0.1  (§29) Local compactness

• $\mathrm{★}$ Def. $X$ is locally compact at $x$ if there is a compact $C$ containing a neighborhood $V$ of $x$:   $x\in V\subset C\subset X$. Locally compact: at every point.

• Ex. Compact $\Rightarrow$ locally compact. $ℝ$ is locally compact ($V=(x-1,x+1)\subset C=[x-1,x+1]$); so is ${ℝ}^n$, using products of intervals.

• Ex. $\mathbb{Q}$ is not locally compact. Nor is ${ℝ}^{\omega }$: a basis neighborhood of $0$ has non-compact closure.

• $\mathrm{★}$ Def. (One-point compactification.) $X$ locally compact Hausdorff, $Y=X\cup \{\mathrm{\infty }\}$ with ${𝒯}_{\mathrm{\infty }}$ consisting of (1) the open $U\subset X$, and (2) the sets $Y-C$ for $C\subset X$ compact.

• $\mathrm{★}$ Thm. $Y$ is compact Hausdorff, $X\subset Y$ is a subspace, and $Y-X$ is one point.

  Proof. Topology: three cases each for $\cap$ and $\cup$; needs $C_1\cup C_2$ compact, $X-C$ open (Hausdorff), and $\bigcap C_{\beta }$, $C-U$ compact (closed in a compact set). Compact: any cover has some $V=Y-C$ containing $\mathrm{\infty }$; cover the compact $C$ by finitely many others. Hausdorff at $\mathrm{\infty }$: use local compactness at $x$.

• Prop. (Converse.) If $X\subset Y$ with $Y$ compact Hausdorff and $Y-X$ a single point, then $X$ is locally compact Hausdorff.

• Prop. (Uniqueness.) Any two such $Y$ are homeomorphic rel $X$ — hence the one-point compactification.

• Ex. ${(0,1)}^{+}\cong S^1$, hence ${ℝ}^{+}\cong S^1$.   $B{(0,1)}^{+}\cong S^n$, hence ${({ℝ}^n)}^{+}\cong S^n$.

• Caution. $[0,1]$ is a different compactification of $(0,1)$ — it adds two points. Likewise $\overline{B}(0,1)$ adds $S^{n-1}$.

• $\mathrm{★}$ Thm. $X$ Hausdorff. Then $X$ is locally compact $\iff$ for each $x$ and each neighborhood $U$ of $x$ there is a neighborhood $V$ with $x\in V\subset \overline{V}\subset U$ and $\overline{V}$ compact.

  Proof. $\Leftarrow$ take $U=X$. $\Rightarrow$ work inside $Y=X\cup \{\mathrm{\infty }\}$: separate $x$ from the compact $K=Y-U$.

• Cor. Open and closed subspaces of a locally compact Hausdorff space are locally compact.

• Cor. $X$ is homeomorphic to an open subspace of a compact Hausdorff space $\iff$ $X$ is locally compact Hausdorff.

1  Countability and Separation Axioms

Goal of the chapter. Urysohn metrization: one countability axiom (second countable) plus one separation axiom (regular) yield enough maps $X\to ℝ$ to embed $X$ in a metric space.

1.1  (§30) The countability axioms

• Rmk. [Mk §7] Recall: subsets, finite or countable unions, and finite products of countable sets are countable. $\mathbb{Q}$ is countable; $ℝ$ is not.

• Def. First-countable: a countable neighborhood basis at each point. Second-countable: a countable basis for the topology.

• Lem. Second-countable $\Rightarrow$ first-countable

  Proof. take ${ℬ}_x=\{U\in ℬ\mid x\in U\}$.

  Every metric space is first-countable.

• Ex. Second-countable: $ℝ$ (intervals $(a,b)$, $a,b\in \mathbb{Q}$); ${ℝ}^n$ (rational boxes); even ${ℝ}^{\omega }$ (rational boxes, $U_n=ℝ$ for all but finitely many $n$).

• Caution. Not every metric space is second-countable: ${ℝ}^{\omega }$ in the uniform topology.

• Thm. Subspaces, and countable products, of first- (resp. second-) countable spaces are first- (resp. second-) countable.

• Def. $A\subset X$ is dense if $\overline{A}=X$, i.e. every nonempty open set meets $A$. Ex. $\mathbb{Q}$ dense in $ℝ$.

• $\mathrm{★}$ Thm. $X$ second-countable $\Rightarrow$ (1) every open cover has a countable subcover (Lindelöf); (2) $X$ has a countable dense subset (separable).

  Proof. Fix a countable basis $\{B_n\}$. (1) For each $n$ pick $A_n\in 𝒜$ with $B_n\subset A_n$ when possible. (2) Pick $x_n\in B_n$ for each nonempty $B_n$.

1.2  (§31–§32) The separation axioms; normal spaces [Mk §31–32]

• Def. Assume one-point sets are closed ($T_1$). Then:

  • $X$ is regular if a point and a disjoint closed set can be separated by disjoint open sets;

  • $X$ is normal if two disjoint closed sets can be so separated.

  Normal $\Rightarrow$ regular $\Rightarrow$ Hausdorff.

• Thm. [Mk Thm 32.1] A regular space with a countable basis is normal. (This is the input to Urysohn metrization.)

• Rmk. Full treatment, examples and counterexamples: work through Munkres §31–32.