English
Let f: X → Y be a map and {U_i} an open cover of Range f. Then GeneralizingMap f iff for all i, GeneralizingMap ((U_i).carrier.restrictPreimage f).
Русский
Пусть f: X → Y, а {U_i} — открытое покрытие Range f. Тогда GeneralizingMap f эквивалентно тому, что для каждого i верно GeneralizingMap ((U_i).carrier.restrictPreimage f).
LaTeX
$$$\operatorname{GeneralizingMap}(f) \iff \forall i,\ \operatorname{GeneralizingMap}\left((U_i).carrier\restrictPreimage f\right)$$$
Lean4
nonrec theorem isLocallyClosed_iff {s : Set X} {f : X → Y} (hf : IsInducing f) :
IsLocallyClosed s ↔ ∃ s' : Set Y, IsLocallyClosed s' ∧ f ⁻¹' s' = s :=
by
simp_rw [IsLocallyClosed, hf.isOpen_iff, hf.isClosed_iff]
constructor
· rintro ⟨_, _, ⟨U, hU, rfl⟩, ⟨Z, hZ, rfl⟩, rfl⟩
exact ⟨_, ⟨U, Z, hU, hZ, rfl⟩, rfl⟩
· rintro ⟨_, ⟨U, Z, hU, hZ, rfl⟩, rfl⟩
exact ⟨_, _, ⟨U, hU, rfl⟩, ⟨Z, hZ, rfl⟩, rfl⟩
