English
A homeomorphism f induces an equivalence between the compact sets of the domain and codomain, i.e., Compacts α ≃ Compacts β.
Русский
Гомеоморфизм f порождает эквивалентность между компактными множествах домена и кодомома, то есть Compacts α ≃ Compacts β.
LaTeX
$$$\\text{If } f : \\alpha \\simeq_{\\/} β \text{ is aHomeomorph, then } \\mathrm{Compacts}(\\alpha) \\simeq \\mathrm{Compacts}(\\beta).$$$
Lean4
/-- A homeomorphism induces an equivalence on compact sets, by taking the image. -/
@[simps]
protected def equiv (f : α ≃ₜ β) : Compacts α ≃ Compacts β
where
toFun := Compacts.map f f.continuous
invFun := Compacts.map _ f.symm.continuous
left_inv
s := by
ext1
simp only [coe_map, ← image_comp, f.symm_comp_self, image_id]
right_inv
s := by
ext1
simp only [coe_map, ← image_comp, f.self_comp_symm, image_id]
