Product Topology

Statement

Given a family of topological spaces $(X_i)_{i \in I}$, the product topology on $\prod_{i \in I} X_i$ is the coarsest topology making every projection $\pi_i : \prod_j X_j \to X_i$ continuous. Its basic open sets are finite intersections of sets of the form $\pi_i^{-1}(U_i)$ for $U_i$ open in $X_i$.

Universal property: A function $f : Z \to \prod_{i \in I} X_i$ is continuous if and only if every coordinate function $\pi_i \circ f : Z \to X_i$ is continuous:

Visualization

Basis for the product topology on $S^1 \times S^1$ (torus):

         S¹ × S¹  (torus)
  θ₂
  2π  ─────────────────────┐
   |  │░░░░░░░░░░░░░░░░░░░│
   |  │░░░┌───┐░░░░░░░░░░│   basic open set:
   |  │░░░│ U │░░░░░░░░░░│   π₁⁻¹(arc₁) ∩ π₂⁻¹(arc₂)
   |  │░░░└───┘░░░░░░░░░░│   = small box on torus
   |  │░░░░░░░░░░░░░░░░░░│
  0   └────────────────────┘
      0    arc₁           2π  θ₁

A map $f : \mathbb{R} \to S^1 \times S^1$, $f(t) = (\cos t, \cos 2t, \sin t, \sin 2t)$ (written in coordinates) is continuous iff both $t \mapsto (\cos t, \sin t)$ and $t \mapsto (\cos 2t, \sin 2t)$ are continuous — which they are, being compositions of continuous functions.

Finite-dimensional case — checking continuity into $\mathbb{R}^2$:

Component	Map	Continuous?
$\pi_1 \circ f$	$t \mapsto t^2$	Yes (polynomial)
$\pi_2 \circ f$	$t \mapsto \sin t$	Yes (trig)
$f$	$t \mapsto (t^2, \sin t)$	Yes (by the theorem)

Proof Sketch

1. ($\Rightarrow$) If $f$ is continuous, each $\pi_i \circ f$ is continuous as a composition of continuous functions ($\pi_i$ is continuous by definition of the product topology).

2. ($\Leftarrow$) Suppose each $\pi_i \circ f$ is continuous. It suffices to check that preimages of subbasic open sets are open. A subbasic open set is $\pi_i^{-1}(U_i)$ for some $U_i$ open in $X_i$. Then:

which is open by continuity of $\pi_i \circ f$.

3. Since the product topology is generated by subbasic sets of this form, $f$ is continuous.

4. This universal property characterises the product topology among all topologies on $\prod_i X_i$: it is the unique coarsest topology with this property.

Connections

• Tychonoff's TheoremTychonoff's Theorem$$\prod_{i \in I} X_i \text{ compact} \iff \forall i,\; X_i \text{ compact}$$An arbitrary product of compact spaces is compact in the product topologyRead more → — the product of compact spaces is compact; the product topology is the natural setting for this result. Tychonoff's theorem uses the Alexander subbase theorem on exactly the subbasic sets described above.
• Heine–Borel TheoremHeine–Borel Theorem$$K \subseteq \mathbb{R}^n \;\text{compact} \iff K \;\text{closed and bounded}$$In ℝⁿ, a subset is compact if and only if it is closed and boundedRead more → — $[a,b]^n \subseteq \mathbb{R}^n$ is compact as a finite product of compact intervals, a direct application of Tychonoff in the finite case. Heine–Borel then characterises all compact subsets of $\mathbb{R}^n$.
• Cayley–Hamilton TheoremCayley–Hamilton Theorem$$p_A(A) = 0 \;\text{where}\; p_A(\lambda) = \det(\lambda I - A)$$Every square matrix satisfies its own characteristic polynomialRead more → — continuous maps between matrix spaces rely on the product topology on $\mathbb{R}^{n \times n} \cong \mathbb{R}^{n^2}$; entries of a product of matrices depend continuously on entries of factors.

import Mathlib.Topology.Constructions

/-- Continuity into a Pi type is equivalent to continuity of each component. -/
theorem continuous_into_product_iff
    {Z : Type*} {ι : Type*} {X : ι → Type*}
    [TopologicalSpace Z] [∀ i, TopologicalSpace (X i)]
    (f : Z → ∀ i, X i) :
    Continuous f ↔ ∀ i, Continuous (fun z => f z i) :=
  continuous_pi_iff