Specific classes of maps between topological spaces #

This file introduces the following properties of a map f : X → Y→ Y between topological spaces:

IsOpenMap f means the image of an open set under f is open.
IsClosedMap 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.
OpenEmbedding 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.
ClosedEmbedding 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.
QuotientMap 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 #

Tags #

open map, closed map, embedding, quotient map, identification map

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theorem inducing_iff {α : Type u_1} {β : Type u_2} [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : α → β) :

Inducing f ↔ tα = TopologicalSpace.induced f tβ

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structure Inducing {α : Type u_1} {β : Type u_2} [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : α → β) :

Prop

The topology on the domain is equal to the induced topology.
induced : tα = TopologicalSpace.induced f tβ

A function f : α → β→ 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.

Instances For

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theorem inducing_induced {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace β] (f : α → β) :

Inducing f

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theorem inducing_id {α : Type u_1} [inst : TopologicalSpace α] :

Inducing id

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theorem Inducing.comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Inducing g) (hf : Inducing f) :

Inducing (g ∘ f)

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theorem inducing_of_inducing_compose {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {f : α → β} {g : β → γ} (hf : Continuous f) (hg : Continuous g) (hgf : Inducing (g ∘ f)) :

Inducing f

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theorem inducing_iff_nhds {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} :

Inducing f ↔ ∀ (a : α), nhds a = Filter.comap f (nhds (f a))

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theorem Inducing.nhds_eq_comap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) (a : α) :

nhds a = Filter.comap f (nhds (f a))

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theorem Inducing.nhdsSet_eq_comap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) (s : Set α) :

nhdsSet s = Filter.comap f (nhdsSet (f '' s))

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theorem Inducing.map_nhds_eq {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) (a : α) :

Filter.map f (nhds a) = nhdsWithin (f a) (Set.range f)

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theorem Inducing.map_nhds_of_mem {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) (a : α) (h : Set.range f ∈ nhds (f a)) :

Filter.map f (nhds a) = nhds (f a)

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theorem Inducing.mapClusterPt_iff {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) {a : α} {l : Filter α} :

MapClusterPt (f a) l f ↔ ClusterPt a l

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theorem Inducing.image_mem_nhdsWithin {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) {a : α} {s : Set α} (hs : s ∈ nhds a) :

f '' s ∈ nhdsWithin (f a) (Set.range f)

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theorem Inducing.tendsto_nhds_iff {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {ι : Type u_1} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β} (hg : Inducing g) :

Filter.Tendsto f a (nhds b) ↔ Filter.Tendsto (g ∘ f) a (nhds (g b))

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theorem Inducing.continuousAt_iff {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {f : α → β} {g : β → γ} (hg : Inducing g) {x : α} :

ContinuousAt f x ↔ ContinuousAt (g ∘ f) x

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theorem Inducing.continuous_iff {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {f : α → β} {g : β → γ} (hg : Inducing g) :

Continuous f ↔ Continuous (g ∘ f)

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theorem Inducing.continuousAt_iff' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {f : α → β} {g : β → γ} (hf : Inducing f) {x : α} (h : Set.range f ∈ nhds (f x)) :

ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)

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theorem Inducing.continuous {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) :

Continuous f

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theorem Inducing.inducing_iff {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {f : α → β} {g : β → γ} (hg : Inducing g) :

Inducing f ↔ Inducing (g ∘ f)

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theorem Inducing.closure_eq_preimage_closure_image {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) (s : Set α) :

closure s = f ⁻¹' closure (f '' s)

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theorem Inducing.isClosed_iff {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) {s : Set α} :

IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s

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theorem Inducing.isClosed_iff' {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) {s : Set α} :

IsClosed s ↔ ∀ (x : α), f x ∈ closure (f '' s) → x ∈ s

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theorem Inducing.isClosed_preimage {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (h : Inducing f) (s : Set β) (hs : IsClosed s) :

IsClosed (f ⁻¹' s)

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theorem Inducing.isOpen_iff {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) {s : Set α} :

IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s

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theorem Inducing.dense_iff {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) {s : Set α} :

Dense s ↔ ∀ (x : α), f x ∈ closure (f '' s)

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theorem embedding_iff {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) :

Embedding f ↔ Inducing f ∧ Function.Injective f

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structure Embedding {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) extends Inducing :

Prop

A topological embedding is injective.
inj : Function.Injective f

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.

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theorem Function.Injective.embedding_induced {α : Type u_2} {β : Type u_1} [t : TopologicalSpace β] {f : α → β} (hf : Function.Injective f) :

Embedding f

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theorem Embedding.mk' {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) (inj : Function.Injective f) (induced : ∀ (a : α), Filter.comap f (nhds (f a)) = nhds a) :

Embedding f

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theorem embedding_id {α : Type u_1} [inst : TopologicalSpace α] :

Embedding id

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theorem Embedding.comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : Embedding g) (hf : Embedding f) :

Embedding (g ∘ f)

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theorem embedding_of_embedding_compose {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {f : α → β} {g : β → γ} (hf : Continuous f) (hg : Continuous g) (hgf : Embedding (g ∘ f)) :

Embedding f

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theorem Function.LeftInverse.embedding {α : Type u_2} {β : Type u_1} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} {g : β → α} (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) :

Embedding g

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theorem Embedding.map_nhds_eq {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Embedding f) (a : α) :

Filter.map f (nhds a) = nhdsWithin (f a) (Set.range f)

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theorem Embedding.map_nhds_of_mem {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Embedding f) (a : α) (h : Set.range f ∈ nhds (f a)) :

Filter.map f (nhds a) = nhds (f a)

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theorem Embedding.tendsto_nhds_iff {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {ι : Type u_1} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β} (hg : Embedding g) :

Filter.Tendsto f a (nhds b) ↔ Filter.Tendsto (g ∘ f) a (nhds (g b))

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theorem Embedding.continuous_iff {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {f : α → β} {g : β → γ} (hg : Embedding g) :

Continuous f ↔ Continuous (g ∘ f)

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theorem Embedding.continuous {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Embedding f) :

Continuous f

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theorem Embedding.closure_eq_preimage_closure_image {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {e : α → β} (he : Embedding e) (s : Set α) :

closure s = e ⁻¹' closure (e '' s)

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theorem Embedding.discreteTopology {X : Type u_1} {Y : Type u_2} [inst : TopologicalSpace X] [tY : TopologicalSpace Y] [inst : DiscreteTopology Y] {f : X → Y} (hf : Embedding f) :

DiscreteTopology X

The topology induced under an inclusion f : X → Y→ Y from the discrete topological space Y is the discrete topology on X.

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def QuotientMap {α : Type u_1} {β : Type u_2} [tα : TopologicalSpace α] [tβ : TopologicalSpace β] (f : α → β) :

Prop

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.

Equations

QuotientMap f = (Function.Surjective f ∧ tβ = TopologicalSpace.coinduced f tα)

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theorem quotientMap_iff {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} :

QuotientMap f ↔ Function.Surjective f ∧ ∀ (s : Set β), IsOpen s ↔ IsOpen (f ⁻¹' s)

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theorem QuotientMap.id {α : Type u_1} [inst : TopologicalSpace α] :

QuotientMap id

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theorem QuotientMap.comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : QuotientMap g) (hf : QuotientMap f) :

QuotientMap (g ∘ f)

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theorem QuotientMap.of_quotientMap_compose {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hf : Continuous f) (hg : Continuous g) (hgf : QuotientMap (g ∘ f)) :

QuotientMap g

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theorem QuotientMap.of_inverse {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} {g : β → α} (hf : Continuous f) (hg : Continuous g) (h : Function.LeftInverse g f) :

QuotientMap g

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theorem QuotientMap.continuous_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hf : QuotientMap f) :

Continuous g ↔ Continuous (g ∘ f)

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theorem QuotientMap.continuous {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : QuotientMap f) :

Continuous f

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theorem QuotientMap.surjective {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : QuotientMap f) :

Function.Surjective f

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theorem QuotientMap.isOpen_preimage {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : QuotientMap f) {s : Set β} :

IsOpen (f ⁻¹' s) ↔ IsOpen s

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theorem QuotientMap.isClosed_preimage {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : QuotientMap f) {s : Set β} :

IsClosed (f ⁻¹' s) ↔ IsClosed s

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def IsOpenMap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) :

Prop

A map f : α → β→ β is said to be an open map, if the image of any open U : set α is open in β.

Equations

IsOpenMap f = ∀ (U : Set α), IsOpen U → IsOpen (f '' U)

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theorem IsOpenMap.id {α : Type u_1} [inst : TopologicalSpace α] :

IsOpenMap id

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theorem IsOpenMap.comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : IsOpenMap g) (hf : IsOpenMap f) :

IsOpenMap (g ∘ f)

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theorem IsOpenMap.isOpen_range {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) :

IsOpen (Set.range f)

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theorem IsOpenMap.image_mem_nhds {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) {x : α} {s : Set α} (hx : s ∈ nhds x) :

f '' s ∈ nhds (f x)

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theorem IsOpenMap.range_mem_nhds {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) (x : α) :

Set.range f ∈ nhds (f x)

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theorem IsOpenMap.mapsTo_interior {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) {s : Set α} {t : Set β} (h : Set.MapsTo f s t) :

Set.MapsTo f (interior s) (interior t)

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theorem IsOpenMap.image_interior_subset {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) (s : Set α) :

f '' interior s ⊆ interior (f '' s)

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theorem IsOpenMap.nhds_le {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) (a : α) :

nhds (f a) ≤ Filter.map f (nhds a)

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theorem IsOpenMap.of_nhds_le {α : Type u_2} {β : Type u_1} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : ∀ (a : α), nhds (f a) ≤ Filter.map f (nhds a)) :

IsOpenMap f

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theorem IsOpenMap.of_sections {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (h : ∀ (x : α), ∃ g, ContinuousAt g (f x) ∧ g (f x) = x ∧ Function.RightInverse g f) :

IsOpenMap f

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theorem IsOpenMap.of_inverse {α : Type u_2} {β : Type u_1} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} {f' : β → α} (h : Continuous f') (l_inv : Function.LeftInverse f f') (r_inv : Function.RightInverse f f') :

IsOpenMap f

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theorem IsOpenMap.to_quotientMap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (open_map : IsOpenMap f) (cont : Continuous f) (surj : Function.Surjective f) :

QuotientMap f

A continuous surjective open map is a quotient map.

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theorem IsOpenMap.interior_preimage_subset_preimage_interior {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) {s : Set β} :

interior (f ⁻¹' s) ⊆ f ⁻¹' interior s

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theorem IsOpenMap.preimage_interior_eq_interior_preimage {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf₁ : IsOpenMap f) (hf₂ : Continuous f) (s : Set β) :

f ⁻¹' interior s = interior (f ⁻¹' s)

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theorem IsOpenMap.preimage_closure_subset_closure_preimage {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) {s : Set β} :

f ⁻¹' closure s ⊆ closure (f ⁻¹' s)

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theorem IsOpenMap.preimage_closure_eq_closure_preimage {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) (hfc : Continuous f) (s : Set β) :

f ⁻¹' closure s = closure (f ⁻¹' s)

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theorem IsOpenMap.preimage_frontier_subset_frontier_preimage {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) {s : Set β} :

f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s)

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theorem IsOpenMap.preimage_frontier_eq_frontier_preimage {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsOpenMap f) (hfc : Continuous f) (s : Set β) :

f ⁻¹' frontier s = frontier (f ⁻¹' s)

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theorem isOpenMap_iff_nhds_le {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} :

IsOpenMap f ↔ ∀ (a : α), nhds (f a) ≤ Filter.map f (nhds a)

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theorem isOpenMap_iff_interior {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} :

IsOpenMap f ↔ ∀ (s : Set α), f '' interior s ⊆ interior (f '' s)

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theorem Inducing.isOpenMap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hi : Inducing f) (ho : IsOpen (Set.range f)) :

IsOpenMap f

An inducing map with an open range is an open map.

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def IsClosedMap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) :

Prop

A map f : α → β→ β is said to be a closed map, if the image of any closed U : set α is closed in β.

Equations

IsClosedMap f = ∀ (U : Set α), IsClosed U → IsClosed (f '' U)

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theorem IsClosedMap.id {α : Type u_1} [inst : TopologicalSpace α] :

IsClosedMap id

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theorem IsClosedMap.comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : IsClosedMap g) (hf : IsClosedMap f) :

IsClosedMap (g ∘ f)

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theorem IsClosedMap.closure_image_subset {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsClosedMap f) (s : Set α) :

closure (f '' s) ⊆ f '' closure s

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theorem IsClosedMap.of_inverse {α : Type u_2} {β : Type u_1} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} {f' : β → α} (h : Continuous f') (l_inv : Function.LeftInverse f f') (r_inv : Function.RightInverse f f') :

IsClosedMap f

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theorem IsClosedMap.of_nonempty {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (h : ∀ (s : Set α), IsClosed s → Set.Nonempty s → IsClosed (f '' s)) :

IsClosedMap f

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theorem IsClosedMap.closed_range {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : IsClosedMap f) :

IsClosed (Set.range f)

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theorem Inducing.isClosedMap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : Inducing f) (h : IsClosed (Set.range f)) :

IsClosedMap f

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theorem isClosedMap_iff_closure_image {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} :

IsClosedMap f ↔ ∀ (s : Set α), closure (f '' s) ⊆ f '' closure s

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theorem openEmbedding_iff {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) :

OpenEmbedding f ↔ Embedding f ∧ IsOpen (Set.range f)

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structure OpenEmbedding {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) extends Embedding :

Prop

The range of an open embedding is an open set.
open_range : IsOpen (Set.range f)

An open embedding is an embedding with open image.

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theorem OpenEmbedding.isOpenMap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : OpenEmbedding f) :

IsOpenMap f

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theorem OpenEmbedding.map_nhds_eq {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : OpenEmbedding f) (a : α) :

Filter.map f (nhds a) = nhds (f a)

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theorem OpenEmbedding.open_iff_image_open {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : OpenEmbedding f) {s : Set α} :

IsOpen s ↔ IsOpen (f '' s)

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theorem OpenEmbedding.tendsto_nhds_iff {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {ι : Type u_1} {f : ι → β} {g : β → γ} {a : Filter ι} {b : β} (hg : OpenEmbedding g) :

Filter.Tendsto f a (nhds b) ↔ Filter.Tendsto (g ∘ f) a (nhds (g b))

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theorem OpenEmbedding.tendsto_nhds_iff' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : OpenEmbedding f) {g : β → γ} {l : Filter γ} {a : α} :

Filter.Tendsto (g ∘ f) (nhds a) l ↔ Filter.Tendsto g (nhds (f a)) l

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theorem OpenEmbedding.continuous {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : OpenEmbedding f) :

Continuous f

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theorem OpenEmbedding.open_iff_preimage_open {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : OpenEmbedding f) {s : Set β} (hs : s ⊆ Set.range f) :

IsOpen s ↔ IsOpen (f ⁻¹' s)

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theorem openEmbedding_of_embedding_open {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (h₁ : Embedding f) (h₂ : IsOpenMap f) :

OpenEmbedding f

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theorem openEmbedding_iff_embedding_open {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} :

OpenEmbedding f ↔ Embedding f ∧ IsOpenMap f

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theorem openEmbedding_of_continuous_injective_open {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (h₁ : Continuous f) (h₂ : Function.Injective f) (h₃ : IsOpenMap f) :

OpenEmbedding f

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theorem openEmbedding_iff_continuous_injective_open {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} :

OpenEmbedding f ↔ Continuous f ∧ Function.Injective f ∧ IsOpenMap f

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theorem openEmbedding_id {α : Type u_1} [inst : TopologicalSpace α] :

OpenEmbedding id

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theorem OpenEmbedding.comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : OpenEmbedding g) (hf : OpenEmbedding f) :

OpenEmbedding (g ∘ f)

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theorem OpenEmbedding.isOpenMap_iff {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : OpenEmbedding g) :

IsOpenMap f ↔ IsOpenMap (g ∘ f)

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theorem OpenEmbedding.of_comp_iff {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] (f : α → β) {g : β → γ} (hg : OpenEmbedding g) :

OpenEmbedding (g ∘ f) ↔ OpenEmbedding f

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theorem OpenEmbedding.of_comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] (f : α → β) {g : β → γ} (hg : OpenEmbedding g) (h : OpenEmbedding (g ∘ f)) :

OpenEmbedding f

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theorem closedEmbedding_iff {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) :

ClosedEmbedding f ↔ Embedding f ∧ IsClosed (Set.range f)

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structure ClosedEmbedding {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] (f : α → β) extends Embedding :

Prop

The range of a closed embedding is a closed set.
closed_range : IsClosed (Set.range f)

A closed embedding is an embedding with closed image.

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theorem ClosedEmbedding.tendsto_nhds_iff {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} {ι : Type u_1} {g : ι → α} {a : Filter ι} {b : α} (hf : ClosedEmbedding f) :

Filter.Tendsto g a (nhds b) ↔ Filter.Tendsto (f ∘ g) a (nhds (f b))

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theorem ClosedEmbedding.continuous {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : ClosedEmbedding f) :

Continuous f

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theorem ClosedEmbedding.isClosedMap {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : ClosedEmbedding f) :

IsClosedMap f

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theorem ClosedEmbedding.closed_iff_image_closed {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : ClosedEmbedding f) {s : Set α} :

IsClosed s ↔ IsClosed (f '' s)

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theorem ClosedEmbedding.closed_iff_preimage_closed {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : ClosedEmbedding f) {s : Set β} (hs : s ⊆ Set.range f) :

IsClosed s ↔ IsClosed (f ⁻¹' s)

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theorem closedEmbedding_of_embedding_closed {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (h₁ : Embedding f) (h₂ : IsClosedMap f) :

ClosedEmbedding f

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theorem closedEmbedding_of_continuous_injective_closed {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (h₁ : Continuous f) (h₂ : Function.Injective f) (h₃ : IsClosedMap f) :

ClosedEmbedding f

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theorem closedEmbedding_id {α : Type u_1} [inst : TopologicalSpace α] :

ClosedEmbedding id

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theorem ClosedEmbedding.comp {α : Type u_3} {β : Type u_1} {γ : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] [inst : TopologicalSpace γ] {g : β → γ} {f : α → β} (hg : ClosedEmbedding g) (hf : ClosedEmbedding f) :

ClosedEmbedding (g ∘ f)

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theorem ClosedEmbedding.closure_image_eq {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst : TopologicalSpace β] {f : α → β} (hf : ClosedEmbedding f) (s : Set α) :

closure (f '' s) = f '' closure s