sigmaProdDistrib — Mathlib

English

If ι has a unique element, then the space of functions ι → X is homeomorphic to X.

Русский

Если у множества индексов ι есть единственный элемент, тогда пространство функций ι → X гомеоморфно X.

LaTeX

$$$\forall \iota [Unique \iota],\ (\iota \to X) \cong_{Top} X.$$$

Lean4

/-- `(Σ i, X i) × Y` is homeomorphic to `Σ i, (X i × Y)`. -/
@[simps! apply symm_apply toEquiv]
def sigmaProdDistrib : (Σ i, X i) × Y ≃ₜ Σ i, X i × Y :=
  Homeomorph.symm <|
    (Equiv.sigmaProdDistrib X Y).symm.toHomeomorphOfContinuousOpen
      (continuous_sigma fun _ => continuous_sigmaMk.fst'.prodMk continuous_snd)
      (isOpenMap_sigma.2 fun _ => isOpenMap_sigmaMk.prodMap IsOpenMap.id)

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