Specific classes of maps between topological spaces

This file introduces the following properties of a map f : X → 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.)

• IsInducing 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.

• IsEmbedding f means f is inducing and also injective. Equivalently, f identifies X with a subspace of Y.

• IsOpenEmbedding 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.

• IsClosedEmbedding 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.

• IsQuotientMap f is the dual condition to IsEmbedding 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

theorem Topology.IsInducing.induced {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] (f : X → Y) : IsInducing f

theorem Topology.IsInducing.id {X : Type u_1} [TopologicalSpace X] : IsInducing id

theorem Topology.IsInducing.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : IsInducing g) (hf : IsInducing f) : IsInducing (g ∘ f)

theorem Topology.IsInducing.of_comp_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : IsInducing g) : IsInducing (g ∘ f) ↔ IsInducing f

theorem Topology.IsInducing.of_comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hf : Continuous f) (hg : Continuous g) (hgf : IsInducing (g ∘ f)) : IsInducing f

theorem Topology.isInducing_iff_nhds {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] : IsInducing f ↔ ∀ (x : X), nhds x = Filter.comap f (nhds (f x))

theorem Topology.IsInducing.nhds_eq_comap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) (x : X) : nhds x = Filter.comap f (nhds (f x))

theorem Topology.IsInducing.basis_nhds {X : Type u_1} {Y : Type u_2} {ι : Type u_4} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] {p : ι → Prop} {s : ι → Set Y} (hf : IsInducing f) {x : X} (h_basis : (nhds (f x)).HasBasis p s) : (nhds x).HasBasis p (Set.preimage f ∘ s)

theorem Topology.IsInducing.nhdsSet_eq_comap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) (s : Set X) : nhdsSet s = Filter.comap f (nhdsSet (f '' s))

theorem Topology.IsInducing.map_nhds_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) (x : X) : Filter.map f (nhds x) = nhdsWithin (f x) (Set.range f)

theorem Topology.IsInducing.map_nhds_of_mem {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) (x : X) (h : Set.range f ∈ nhds (f x)) : Filter.map f (nhds x) = nhds (f x)

theorem Topology.IsInducing.mapClusterPt_iff {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) {x : X} {l : Filter X} : MapClusterPt (f x) l f ↔ ClusterPt x l

theorem Topology.IsInducing.image_mem_nhdsWithin {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) {x : X} {s : Set X} (hs : s ∈ nhds x) : f '' s ∈ nhdsWithin (f x) (Set.range f)

theorem Topology.IsInducing.tendsto_nhds_iff {Y : Type u_2} {Z : Type u_3} {ι : Type u_4} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace Z] {f : ι → Y} {l : Filter ι} {y : Y} (hg : IsInducing g) : Filter.Tendsto f l (nhds y) ↔ Filter.Tendsto (g ∘ f) l (nhds (g y))

theorem Topology.IsInducing.continuousAt_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : IsInducing g) {x : X} : ContinuousAt f x ↔ ContinuousAt (g ∘ f) x

theorem Topology.IsInducing.continuous_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hg : IsInducing g) : Continuous f ↔ Continuous (g ∘ f)

theorem Topology.IsInducing.continuousAt_iff' {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace X] [TopologicalSpace Z] (hf : IsInducing f) {x : X} (h : Set.range f ∈ nhds (f x)) : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)

theorem Topology.IsInducing.continuous {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) : Continuous f

theorem Topology.IsInducing.closure_eq_preimage_closure_image {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) (s : Set X) : closure s = f ⁻¹' closure (f '' s)

theorem Topology.IsInducing.isClosed_iff {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) {s : Set X} : IsClosed s ↔ ∃ (t : Set Y), IsClosed t ∧ f ⁻¹' t = s

theorem Topology.IsInducing.isClosed_iff' {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) {s : Set X} : IsClosed s ↔ ∀ (x : X), f x ∈ closure (f '' s) → x ∈ s

theorem Topology.IsInducing.isClosed_preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (h : IsInducing f) (s : Set Y) (hs : IsClosed s) : IsClosed (f ⁻¹' s)

theorem Topology.IsInducing.isOpen_iff {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) {s : Set X} : IsOpen s ↔ ∃ (t : Set Y), IsOpen t ∧ f ⁻¹' t = s

theorem Topology.IsInducing.setOf_isOpen {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) : {s : Set X | IsOpen s} = Set.preimage f '' {t : Set Y | IsOpen t}

theorem Topology.IsInducing.dense_iff {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace Y] [TopologicalSpace X] (hf : IsInducing f) {s : Set X} : Dense s ↔ ∀ (x : X), f x ∈ closure (f '' s)

theorem Topology.IsInducing.of_subsingleton {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] [TopologicalSpace X] [Subsingleton X] (f : X → Y) : IsInducing f

theorem Topology.IsEmbedding.induced {X : Type u_1} {Y : Type u_2} {f : X → Y} [t : TopologicalSpace Y] (hf : Function.Injective f) : IsEmbedding f

theorem Function.Injective.isEmbedding_induced {X : Type u_1} {Y : Type u_2} {f : X → Y} [t : TopologicalSpace Y] (hf : Injective f) : Topology.IsEmbedding f

Alias of Topology.IsEmbedding.induced.

theorem Topology.IsEmbedding.isInducing {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsEmbedding f) : IsInducing f

theorem Topology.IsEmbedding.mk' {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (f : X → Y) (inj : Function.Injective f) (induced : ∀ (x : X), Filter.comap f (nhds (f x)) = nhds x) : IsEmbedding f

theorem Topology.IsEmbedding.id {X : Type u_1} [TopologicalSpace X] : IsEmbedding id

theorem Topology.IsEmbedding.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsEmbedding g) (hf : IsEmbedding f) : IsEmbedding (g ∘ f)

theorem Topology.IsEmbedding.of_comp_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsEmbedding g) : IsEmbedding (g ∘ f) ↔ IsEmbedding f

theorem Topology.IsEmbedding.of_comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hf : Continuous f) (hg : Continuous g) (hgf : IsEmbedding (g ∘ f)) : IsEmbedding f

theorem Topology.IsEmbedding.of_leftInverse {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {g : Y → X} (h : Function.LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : IsEmbedding g

theorem Function.LeftInverse.isEmbedding {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {g : Y → X} (h : LeftInverse f g) (hf : Continuous f) (hg : Continuous g) : Topology.IsEmbedding g

Alias of Topology.IsEmbedding.of_leftInverse.

theorem Topology.IsEmbedding.map_nhds_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsEmbedding f) (x : X) : Filter.map f (nhds x) = nhdsWithin (f x) (Set.range f)

theorem Topology.IsEmbedding.map_nhds_of_mem {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsEmbedding f) (x : X) (h : Set.range f ∈ nhds (f x)) : Filter.map f (nhds x) = nhds (f x)

theorem Topology.IsEmbedding.tendsto_nhds_iff {Y : Type u_2} {Z : Type u_3} {ι : Type u_4} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace Z] {f : ι → Y} {l : Filter ι} {y : Y} (hg : IsEmbedding g) : Filter.Tendsto f l (nhds y) ↔ Filter.Tendsto (g ∘ f) l (nhds (g y))

theorem Topology.IsEmbedding.continuous_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsEmbedding g) : Continuous f ↔ Continuous (g ∘ f)

theorem Topology.IsEmbedding.continuous {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsEmbedding f) : Continuous f

theorem Topology.IsEmbedding.closure_eq_preimage_closure_image {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsEmbedding f) (s : Set X) : closure s = f ⁻¹' closure (f '' s)

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

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

See also DiscreteTopology.of_continuous_injective.

theorem Topology.IsEmbedding.of_subsingleton {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Subsingleton X] (f : X → Y) : IsEmbedding f

theorem Topology.isQuotientMap_iff {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsQuotientMap f ↔ Function.Surjective f ∧ ∀ (s : Set Y), IsOpen s ↔ IsOpen (f ⁻¹' s)

theorem Topology.isQuotientMap_iff_isClosed {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsQuotientMap f ↔ Function.Surjective f ∧ ∀ (s : Set Y), IsClosed s ↔ IsClosed (f ⁻¹' s)

theorem Topology.IsQuotientMap.id {X : Type u_1} [TopologicalSpace X] : IsQuotientMap id

theorem Topology.IsQuotientMap.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsQuotientMap g) (hf : IsQuotientMap f) : IsQuotientMap (g ∘ f)

theorem Topology.IsQuotientMap.of_comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hf : Continuous f) (hg : Continuous g) (hgf : IsQuotientMap (g ∘ f)) : IsQuotientMap g

theorem Topology.IsQuotientMap.of_inverse {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] {g : Y → X} (hf : Continuous f) (hg : Continuous g) (h : Function.LeftInverse g f) : IsQuotientMap g

theorem Topology.IsQuotientMap.continuous_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hf : IsQuotientMap f) : Continuous g ↔ Continuous (g ∘ f)

theorem Topology.IsQuotientMap.continuous {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsQuotientMap f) : Continuous f

theorem Topology.IsQuotientMap.isOpen_preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsQuotientMap f) {s : Set Y} : IsOpen (f ⁻¹' s) ↔ IsOpen s

theorem Topology.IsQuotientMap.isClosed_preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsQuotientMap f) {s : Set Y} : IsClosed (f ⁻¹' s) ↔ IsClosed s

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

theorem IsOpenMap.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsOpenMap g) (hf : IsOpenMap f) : IsOpenMap (g ∘ f)

theorem IsOpenMap.isOpen_range {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) : IsOpen (Set.range f)

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

theorem IsOpenMap.range_mem_nhds {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (x : X) : Set.range f ∈ nhds (f x)

theorem IsOpenMap.mapsTo_interior {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {s : Set X} {t : Set Y} (h : Set.MapsTo f s t) : Set.MapsTo f (interior s) (interior t)

theorem IsOpenMap.image_interior_subset {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (s : Set X) : f '' interior s ⊆ interior (f '' s)

theorem IsOpenMap.nhds_le {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (x : X) : nhds (f x) ≤ Filter.map f (nhds x)

theorem IsOpenMap.map_nhds_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {x : X} (hf' : ContinuousAt f x) : Filter.map f (nhds x) = nhds (f x)

theorem IsOpenMap.map_nhdsSet_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (hf' : Continuous f) (s : Set X) : Filter.map f (nhdsSet s) = nhdsSet (f '' s)

theorem IsOpenMap.of_nhds_le {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : ∀ (x : X), nhds (f x) ≤ Filter.map f (nhds x)) : IsOpenMap f

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

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

theorem IsOpenMap.isQuotientMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (open_map : IsOpenMap f) (cont : Continuous f) (surj : Function.Surjective f) : Topology.IsQuotientMap f

A continuous surjective open map is a quotient map.

theorem IsOpenMap.interior_preimage_subset_preimage_interior {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {s : Set Y} : interior (f ⁻¹' s) ⊆ f ⁻¹' interior s

theorem IsOpenMap.preimage_interior_eq_interior_preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf₁ : IsOpenMap f) (hf₂ : Continuous f) (s : Set Y) : f ⁻¹' interior s = interior (f ⁻¹' s)

theorem IsOpenMap.preimage_closure_subset_closure_preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {s : Set Y} : f ⁻¹' closure s ⊆ closure (f ⁻¹' s)

theorem IsOpenMap.preimage_closure_eq_closure_preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) : f ⁻¹' closure s = closure (f ⁻¹' s)

theorem IsOpenMap.preimage_frontier_subset_frontier_preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {s : Set Y} : f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s)

theorem IsOpenMap.preimage_frontier_eq_frontier_preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) : f ⁻¹' frontier s = frontier (f ⁻¹' s)

theorem IsOpenMap.of_isEmpty {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [h : IsEmpty X] (f : X → Y) : IsOpenMap f

theorem IsOpenMap.clusterPt_comap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {x : X} {l : Filter Y} (h : ClusterPt (f x) l) : ClusterPt x (Filter.comap f l)

theorem isOpenMap_iff_kernImage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenMap f ↔ ∀ {u : Set X}, IsClosed u → IsClosed (Set.kernImage f u)

theorem isOpenMap_iff_nhds_le {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenMap f ↔ ∀ (x : X), nhds (f x) ≤ Filter.map f (nhds x)

theorem isOpenMap_iff_clusterPt_comap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenMap f ↔ ∀ (x : X) (l : Filter Y), ClusterPt (f x) l → ClusterPt x (Filter.comap f l)

theorem isOpenMap_iff_image_interior {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenMap f ↔ ∀ (s : Set X), f '' interior s ⊆ interior (f '' s)

@[deprecated isOpenMap_iff_image_interior (since := "2025-08-30")]

theorem isOpenMap_iff_interior {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenMap f ↔ ∀ (s : Set X), f '' interior s ⊆ interior (f '' s)

Alias of isOpenMap_iff_image_interior.

theorem isOpenMap_iff_closure_kernImage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenMap f ↔ ∀ {s : Set X}, closure (Set.kernImage f s) ⊆ Set.kernImage f (closure s)

A map is open if and only if the Set.kernImage of every closed set is closed.

theorem Topology.IsInducing.isOpenMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hi : IsInducing f) (ho : IsOpen (Set.range f)) : IsOpenMap f

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

theorem Dense.preimage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] {s : Set Y} (hs : Dense s) (hf : IsOpenMap f) : Dense (f ⁻¹' s)

Preimage of a dense set under an open map is dense.

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

theorem IsClosedMap.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsClosedMap g) (hf : IsClosedMap f) : IsClosedMap (g ∘ f)

theorem IsClosedMap.of_comp_surjective {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hf : Function.Surjective f) (hf' : Continuous f) (hfg : IsClosedMap (g ∘ f)) : IsClosedMap g

theorem IsClosedMap.closure_image_subset {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) (s : Set X) : closure (f '' s) ⊆ f '' closure s

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

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

theorem IsClosedMap.isClosed_range {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) : IsClosed (Set.range f)

theorem IsClosedMap.isQuotientMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hcl : IsClosedMap f) (hcont : Continuous f) (hsurj : Function.Surjective f) : Topology.IsQuotientMap f

theorem isClosedMap_iff_kernImage {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f ↔ ∀ {u : Set X}, IsOpen u → IsOpen (Set.kernImage f u)

A map is closed if and only if the Set.kernImage of every open set is open.

One way to understand this result is that f : X → Y is closed if and only if its fibers vary in an upper hemicontinuous way: for any open subset U ⊆ X, the set of all y ∈ Y such that f ⁻¹' {y} ⊆ U is open in Y.

theorem Topology.IsInducing.isClosedMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsInducing f) (h : IsClosed (Set.range f)) : IsClosedMap f

theorem isClosedMap_iff_closure_image {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f ↔ ∀ (s : Set X), closure (f '' s) ⊆ f '' closure s

theorem isClosedMap_iff_kernImage_interior {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f ↔ ∀ {s : Set X}, Set.kernImage f (interior s) ⊆ interior (Set.kernImage f s)

theorem isClosedMap_iff_clusterPt {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f ↔ ∀ (s : Set X) (y : Y), MapClusterPt y (Filter.principal s) f → ∃ (x : X), f x = y ∧ ClusterPt x (Filter.principal s)

A map f : X → Y is closed if and only if for all sets s, any cluster point of f '' s is the image by f of some cluster point of s. If you require this for all filters instead of just principal filters, and also that f is continuous, you get the notion of proper map. See isProperMap_iff_clusterPt.

theorem isClosedMap_iff_comap_nhdsSet_le {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f ↔ ∀ {s : Set Y}, Filter.comap f (nhdsSet s) ≤ nhdsSet (f ⁻¹' s)

theorem IsClosedMap.comap_nhdsSet_le {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f → ∀ {s : Set Y}, Filter.comap f (nhdsSet s) ≤ nhdsSet (f ⁻¹' s)

Alias of the forward direction of isClosedMap_iff_comap_nhdsSet_le.

theorem isClosedMap_iff_comap_nhds_le {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f ↔ ∀ {y : Y}, Filter.comap f (nhds y) ≤ nhdsSet (f ⁻¹' {y})

theorem IsClosedMap.comap_nhds_le {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsClosedMap f → ∀ {y : Y}, Filter.comap f (nhds y) ≤ nhdsSet (f ⁻¹' {y})

Alias of the forward direction of isClosedMap_iff_comap_nhds_le.

theorem IsClosedMap.comap_nhds_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) (hf' : Continuous f) (y : Y) : Filter.comap f (nhds y) = nhdsSet (f ⁻¹' {y})

theorem IsClosedMap.comap_nhdsSet_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) (hf' : Continuous f) (s : Set Y) : Filter.comap f (nhdsSet s) = nhdsSet (f ⁻¹' s)

theorem IsClosedMap.eventually_nhds_fiber {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) {p : X → Prop} (y₀ : Y) (H : ∀ x₀ ∈ f ⁻¹' {y₀}, ∀ᶠ (x : X) in nhds x₀, p x) : ∀ᶠ (y : Y) in nhds y₀, ∀ x ∈ f ⁻¹' {y}, p x

Assume f is a closed map. If some property p holds around every point in the fiber of f at y₀, then for any y close enough to y₀ we have that p holds on the fiber at y.

theorem IsClosedMap.frequently_nhds_fiber {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) {p : X → Prop} (y₀ : Y) (H : ∃ᶠ (y : Y) in nhds y₀, ∃ x ∈ f ⁻¹' {y}, p x) : ∃ x₀ ∈ f ⁻¹' {y₀}, ∃ᶠ (x : X) in nhds x₀, p x

Assume f is a closed map. If there are points y arbitrarily close to y₀ such that p holds for at least some x ∈ f ⁻¹' {y}, then one can find x₀ ∈ f ⁻¹' {y₀} such that there are points x arbitrarily close to x₀ which satisfy p.

theorem IsClosedMap.closure_image_eq_of_continuous {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (f_closed : IsClosedMap f) (f_cont : Continuous f) (s : Set X) : closure (f '' s) = f '' closure s

theorem IsClosedMap.lift'_closure_map_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (f_closed : IsClosedMap f) (f_cont : Continuous f) (F : Filter X) : (Filter.map f F).lift' closure = Filter.map f (F.lift' closure)

theorem IsClosedMap.mapClusterPt_iff_lift'_closure {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] {F : Filter X} (f_closed : IsClosedMap f) (f_cont : Continuous f) {y : Y} : MapClusterPt y F f ↔ (F.lift' closure ⊓ Filter.principal (f ⁻¹' {y})).NeBot

theorem Topology.IsOpenEmbedding.isEmbedding {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenEmbedding f) : IsEmbedding f

theorem Topology.IsOpenEmbedding.isInducing {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenEmbedding f) : IsInducing f

theorem Topology.IsOpenEmbedding.isOpenMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenEmbedding f) : IsOpenMap f

theorem Topology.IsOpenEmbedding.map_nhds_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenEmbedding f) (x : X) : Filter.map f (nhds x) = nhds (f x)

theorem Topology.IsOpenEmbedding.isOpen_iff_image_isOpen {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenEmbedding f) {s : Set X} : IsOpen s ↔ IsOpen (f '' s)

theorem Topology.IsOpenEmbedding.tendsto_nhds_iff {Y : Type u_2} {Z : Type u_3} {ι : Type u_4} {g : Y → Z} [TopologicalSpace Y] [TopologicalSpace Z] {f : ι → Y} {l : Filter ι} {y : Y} (hg : IsOpenEmbedding g) : Filter.Tendsto f l (nhds y) ↔ Filter.Tendsto (g ∘ f) l (nhds (g y))

theorem Topology.IsOpenEmbedding.tendsto_nhds_iff' {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenEmbedding f) {l : Filter Z} {x : X} : Filter.Tendsto (g ∘ f) (nhds x) l ↔ Filter.Tendsto g (nhds (f x)) l

theorem Topology.IsOpenEmbedding.continuousAt_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hf : IsOpenEmbedding f) {x : X} : ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)

theorem Topology.IsOpenEmbedding.continuous {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenEmbedding f) : Continuous f

theorem Topology.IsOpenEmbedding.isOpen_iff_preimage_isOpen {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenEmbedding f) {s : Set Y} (hs : s ⊆ Set.range f) : IsOpen s ↔ IsOpen (f ⁻¹' s)

theorem Topology.IsOpenEmbedding.of_isEmbedding_isOpenMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : IsEmbedding f) (h₂ : IsOpenMap f) : IsOpenEmbedding f

theorem Topology.IsEmbedding.isOpenEmbedding_of_surjective {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsEmbedding f) (hsurj : Function.Surjective f) : IsOpenEmbedding f

A surjective embedding is an IsOpenEmbedding.

theorem Topology.IsOpenEmbedding.of_isEmbedding {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsEmbedding f) (hsurj : Function.Surjective f) : IsOpenEmbedding f

Alias of Topology.IsEmbedding.isOpenEmbedding_of_surjective.

---

A surjective embedding is an IsOpenEmbedding.

theorem Topology.isOpenEmbedding_iff_isEmbedding_isOpenMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenEmbedding f ↔ IsEmbedding f ∧ IsOpenMap f

theorem Topology.IsOpenEmbedding.of_continuous_injective_isOpenMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : Continuous f) (h₂ : Function.Injective f) (h₃ : IsOpenMap f) : IsOpenEmbedding f

theorem Topology.isOpenEmbedding_iff_continuous_injective_isOpenMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] : IsOpenEmbedding f ↔ Continuous f ∧ Function.Injective f ∧ IsOpenMap f

theorem Topology.IsOpenEmbedding.id {X : Type u_1} [TopologicalSpace X] : IsOpenEmbedding id

theorem Topology.IsOpenEmbedding.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsOpenEmbedding g) (hf : IsOpenEmbedding f) : IsOpenEmbedding (g ∘ f)

theorem Topology.IsOpenEmbedding.isOpenMap_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsOpenEmbedding g) : IsOpenMap f ↔ IsOpenMap (g ∘ f)

theorem Topology.IsOpenEmbedding.of_comp_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X → Y) (hg : IsOpenEmbedding g) : IsOpenEmbedding (g ∘ f) ↔ IsOpenEmbedding f

theorem Topology.IsOpenEmbedding.of_comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X → Y) (hg : IsOpenEmbedding g) (h : IsOpenEmbedding (g ∘ f)) : IsOpenEmbedding f

theorem Topology.IsOpenEmbedding.of_isEmpty {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [IsEmpty X] (f : X → Y) : IsOpenEmbedding f

theorem Topology.IsOpenEmbedding.image_mem_nhds {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : IsOpenEmbedding f) {s : Set X} {x : X} : f '' s ∈ nhds (f x) ↔ s ∈ nhds x

theorem Topology.IsClosedEmbedding.isEmbedding {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedEmbedding f) : IsEmbedding f

theorem Topology.IsClosedEmbedding.isInducing {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedEmbedding f) : IsInducing f

theorem Topology.IsClosedEmbedding.continuous {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedEmbedding f) : Continuous f

theorem Topology.IsClosedEmbedding.tendsto_nhds_iff {X : Type u_1} {Y : Type u_2} {ι : Type u_4} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] {g : ι → X} {l : Filter ι} {x : X} (hf : IsClosedEmbedding f) : Filter.Tendsto g l (nhds x) ↔ Filter.Tendsto (f ∘ g) l (nhds (f x))

theorem Topology.IsClosedEmbedding.isClosedMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedEmbedding f) : IsClosedMap f

theorem Topology.IsClosedEmbedding.isClosed_iff_image_isClosed {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedEmbedding f) {s : Set X} : IsClosed s ↔ IsClosed (f '' s)

theorem Topology.IsClosedEmbedding.isClosed_iff_preimage_isClosed {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedEmbedding f) {s : Set Y} (hs : s ⊆ Set.range f) : IsClosed s ↔ IsClosed (f ⁻¹' s)

theorem Topology.IsClosedEmbedding.of_isEmbedding_isClosedMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : IsEmbedding f) (h₂ : IsClosedMap f) : IsClosedEmbedding f

theorem Topology.IsClosedEmbedding.of_continuous_injective_isClosedMap {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h₁ : Continuous f) (h₂ : Function.Injective f) (h₃ : IsClosedMap f) : IsClosedEmbedding f

theorem Topology.IsClosedEmbedding.isClosedEmbedding_iff_continuous_injective_isClosedMap {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} : IsClosedEmbedding f ↔ Continuous f ∧ Function.Injective f ∧ IsClosedMap f

theorem Topology.IsClosedEmbedding.id {X : Type u_1} [TopologicalSpace X] : IsClosedEmbedding id

theorem Topology.IsClosedEmbedding.comp {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsClosedEmbedding g) (hf : IsClosedEmbedding f) : IsClosedEmbedding (g ∘ f)

theorem Topology.IsClosedEmbedding.of_comp_iff {X : Type u_1} {Y : Type u_2} {Z : Type u_3} {f : X → Y} {g : Y → Z} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (hg : IsClosedEmbedding g) : IsClosedEmbedding (g ∘ f) ↔ IsClosedEmbedding f

theorem Topology.IsClosedEmbedding.closure_image_eq {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedEmbedding f) (s : Set X) : closure (f '' s) = f '' closure s