surjective' — Mathlib

English

If α is a densely ordered, conditionally complete linear order and f is continuous and injective, then f is either strictly monotone or strictly antitone.

Русский

Пусть α — плотно упорядоченный обобщённый линейный порядок, и f непрерывна и инъективна; тогда f либо строго монотонна, либо строго антимоносонична.

LaTeX

$$StrictMono f ∨ StrictAnti f$$

Lean4

/-- A continuous function which tendsto `Filter.atBot` along `Filter.atTop` and to `Filter.atTop`
along `atBot` is surjective. -/
theorem surjective' {f : α → δ} (hf : Continuous f) (h_top : Tendsto f atBot atTop) (h_bot : Tendsto f atTop atBot) :
    Function.Surjective f :=
  Continuous.surjective (α := αᵒᵈ) hf h_top h_bot

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