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Copyright 2026 The Formal Conjectures Authors.
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you may not use this file except in compliance with the License.
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-/
module
public import Mathlib.Topology.Defs.Induced
public import Mathlib.Topology.Separation.Regular
public import Mathlib.Topology.ContinuousOn@[expose] public sectionuniverse u
A topological space X is an absolute neighborhood retract (ANR) if, whenever it is
embedded as a closed subspace of a normal space Y, there exists a continuous retraction
from some open neighborhood of X in Y onto X.
More precisely, for every closed embedding e : X → Y into a normal space Y, there
exist an open set U ⊆ Y containing the image of e and a continuous map r : Y → X
defined on U such that r ∘ e = id.
class IsAbsoluteNeighborhoodRetract (X : Type u) [TopologicalSpace X] : Prop where
exists_neighborhood_retract (Y : Type u) [TopologicalSpace Y] [NormalSpace Y] (e : X → Y) :
Topology.IsClosedEmbedding e → ∃ (U : Set Y) (r : Y → X),
IsOpen U ∧ Set.range e ⊆ U ∧ ContinuousOn r U ∧ ∀ x : X, r (e x) = x