1  The Tychonoff Theorem

1.1  (§37) The Tychonoff theorem

• $\mathrm{★}$ Thm. (Tychonoff.) For any $J$ and any compact spaces $X_{\alpha }$, the product ${\prod }_{\alpha \in J}{X}_{\alpha }$ is compact in the product topology.

• Rmk. Proved above for finite $J$ (tube lemma). The general case is omitted here; it needs the axiom of choice. Application below instead.

1.2  The profinite integers

• Notation. $a\equiv b(\mathrm{mod}n)$ iff $n\mid b-a$; classes ${[a]}_n=a+n\mathbb{Z}$; ring $\mathbb{Z}/n$ with ${[a]}_n+{[b]}_n={[a+b]}_n$, ${[a]}_n\cdot {[b]}_n={[ab]}_n$; surjection ${\phi }_n:\mathbb{Z}\to \mathbb{Z}/n$.

• Lem. Give each $\mathbb{Z}/n$ and $\mathbb{Z}$ the discrete topology. Then ${\prod }_{n=1}^{\mathrm{\infty }}\mathbb{Z}/n$ is compact Hausdorff, and $\mathrm{\Phi }=({\phi }_1,{\phi }_2,\mathrm{\dots }):\mathbb{Z}\to {\prod }_n\mathbb{Z}/n$ is injective and continuous.

  Proof. Each $\mathbb{Z}/n$ is finite, hence compact; Tychonoff. Injective: if $n\mid b-a$ for all $n$, take $n>|b-a|$.

• Def. For $d\mid m$, reduction ${\rho }_{m,d}:\mathbb{Z}/m\to \mathbb{Z}/d$, ${[a]}_m\mapsto {[a]}_d$; note ${\rho }_{m,d}\circ {\phi }_m={\phi }_d$. Set

  $$\widehat{\mathbb{Z}}=\left\{{(x_n)}_n\in \prod _{n=1}^{\mathrm{\infty }}\mathbb{Z}/n\right|{\rho }_{m,d}(x_m)=x_d\text{ for all }d\mid m\},$$

  the ring of profinite integers (operations termwise; a topological ring).

• $\mathrm{★}$ Lem. $\widehat{\mathbb{Z}}$ is a closed subspace of ${\prod }_n\mathbb{Z}/n$, hence compact Hausdorff.

  Proof. $\widehat{\mathbb{Z}}={\bigcap }_{d\mid m}{f}_{m,d}^{-1}({[0]}_d)$ where $f_{m,d}\left((x_n)\right)={\rho }_{m,d}(x_m)-x_d$ is continuous and $\mathbb{Z}/d$ is discrete.

• Lem. $\mathrm{\Phi }$ corestricts to an injective continuous ring map $\mathrm{\Psi }:\mathbb{Z}\to \widehat{\mathbb{Z}}$, with dense image.

  Proof. Given $p\in \widehat{\mathbb{Z}}$ and a basis neighborhood constrained at finitely many $n$, let $N$ be a common multiple of those $n$ and choose $a$ with ${[a]}_N=x_N$; then ${[a]}_n=x_n$ for each constrained $n$.

• Def. The subspace topology on $\mathbb{Z}$ from $\widehat{\mathbb{Z}}$ is the Fürstenberg topology. ($\mathrm{\Psi }$ is not an embedding of discrete $\mathbb{Z}$.)

• $\mathrm{★}$ Thm. (Euclid, ca. 300 BC.) There are infinitely many primes.

  Proof. Fürstenberg’s 1955 argument. In the Fürstenberg topology every nonempty open set is infinite, and each $p\mathbb{Z}$ is closed. If there were finitely many primes, $A={\bigcup }_{p}p\mathbb{Z}$ would be closed, so its complement $\{\pm 1\}$ would be open and finite — contradiction.