# mathlib3documentation

topology.is_locally_homeomorph

# Local homeomorphisms #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

This file defines local homeomorphisms.

## Main definitions #

• is_locally_homeomorph: A function f : X → Y satisfies is_locally_homeomorph if for each point x : X, the restriction of f to some open neighborhood U of x gives a homeomorphism between U and an open subset of Y.

Note that is_locally_homeomorph is a global condition. This is in contrast to local_homeomorph, which is a homeomorphism between specific open subsets.

def is_locally_homeomorph_on {X : Type u_1} {Y : Type u_2} (f : X Y) (s : set X) :
Prop

A function f : X → Y satisfies is_locally_homeomorph_on f s if each x ∈ s is contained in the source of some e : local_homeomorph X Y with f = e.

Equations
theorem is_locally_homeomorph_on.mk {X : Type u_1} {Y : Type u_2} (f : X Y) (s : set X) (h : (x : X), x s ( (e : Y), (y : X), f y = e y)) :

Proves that f satisfies is_locally_homeomorph_on f s. The condition h is weaker than the definition of is_locally_homeomorph_on f s, since it only requires e : local_homeomorph X Y to agree with f on its source e.source, as opposed to on the whole space X.

theorem is_locally_homeomorph_on.map_nhds_eq {X : Type u_1} {Y : Type u_2} {f : X Y} {s : set X} (hf : s) {x : X} (hx : x s) :
(nhds x) = nhds (f x)
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theorem is_locally_homeomorph_on.continuous_at {X : Type u_1} {Y : Type u_2} {f : X Y} {s : set X} (hf : s) {x : X} (hx : x s) :
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theorem is_locally_homeomorph_on.continuous_on {X : Type u_1} {Y : Type u_2} {f : X Y} {s : set X} (hf : s) :
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theorem is_locally_homeomorph_on.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {g : Y Z} {f : X Y} {s : set X} {t : set Y} (hg : t) (hf : s) (h : s t) :
s
def is_locally_homeomorph {X : Type u_1} {Y : Type u_2} (f : X Y) :
Prop

A function f : X → Y satisfies is_locally_homeomorph f if each x : x is contained in the source of some e : local_homeomorph X Y with f = e.

Equations
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theorem is_locally_homeomorph.is_locally_homeomorph_on {X : Type u_1} {Y : Type u_2} {f : X Y} (hf : is_locally_homeomorph f) :
theorem is_locally_homeomorph.mk {X : Type u_1} {Y : Type u_2} (f : X Y) (h : (x : X), (e : Y), (y : X), f y = e y) :

Proves that f satisfies is_locally_homeomorph f. The condition h is weaker than the definition of is_locally_homeomorph f, since it only requires e : local_homeomorph X Y to agree with f on its source e.source, as opposed to on the whole space X.

theorem is_locally_homeomorph.map_nhds_eq {X : Type u_1} {Y : Type u_2} {f : X Y} (hf : is_locally_homeomorph f) (x : X) :
(nhds x) = nhds (f x)
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theorem is_locally_homeomorph.continuous {X : Type u_1} {Y : Type u_2} {f : X Y} (hf : is_locally_homeomorph f) :
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theorem is_locally_homeomorph.is_open_map {X : Type u_1} {Y : Type u_2} {f : X Y} (hf : is_locally_homeomorph f) :
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theorem is_locally_homeomorph.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {g : Y Z} {f : X Y} (hg : is_locally_homeomorph g) (hf : is_locally_homeomorph f) :