nhds_eq_comap_inf_principal — Mathlib

English

If an OpenPartialHomeomorph has source equal to X and target equal to Y (i.e., source = univ and target = univ), it induces a homeomorphism between the whole spaces X and Y.

Русский

Если источник равен всей области X и цель — всей области Y, то он порождает гомоморфизм между весьми пространствами X и Y.

LaTeX

$$$\exists e: OpenPartialHomeomorph X Y, (e.source = univ) \land (e.target = univ) \Rightarrow X \simeq Y$$$

Lean4

theorem nhds_eq_comap_inf_principal {x} (hx : x ∈ e.source) : 𝓝 x = comap e (𝓝 (e x)) ⊓ 𝓟 e.source :=
  by
  lift x to e.source using hx
  rw [← e.open_source.nhdsWithin_eq x.2, ← map_nhds_subtype_val, ← map_comap_setCoe_val,
    e.toHomeomorphSourceTarget.nhds_eq_comap, nhds_subtype_eq_comap]
  simp only [Function.comp_def, toHomeomorphSourceTarget_apply_coe, comap_comap]

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