isConnected_univ_pi — Mathlib

English

IsConnected(univ.pi s) iff ∀ i, IsConnected(s(i)).

Русский

Связанность унив.пi равняется связности каждого компонента.

LaTeX

$$$\\mathrm{IsConnected}(\\mathrm{univ.pi}\\; s) \\iff \\forall i, \\mathrm{IsConnected}(s(i))$$$

Lean4

@[simp]
theorem isConnected_univ_pi [∀ i, TopologicalSpace (X i)] {s : ∀ i, Set (X i)} :
    IsConnected (pi univ s) ↔ ∀ i, IsConnected (s i) :=
  by
  simp only [IsConnected, ← univ_pi_nonempty_iff, forall_and, and_congr_right_iff]
  refine fun hne => ⟨fun hc i => ?_, isPreconnected_univ_pi⟩
  rw [← eval_image_univ_pi hne]
  exact hc.image _ (continuous_apply _).continuousOn

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