isProperMap_iff_isClosedMap_of_inj

English

If the diagram satisfies h, hf, hp, hq and inducing, then p is quotient map.

Русский

Если диаграмма удовлетворяет условиям, то p — квотирующее отображение.

LaTeX

$$$\mathrm{IsQuotientMap}(p)$$$

Lean4

/-- An injective and continuous function is proper if and only if it is closed. -/
theorem isProperMap_iff_isClosedMap_of_inj (f_cont : Continuous f) (f_inj : f.Injective) :
    IsProperMap f ↔ IsClosedMap f := by
  refine ⟨fun h ↦ h.isClosedMap, fun h ↦ ?_⟩
  rw [isProperMap_iff_isClosedMap_and_compact_fibers]
  exact ⟨f_cont, h, fun y ↦ (subsingleton_singleton.preimage f_inj).isCompact⟩

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