Martin Escardo 2011. We now iterate the proof of the fact that binary products preserve compactness, to get that countable (dependent) products preserve compactness. The function countable-Tychonoff requires explicit indication of termination. \begin{code} {-# OPTIONS --without-K --exact-split #-} open import UF-FunExt module CountableTychonoff (fe : FunExt) where open import SpartanMLTT open import CompactTypes open import Sequence fe binary-Tychonoff' :{X : ℕ → 𝓤 ̇ } → compact∙ (X 0) → compact∙ ((n : ℕ) → X (succ n)) → compact∙ ((n : ℕ) → X n) binary-Tychonoff' ε δ = retractions-preserve-compactness cons-has-section' (binary-Tychonoff ε δ) \end{code} The following needs disabling of termination checking. It terminates on the assumption that functions are continuous, but doesn't use their moduli of continuity. Put another way, we get an infinite proof, so to speak, but whenever it is applied to compute a ground value, a finite portion of the proof is used, because of continuity. We emphasize that although continuity is used at the meta-level to justify termination, the result is not anti-classical. In fact, classically, all sets are compact! This module just enlarges the constructive universe a bit, using Brouwerian ideas, but being compatible with Bishop in the sense of not contradicting classical mathematics. Because the proof of termination is not constructive, if you are a strict constructivist maybe you won't be convinced that this proof-program always terminates (when used to define a value of ground type). However, you will have a hard time exhibiting a counter-example: the assumption of existence of a non-continuous function allows you to constructively prove LPO! (With the termination checker enabled.) (I plan to actually write down this proof in Agda.) \begin{code} {-# TERMINATING #-} countable-Tychonoff :{X : ℕ → 𝓤 ̇ } → ((n : ℕ) → compact∙ (X n)) → compact∙ ((n : ℕ) → X n) countable-Tychonoff {X} ε = binary-Tychonoff' (head ε) (countable-Tychonoff (tail ε)) \end{code} A corollary is that the Cantor space is compact. See the module CantorCompact.