mathlib documentation

topology.is_locally_homeomorph

Local homeomorphisms #

This file defines local homeomorphisms.

Main definitions #

def is_locally_homeomorph_on {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] (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} [topological_space X] [topological_space Y] (f : X Y) (s : set X) (h : (x : X), x s ( (e : local_homeomorph X Y), x e.to_local_equiv.source (y : X), y e.to_local_equiv.source 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} [topological_space X] [topological_space Y] {f : X Y} {s : set X} (hf : is_locally_homeomorph_on f s) {x : X} (hx : x s) :
filter.map f (nhds x) = nhds (f x)
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theorem is_locally_homeomorph_on.continuous_at {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] {f : X Y} {s : set X} (hf : is_locally_homeomorph_on f 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} [topological_space X] [topological_space Y] {f : X Y} {s : set X} (hf : is_locally_homeomorph_on f s) :
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theorem is_locally_homeomorph_on.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [topological_space X] [topological_space Y] [topological_space Z] {g : Y Z} {f : X Y} {s : set X} {t : set Y} (hg : is_locally_homeomorph_on g t) (hf : is_locally_homeomorph_on f s) (h : set.maps_to f s t) :
def is_locally_homeomorph {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] (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
theorem is_locally_homeomorph.mk {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] (f : X Y) (h : (x : X), (e : local_homeomorph X Y), x e.to_local_equiv.source (y : X), y e.to_local_equiv.source 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} [topological_space X] [topological_space Y] {f : X Y} (hf : is_locally_homeomorph f) (x : X) :
filter.map f (nhds x) = nhds (f x)
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theorem is_locally_homeomorph.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [topological_space X] [topological_space Y] [topological_space Z] {g : Y Z} {f : X Y} (hg : is_locally_homeomorph g) (hf : is_locally_homeomorph f) :