comap — Mathlib

English

Preimage of an open neighborhood under a continuous map is a lattice homomorphism between open neighborhood lattices.

Русский

Обратное образ OpenNhdsOf под непрерывным отображением является гомоморфизм решёток граней открытых окрестностей.

LaTeX

$$$\\\\text{For } f: C(\\\\alpha,\\\\beta) \\\\text{ continuous},\\\\ f^{-1} : OpenNhdsOf(f(x)) \\\\to OpenNhdsOf(x) \\\\text{ is a lattice homomorphism.}$$$

Lean4

/-- Preimage of an open neighborhood of `f x` under a continuous map `f` as a `LatticeHom`. -/
def comap (f : C(α, β)) (x : α) : LatticeHom (OpenNhdsOf (f x)) (OpenNhdsOf x)
    where
  toFun U := ⟨Opens.comap f U.1, U.mem⟩
  map_sup' _ _ := rfl
  map_inf' _ _ := rfl

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