between them. Inductively insert $U_{(2b+1)/2^n}$ between $U_{2b/2^n}$ and $U_{(2b+2)/2^n}$. Result: ${\overline{U}}_p\subset U_q$ whenever . Extend by $U_r=\mathrm{\varnothing }$ for and $U_r=X$ for $r>1$.

  3. Define $D(x)=\{r\text{ dyadic}\mid x\in U_r\}$ and $f(x)=infD(x)\in [0,1].$

  4. Three claims. (a) $x\in {\overline{U}}_r\Rightarrow f(x)\le r$;  (b) $x\notin U_r\Rightarrow f(x)\ge r$;  (c) $f$ is continuous.

  Proof. For (c): given $(c,d)\ni f(x)$, pick dyadic ; then $U=U_q-{\overline{U}}_p$ is a neighborhood of $x$ with $f(U)\subset (c,d)$.

• Def. $A,B$ are separated by a continuous function if some $f:X\to ℝ$ is $0$ on $A$ and $1$ on $B$. (Forces $A\cap B=\mathrm{\varnothing }$.)

• Def. $X$ is completely regular ($T_{3⁤\frac{1}{2}}$) if points are closed and each point and disjoint closed set are separated by a continuous function.

• Lem. Normal $\Rightarrow$ completely regular $\Rightarrow$ regular.

  Proof. First: Urysohn with $A=\{x\}$. Second: $U=f^{-1}[0,\frac{1}{2})$, $V=f^{-1}(\frac{1}{2},1]$.

• Thm. Subspaces and products of completely regular spaces are completely regular. Caution. The same is not true for normality.

0.2  (§34) The Urysohn metrization theorem

• $\mathrm{★}$ Thm. Every second-countable regular space is metrizable.

• Proof: embed $X$ into ${ℝ}^{\omega }$ (product topology, which is metrizable).

  1. Claim 1: a countable separating family. There are maps $f_k:X\to [0,1]$ such that for every $p\in U$ open, some $f_k$ has $f_k(p)=1$ and $f_k(X-U)\subset \{0\}$.

  2. Construction. $\{B_n\}$ a countable basis. For each pair $(m,n)$ with ${\overline{B}}_m\subset B_n$, use Urysohn (X is normal by Thm 32.1) to get $g_{m,n}$ with $g_{m,n}({\overline{B}}_m)=\{1\}$, $g_{m,n}(X-B_n)=\{0\}$. Reindex as $\{f_k\}$.

  3. Claim 2: $F(x)=(f_1(x),f_2(x),\mathrm{\dots })$ is an embedding $X\to {ℝ}^{\omega }$. Continuous (components are); injective (separate $x$ from $y$ using $U=X-\{y\}$).

  4. Claim 3: $F$ is open onto its image $Z$. Given $q=F(p)\in F(U)$, choose $k$ with $f_k(p)=1$, $f_k(X-U)=\{0\}$, and set $W=Z\cap {\pi }_k^{-1}(0,\mathrm{\infty })$. Then $q\in W\subset F(U)$.

  Note. Regularity is used exactly once — to produce $p\in B_m\subset {\overline{B}}_m\subset B_n$, which is what makes the pair $(m,n)$ available.

0.3  (§35–§36) Tietze extension; manifolds

• $\mathrm{★}$ Thm. (Tietze.) $A$ closed in a normal $X$. Every map $A\to [0,1]$ (resp. $A\to ℝ$) extends to a map $X\to [0,1]$ (resp. $X\to ℝ$). [Mk §35]

• $\mathrm{★}$ Def. An $m$-manifold is a second-countable Hausdorff space in which every point has a neighborhood homeomorphic to an open subset of ${ℝ}^m$ (equivalently, to ${ℝ}^m$ itself).

• Notation. $m=0$: discrete countable set. $m=1$: curve. $m=2$: surface.

• Thm. A compact $m$-manifold embeds in ${ℝ}^n$ for some $n$. [Mk §36]