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/- | |
Copyright (c) 2017 Johannes Hölzl. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot | |
-/ | |
import topology.order | |
/-! | |
# Specific classes of maps between topological spaces | |
This file introduces the following properties of a map `f : X → Y` between topological spaces: | |
* `is_open_map f` means the image of an open set under `f` is open. | |
* `is_closed_map f` means the image of a closed set under `f` is closed. | |
(Open and closed maps need not be continuous.) | |
* `inducing f` means the topology on `X` is the one induced via `f` from the topology on `Y`. | |
These behave like embeddings except they need not be injective. Instead, points of `X` which | |
are identified by `f` are also inseparable in the topology on `X`. | |
* `embedding f` means `f` is inducing and also injective. Equivalently, `f` identifies `X` with | |
a subspace of `Y`. | |
* `open_embedding f` means `f` is an embedding with open image, so it identifies `X` with an | |
open subspace of `Y`. Equivalently, `f` is an embedding and an open map. | |
* `closed_embedding f` similarly means `f` is an embedding with closed image, so it identifies | |
`X` with a closed subspace of `Y`. Equivalently, `f` is an embedding and a closed map. | |
* `quotient_map f` is the dual condition to `embedding f`: `f` is surjective and the topology | |
on `Y` is the one coinduced via `f` from the topology on `X`. Equivalently, `f` identifies | |
`Y` with a quotient of `X`. Quotient maps are also sometimes known as identification maps. | |
## References | |
* <https://en.wikipedia.org/wiki/Open_and_closed_maps> | |
* <https://en.wikipedia.org/wiki/Embedding#General_topology> | |
* <https://en.wikipedia.org/wiki/Quotient_space_(topology)#Quotient_map> | |
## Tags | |
open map, closed map, embedding, quotient map, identification map | |
-/ | |
open set filter function | |
open_locale topological_space filter | |
variables {α : Type*} {β : Type*} {γ : Type*} {δ : Type*} | |
section inducing | |
/-- A function `f : α → β` between topological spaces is inducing if the topology on `α` is induced | |
by the topology on `β` through `f`, meaning that a set `s : set α` is open iff it is the preimage | |
under `f` of some open set `t : set β`. -/ | |
@[mk_iff] | |
structure inducing [tα : topological_space α] [tβ : topological_space β] (f : α → β) : Prop := | |
(induced : tα = tβ.induced f) | |
variables [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] | |
lemma inducing_id : inducing (@id α) := | |
⟨induced_id.symm⟩ | |
protected lemma inducing.comp {g : β → γ} {f : α → β} (hg : inducing g) (hf : inducing f) : | |
inducing (g ∘ f) := | |
⟨by rw [hf.induced, hg.induced, induced_compose]⟩ | |
lemma inducing_of_inducing_compose {f : α → β} {g : β → γ} (hf : continuous f) (hg : continuous g) | |
(hgf : inducing (g ∘ f)) : inducing f := | |
⟨le_antisymm | |
(by rwa ← continuous_iff_le_induced) | |
(by { rw [hgf.induced, ← continuous_iff_le_induced], apply hg.comp continuous_induced_dom })⟩ | |
lemma inducing_iff_nhds {f : α → β} : inducing f ↔ ∀ a, 𝓝 a = comap f (𝓝 (f a)) := | |
(inducing_iff _).trans (induced_iff_nhds_eq f) | |
lemma inducing.nhds_eq_comap {f : α → β} (hf : inducing f) : | |
∀ (a : α), 𝓝 a = comap f (𝓝 $ f a) := | |
inducing_iff_nhds.1 hf | |
lemma inducing.map_nhds_eq {f : α → β} (hf : inducing f) (a : α) : | |
(𝓝 a).map f = 𝓝[range f] (f a) := | |
hf.induced.symm ▸ map_nhds_induced_eq a | |
lemma inducing.map_nhds_of_mem {f : α → β} (hf : inducing f) (a : α) (h : range f ∈ 𝓝 (f a)) : | |
(𝓝 a).map f = 𝓝 (f a) := | |
hf.induced.symm ▸ map_nhds_induced_of_mem h | |
lemma inducing.image_mem_nhds_within {f : α → β} (hf : inducing f) {a : α} {s : set α} | |
(hs : s ∈ 𝓝 a) : f '' s ∈ 𝓝[range f] (f a) := | |
hf.map_nhds_eq a ▸ image_mem_map hs | |
lemma inducing.tendsto_nhds_iff {ι : Type*} | |
{f : ι → β} {g : β → γ} {a : filter ι} {b : β} (hg : inducing g) : | |
tendsto f a (𝓝 b) ↔ tendsto (g ∘ f) a (𝓝 (g b)) := | |
by rw [hg.nhds_eq_comap, tendsto_comap_iff] | |
lemma inducing.continuous_at_iff {f : α → β} {g : β → γ} (hg : inducing g) {x : α} : | |
continuous_at f x ↔ continuous_at (g ∘ f) x := | |
by simp_rw [continuous_at, inducing.tendsto_nhds_iff hg] | |
lemma inducing.continuous_iff {f : α → β} {g : β → γ} (hg : inducing g) : | |
continuous f ↔ continuous (g ∘ f) := | |
by simp_rw [continuous_iff_continuous_at, hg.continuous_at_iff] | |
lemma inducing.continuous_at_iff' {f : α → β} {g : β → γ} (hf : inducing f) {x : α} | |
(h : range f ∈ 𝓝 (f x)) : continuous_at (g ∘ f) x ↔ continuous_at g (f x) := | |
by { simp_rw [continuous_at, filter.tendsto, ← hf.map_nhds_of_mem _ h, filter.map_map] } | |
protected lemma inducing.continuous {f : α → β} (hf : inducing f) : continuous f := | |
hf.continuous_iff.mp continuous_id | |
protected lemma inducing.inducing_iff {f : α → β} {g : β → γ} (hg : inducing g) : | |
inducing f ↔ inducing (g ∘ f) := | |
begin | |
refine ⟨λ h, hg.comp h, λ hgf, inducing_of_inducing_compose _ hg.continuous hgf⟩, | |
rw hg.continuous_iff, | |
exact hgf.continuous | |
end | |
lemma inducing.closure_eq_preimage_closure_image {f : α → β} (hf : inducing f) (s : set α) : | |
closure s = f ⁻¹' closure (f '' s) := | |
by { ext x, rw [set.mem_preimage, ← closure_induced, hf.induced] } | |
lemma inducing.is_closed_iff {f : α → β} (hf : inducing f) {s : set α} : | |
is_closed s ↔ ∃ t, is_closed t ∧ f ⁻¹' t = s := | |
by rw [hf.induced, is_closed_induced_iff] | |
lemma inducing.is_closed_iff' {f : α → β} (hf : inducing f) {s : set α} : | |
is_closed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := | |
by rw [hf.induced, is_closed_induced_iff'] | |
lemma inducing.is_closed_preimage {f : α → β} (h : inducing f) (s : set β) (hs : is_closed s) : | |
is_closed (f ⁻¹' s) := | |
(inducing.is_closed_iff h).mpr ⟨s, hs, rfl⟩ | |
lemma inducing.is_open_iff {f : α → β} (hf : inducing f) {s : set α} : | |
is_open s ↔ ∃ t, is_open t ∧ f ⁻¹' t = s := | |
by rw [hf.induced, is_open_induced_iff] | |
lemma inducing.dense_iff {f : α → β} (hf : inducing f) {s : set α} : | |
dense s ↔ ∀ x, f x ∈ closure (f '' s) := | |
by simp only [dense, hf.closure_eq_preimage_closure_image, mem_preimage] | |
end inducing | |
section embedding | |
/-- A function between topological spaces is an embedding if it is injective, | |
and for all `s : set α`, `s` is open iff it is the preimage of an open set. -/ | |
@[mk_iff] structure embedding [tα : topological_space α] [tβ : topological_space β] (f : α → β) | |
extends inducing f : Prop := | |
(inj : injective f) | |
lemma function.injective.embedding_induced [t : topological_space β] | |
{f : α → β} (hf : injective f) : | |
@_root_.embedding α β (t.induced f) t f := | |
{ induced := rfl, | |
inj := hf } | |
variables [topological_space α] [topological_space β] [topological_space γ] | |
lemma embedding.mk' (f : α → β) (inj : injective f) | |
(induced : ∀ a, comap f (𝓝 (f a)) = 𝓝 a) : embedding f := | |
⟨inducing_iff_nhds.2 (λ a, (induced a).symm), inj⟩ | |
lemma embedding_id : embedding (@id α) := | |
⟨inducing_id, assume a₁ a₂ h, h⟩ | |
lemma embedding.comp {g : β → γ} {f : α → β} (hg : embedding g) (hf : embedding f) : | |
embedding (g ∘ f) := | |
{ inj:= assume a₁ a₂ h, hf.inj $ hg.inj h, | |
..hg.to_inducing.comp hf.to_inducing } | |
lemma embedding_of_embedding_compose {f : α → β} {g : β → γ} (hf : continuous f) (hg : continuous g) | |
(hgf : embedding (g ∘ f)) : embedding f := | |
{ induced := (inducing_of_inducing_compose hf hg hgf.to_inducing).induced, | |
inj := assume a₁ a₂ h, hgf.inj $ by simp [h, (∘)] } | |
protected lemma function.left_inverse.embedding {f : α → β} {g : β → α} | |
(h : left_inverse f g) (hf : continuous f) (hg : continuous g) : | |
embedding g := | |
embedding_of_embedding_compose hg hf $ h.comp_eq_id.symm ▸ embedding_id | |
lemma embedding.map_nhds_eq {f : α → β} (hf : embedding f) (a : α) : | |
(𝓝 a).map f = 𝓝[range f] (f a) := | |
hf.1.map_nhds_eq a | |
lemma embedding.map_nhds_of_mem {f : α → β} | |
(hf : embedding f) (a : α) (h : range f ∈ 𝓝 (f a)) : (𝓝 a).map f = 𝓝 (f a) := | |
hf.1.map_nhds_of_mem a h | |
lemma embedding.tendsto_nhds_iff {ι : Type*} | |
{f : ι → β} {g : β → γ} {a : filter ι} {b : β} (hg : embedding g) : | |
tendsto f a (𝓝 b) ↔ tendsto (g ∘ f) a (𝓝 (g b)) := | |
hg.to_inducing.tendsto_nhds_iff | |
lemma embedding.continuous_iff {f : α → β} {g : β → γ} (hg : embedding g) : | |
continuous f ↔ continuous (g ∘ f) := | |
inducing.continuous_iff hg.1 | |
lemma embedding.continuous {f : α → β} (hf : embedding f) : continuous f := | |
inducing.continuous hf.1 | |
lemma embedding.closure_eq_preimage_closure_image {e : α → β} (he : embedding e) (s : set α) : | |
closure s = e ⁻¹' closure (e '' s) := | |
he.1.closure_eq_preimage_closure_image s | |
end embedding | |
/-- A function between topological spaces is a quotient map if it is surjective, | |
and for all `s : set β`, `s` is open iff its preimage is an open set. -/ | |
def quotient_map {α : Type*} {β : Type*} [tα : topological_space α] [tβ : topological_space β] | |
(f : α → β) : Prop := | |
surjective f ∧ tβ = tα.coinduced f | |
lemma quotient_map_iff {α β : Type*} [topological_space α] [topological_space β] {f : α → β} : | |
quotient_map f ↔ surjective f ∧ ∀ s : set β, is_open s ↔ is_open (f ⁻¹' s) := | |
and_congr iff.rfl topological_space_eq_iff | |
namespace quotient_map | |
variables [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] | |
{g : β → γ} {f : α → β} | |
protected lemma id : quotient_map (@id α) := | |
⟨assume a, ⟨a, rfl⟩, coinduced_id.symm⟩ | |
protected lemma comp (hg : quotient_map g) (hf : quotient_map f) : | |
quotient_map (g ∘ f) := | |
⟨hg.left.comp hf.left, by rw [hg.right, hf.right, coinduced_compose]⟩ | |
protected lemma of_quotient_map_compose (hf : continuous f) (hg : continuous g) | |
(hgf : quotient_map (g ∘ f)) : quotient_map g := | |
⟨hgf.1.of_comp, | |
le_antisymm | |
(by { rw [hgf.right, ← continuous_iff_coinduced_le], apply continuous_coinduced_rng.comp hf }) | |
(by rwa ← continuous_iff_coinduced_le)⟩ | |
lemma of_inverse {g : β → α} (hf : continuous f) (hg : continuous g) (h : left_inverse g f) : | |
quotient_map g := | |
quotient_map.of_quotient_map_compose hf hg $ h.comp_eq_id.symm ▸ quotient_map.id | |
protected lemma continuous_iff (hf : quotient_map f) : | |
continuous g ↔ continuous (g ∘ f) := | |
by rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.right, coinduced_compose] | |
protected lemma continuous (hf : quotient_map f) : continuous f := | |
hf.continuous_iff.mp continuous_id | |
protected lemma surjective (hf : quotient_map f) : surjective f := hf.1 | |
protected lemma is_open_preimage (hf : quotient_map f) {s : set β} : | |
is_open (f ⁻¹' s) ↔ is_open s := | |
((quotient_map_iff.1 hf).2 s).symm | |
protected lemma is_closed_preimage (hf : quotient_map f) {s : set β} : | |
is_closed (f ⁻¹' s) ↔ is_closed s := | |
by simp only [← is_open_compl_iff, ← preimage_compl, hf.is_open_preimage] | |
end quotient_map | |
/-- A map `f : α → β` is said to be an *open map*, if the image of any open `U : set α` | |
is open in `β`. -/ | |
def is_open_map [topological_space α] [topological_space β] (f : α → β) := | |
∀ U : set α, is_open U → is_open (f '' U) | |
namespace is_open_map | |
variables [topological_space α] [topological_space β] [topological_space γ] {f : α → β} | |
protected lemma id : is_open_map (@id α) := assume s hs, by rwa [image_id] | |
protected lemma comp | |
{g : β → γ} {f : α → β} (hg : is_open_map g) (hf : is_open_map f) : is_open_map (g ∘ f) := | |
by intros s hs; rw [image_comp]; exact hg _ (hf _ hs) | |
lemma is_open_range (hf : is_open_map f) : is_open (range f) := | |
by { rw ← image_univ, exact hf _ is_open_univ } | |
lemma image_mem_nhds (hf : is_open_map f) {x : α} {s : set α} (hx : s ∈ 𝓝 x) : | |
f '' s ∈ 𝓝 (f x) := | |
let ⟨t, hts, ht, hxt⟩ := mem_nhds_iff.1 hx in | |
mem_of_superset (is_open.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts) | |
lemma range_mem_nhds (hf : is_open_map f) (x : α) : range f ∈ 𝓝 (f x) := | |
hf.is_open_range.mem_nhds $ mem_range_self _ | |
lemma maps_to_interior (hf : is_open_map f) {s : set α} {t : set β} (h : maps_to f s t) : | |
maps_to f (interior s) (interior t) := | |
maps_to'.2 $ interior_maximal (h.mono interior_subset subset.rfl).image_subset | |
(hf _ is_open_interior) | |
lemma image_interior_subset (hf : is_open_map f) (s : set α) : | |
f '' interior s ⊆ interior (f '' s) := | |
(hf.maps_to_interior (maps_to_image f s)).image_subset | |
lemma nhds_le (hf : is_open_map f) (a : α) : 𝓝 (f a) ≤ (𝓝 a).map f := | |
le_map $ λ s, hf.image_mem_nhds | |
lemma of_nhds_le (hf : ∀ a, 𝓝 (f a) ≤ map f (𝓝 a)) : is_open_map f := | |
λ s hs, is_open_iff_mem_nhds.2 $ λ b ⟨a, has, hab⟩, | |
hab ▸ hf _ (image_mem_map $ is_open.mem_nhds hs has) | |
lemma of_sections {f : α → β} | |
(h : ∀ x, ∃ g : β → α, continuous_at g (f x) ∧ g (f x) = x ∧ right_inverse g f) : | |
is_open_map f := | |
of_nhds_le $ λ x, let ⟨g, hgc, hgx, hgf⟩ := h x in | |
calc 𝓝 (f x) = map f (map g (𝓝 (f x))) : by rw [map_map, hgf.comp_eq_id, map_id] | |
... ≤ map f (𝓝 (g (f x))) : map_mono hgc | |
... = map f (𝓝 x) : by rw hgx | |
lemma of_inverse {f : α → β} {f' : β → α} | |
(h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') : | |
is_open_map f := | |
of_sections $ λ x, ⟨f', h.continuous_at, r_inv _, l_inv⟩ | |
/-- A continuous surjective open map is a quotient map. -/ | |
lemma to_quotient_map {f : α → β} | |
(open_map : is_open_map f) (cont : continuous f) (surj : surjective f) : | |
quotient_map f := | |
quotient_map_iff.2 ⟨surj, λ s, ⟨λ h, h.preimage cont, λ h, surj.image_preimage s ▸ open_map _ h⟩⟩ | |
lemma interior_preimage_subset_preimage_interior (hf : is_open_map f) {s : set β} : | |
interior (f⁻¹' s) ⊆ f⁻¹' (interior s) := | |
hf.maps_to_interior (maps_to_preimage _ _) | |
lemma preimage_interior_eq_interior_preimage (hf₁ : is_open_map f) (hf₂ : continuous f) | |
(s : set β) : | |
f⁻¹' (interior s) = interior (f⁻¹' s) := | |
subset.antisymm | |
(preimage_interior_subset_interior_preimage hf₂) | |
(interior_preimage_subset_preimage_interior hf₁) | |
lemma preimage_closure_subset_closure_preimage (hf : is_open_map f) {s : set β} : | |
f ⁻¹' (closure s) ⊆ closure (f ⁻¹' s) := | |
begin | |
rw ← compl_subset_compl, | |
simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior] | |
end | |
lemma preimage_closure_eq_closure_preimage (hf : is_open_map f) (hfc : continuous f) (s : set β) : | |
f ⁻¹' (closure s) = closure (f ⁻¹' s) := | |
hf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s) | |
lemma preimage_frontier_subset_frontier_preimage (hf : is_open_map f) {s : set β} : | |
f ⁻¹' (frontier s) ⊆ frontier (f ⁻¹' s) := | |
by simpa only [frontier_eq_closure_inter_closure, preimage_inter] | |
using inter_subset_inter hf.preimage_closure_subset_closure_preimage | |
hf.preimage_closure_subset_closure_preimage | |
lemma preimage_frontier_eq_frontier_preimage (hf : is_open_map f) (hfc : continuous f) (s : set β) : | |
f ⁻¹' (frontier s) = frontier (f ⁻¹' s) := | |
by simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl, | |
hf.preimage_closure_eq_closure_preimage hfc] | |
end is_open_map | |
lemma is_open_map_iff_nhds_le [topological_space α] [topological_space β] {f : α → β} : | |
is_open_map f ↔ ∀(a:α), 𝓝 (f a) ≤ (𝓝 a).map f := | |
⟨λ hf, hf.nhds_le, is_open_map.of_nhds_le⟩ | |
lemma is_open_map_iff_interior [topological_space α] [topological_space β] {f : α → β} : | |
is_open_map f ↔ ∀ s, f '' (interior s) ⊆ interior (f '' s) := | |
⟨is_open_map.image_interior_subset, λ hs u hu, subset_interior_iff_open.mp $ | |
calc f '' u = f '' (interior u) : by rw hu.interior_eq | |
... ⊆ interior (f '' u) : hs u⟩ | |
/-- An inducing map with an open range is an open map. -/ | |
protected lemma inducing.is_open_map [topological_space α] [topological_space β] {f : α → β} | |
(hi : inducing f) (ho : is_open (range f)) : | |
is_open_map f := | |
is_open_map.of_nhds_le $ λ x, (hi.map_nhds_of_mem _ $ is_open.mem_nhds ho $ mem_range_self _).ge | |
section is_closed_map | |
variables [topological_space α] [topological_space β] | |
/-- A map `f : α → β` is said to be a *closed map*, if the image of any closed `U : set α` | |
is closed in `β`. -/ | |
def is_closed_map (f : α → β) := ∀ U : set α, is_closed U → is_closed (f '' U) | |
end is_closed_map | |
namespace is_closed_map | |
variables [topological_space α] [topological_space β] [topological_space γ] | |
open function | |
protected lemma id : is_closed_map (@id α) := assume s hs, by rwa image_id | |
protected lemma comp {g : β → γ} {f : α → β} (hg : is_closed_map g) (hf : is_closed_map f) : | |
is_closed_map (g ∘ f) := | |
by { intros s hs, rw image_comp, exact hg _ (hf _ hs) } | |
lemma closure_image_subset {f : α → β} (hf : is_closed_map f) (s : set α) : | |
closure (f '' s) ⊆ f '' closure s := | |
closure_minimal (image_subset _ subset_closure) (hf _ is_closed_closure) | |
lemma of_inverse {f : α → β} {f' : β → α} | |
(h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') : | |
is_closed_map f := | |
assume s hs, | |
have f' ⁻¹' s = f '' s, by ext x; simp [mem_image_iff_of_inverse r_inv l_inv], | |
this ▸ hs.preimage h | |
lemma of_nonempty {f : α → β} (h : ∀ s, is_closed s → s.nonempty → is_closed (f '' s)) : | |
is_closed_map f := | |
begin | |
intros s hs, cases eq_empty_or_nonempty s with h2s h2s, | |
{ simp_rw [h2s, image_empty, is_closed_empty] }, | |
{ exact h s hs h2s } | |
end | |
lemma closed_range {f : α → β} (hf : is_closed_map f) : is_closed (range f) := | |
@image_univ _ _ f ▸ hf _ is_closed_univ | |
end is_closed_map | |
lemma inducing.is_closed_map [topological_space α] [topological_space β] | |
{f : α → β} (hf : inducing f) (h : is_closed (range f)) : is_closed_map f := | |
begin | |
intros s hs, | |
rcases hf.is_closed_iff.1 hs with ⟨t, ht, rfl⟩, | |
rw image_preimage_eq_inter_range, | |
exact ht.inter h | |
end | |
lemma is_closed_map_iff_closure_image [topological_space α] [topological_space β] {f : α → β} : | |
is_closed_map f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s := | |
⟨is_closed_map.closure_image_subset, λ hs c hc, is_closed_of_closure_subset $ | |
calc closure (f '' c) ⊆ f '' (closure c) : hs c | |
... = f '' c : by rw hc.closure_eq⟩ | |
section open_embedding | |
variables [topological_space α] [topological_space β] [topological_space γ] | |
/-- An open embedding is an embedding with open image. -/ | |
@[mk_iff] | |
structure open_embedding (f : α → β) extends _root_.embedding f : Prop := | |
(open_range : is_open $ range f) | |
lemma open_embedding.is_open_map {f : α → β} (hf : open_embedding f) : is_open_map f := | |
hf.to_embedding.to_inducing.is_open_map hf.open_range | |
lemma open_embedding.map_nhds_eq {f : α → β} (hf : open_embedding f) (a : α) : | |
map f (𝓝 a) = 𝓝 (f a) := | |
hf.to_embedding.map_nhds_of_mem _ $ hf.open_range.mem_nhds $ mem_range_self _ | |
lemma open_embedding.open_iff_image_open {f : α → β} (hf : open_embedding f) | |
{s : set α} : is_open s ↔ is_open (f '' s) := | |
⟨hf.is_open_map s, | |
λ h, begin | |
convert ← h.preimage hf.to_embedding.continuous, | |
apply preimage_image_eq _ hf.inj | |
end⟩ | |
lemma open_embedding.tendsto_nhds_iff {ι : Type*} | |
{f : ι → β} {g : β → γ} {a : filter ι} {b : β} (hg : open_embedding g) : | |
tendsto f a (𝓝 b) ↔ tendsto (g ∘ f) a (𝓝 (g b)) := | |
hg.to_embedding.tendsto_nhds_iff | |
lemma open_embedding.continuous {f : α → β} (hf : open_embedding f) : continuous f := | |
hf.to_embedding.continuous | |
lemma open_embedding.open_iff_preimage_open {f : α → β} (hf : open_embedding f) | |
{s : set β} (hs : s ⊆ range f) : is_open s ↔ is_open (f ⁻¹' s) := | |
begin | |
convert ←hf.open_iff_image_open.symm, | |
rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left] | |
end | |
lemma open_embedding_of_embedding_open {f : α → β} (h₁ : embedding f) | |
(h₂ : is_open_map f) : open_embedding f := | |
⟨h₁, h₂.is_open_range⟩ | |
lemma open_embedding_iff_embedding_open {f : α → β} : | |
open_embedding f ↔ embedding f ∧ is_open_map f := | |
⟨λ h, ⟨h.1, h.is_open_map⟩, λ h, open_embedding_of_embedding_open h.1 h.2⟩ | |
lemma open_embedding_of_continuous_injective_open {f : α → β} (h₁ : continuous f) | |
(h₂ : injective f) (h₃ : is_open_map f) : open_embedding f := | |
begin | |
simp only [open_embedding_iff_embedding_open, embedding_iff, inducing_iff_nhds, *, and_true], | |
exact λ a, le_antisymm (h₁.tendsto _).le_comap | |
(@comap_map _ _ (𝓝 a) _ h₂ ▸ comap_mono (h₃.nhds_le _)) | |
end | |
lemma open_embedding_iff_continuous_injective_open {f : α → β} : | |
open_embedding f ↔ continuous f ∧ injective f ∧ is_open_map f := | |
⟨λ h, ⟨h.continuous, h.inj, h.is_open_map⟩, | |
λ h, open_embedding_of_continuous_injective_open h.1 h.2.1 h.2.2⟩ | |
lemma open_embedding_id : open_embedding (@id α) := | |
⟨embedding_id, is_open_map.id.is_open_range⟩ | |
lemma open_embedding.comp {g : β → γ} {f : α → β} | |
(hg : open_embedding g) (hf : open_embedding f) : open_embedding (g ∘ f) := | |
⟨hg.1.comp hf.1, (hg.is_open_map.comp hf.is_open_map).is_open_range⟩ | |
lemma open_embedding.is_open_map_iff {g : β → γ} {f : α → β} (hg : open_embedding g) : | |
is_open_map f ↔ is_open_map (g ∘ f) := | |
by simp only [is_open_map_iff_nhds_le, ← @map_map _ _ _ _ f g, ← hg.map_nhds_eq, | |
map_le_map_iff hg.inj] | |
lemma open_embedding.of_comp_iff (f : α → β) {g : β → γ} (hg : open_embedding g) : | |
open_embedding (g ∘ f) ↔ open_embedding f := | |
by simp only [open_embedding_iff_continuous_injective_open, ← hg.is_open_map_iff, | |
← hg.1.continuous_iff, hg.inj.of_comp_iff] | |
lemma open_embedding.of_comp (f : α → β) {g : β → γ} (hg : open_embedding g) | |
(h : open_embedding (g ∘ f)) : open_embedding f := | |
(open_embedding.of_comp_iff f hg).1 h | |
end open_embedding | |
section closed_embedding | |
variables [topological_space α] [topological_space β] [topological_space γ] | |
/-- A closed embedding is an embedding with closed image. -/ | |
@[mk_iff] | |
structure closed_embedding (f : α → β) extends _root_.embedding f : Prop := | |
(closed_range : is_closed $ range f) | |
variables {f : α → β} | |
lemma closed_embedding.tendsto_nhds_iff {ι : Type*} | |
{g : ι → α} {a : filter ι} {b : α} (hf : closed_embedding f) : | |
tendsto g a (𝓝 b) ↔ tendsto (f ∘ g) a (𝓝 (f b)) := | |
hf.to_embedding.tendsto_nhds_iff | |
lemma closed_embedding.continuous (hf : closed_embedding f) : continuous f := | |
hf.to_embedding.continuous | |
lemma closed_embedding.is_closed_map (hf : closed_embedding f) : is_closed_map f := | |
hf.to_embedding.to_inducing.is_closed_map hf.closed_range | |
lemma closed_embedding.closed_iff_image_closed (hf : closed_embedding f) | |
{s : set α} : is_closed s ↔ is_closed (f '' s) := | |
⟨hf.is_closed_map s, | |
λ h, begin | |
convert ←continuous_iff_is_closed.mp hf.continuous _ h, | |
apply preimage_image_eq _ hf.inj | |
end⟩ | |
lemma closed_embedding.closed_iff_preimage_closed (hf : closed_embedding f) | |
{s : set β} (hs : s ⊆ range f) : is_closed s ↔ is_closed (f ⁻¹' s) := | |
begin | |
convert ←hf.closed_iff_image_closed.symm, | |
rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left] | |
end | |
lemma closed_embedding_of_embedding_closed (h₁ : embedding f) | |
(h₂ : is_closed_map f) : closed_embedding f := | |
⟨h₁, by convert h₂ univ is_closed_univ; simp⟩ | |
lemma closed_embedding_of_continuous_injective_closed (h₁ : continuous f) | |
(h₂ : injective f) (h₃ : is_closed_map f) : closed_embedding f := | |
begin | |
refine closed_embedding_of_embedding_closed ⟨⟨_⟩, h₂⟩ h₃, | |
apply le_antisymm (continuous_iff_le_induced.mp h₁) _, | |
intro s', | |
change is_open _ ≤ is_open _, | |
rw [←is_closed_compl_iff, ←is_closed_compl_iff], | |
generalize : s'ᶜ = s, | |
rw is_closed_induced_iff, | |
refine λ hs, ⟨f '' s, h₃ s hs, _⟩, | |
rw preimage_image_eq _ h₂ | |
end | |
lemma closed_embedding_id : closed_embedding (@id α) := | |
⟨embedding_id, by convert is_closed_univ; apply range_id⟩ | |
lemma closed_embedding.comp {g : β → γ} {f : α → β} | |
(hg : closed_embedding g) (hf : closed_embedding f) : closed_embedding (g ∘ f) := | |
⟨hg.to_embedding.comp hf.to_embedding, show is_closed (range (g ∘ f)), | |
by rw [range_comp, ←hg.closed_iff_image_closed]; exact hf.closed_range⟩ | |
lemma closed_embedding.closure_image_eq {f : α → β} (hf : closed_embedding f) (s : set α) : | |
closure (f '' s) = f '' closure s := | |
le_antisymm (is_closed_map_iff_closure_image.mp hf.is_closed_map _) | |
(image_closure_subset_closure_image hf.continuous) | |
end closed_embedding | |