ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Continuous.connectedComponentsLift_unique
theorem Continuous.connectedComponentsLift_unique (h : Continuous f) (g : ConnectedComponents α → β)
    (hg : g ∘ (↑) = f) : g = h.connectedComponentsLift 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] [TotallyDisconnectedSpace β] {f : α → β} (h : Continuous f) (g : ConnectedComponents α → β) (hg : g ∘ ConnectedComponents.mk = f)
Goal: g = h.connectedComponentsLift
-- Continuous.connectedComponentsMap_continuous
theorem Continuous.connectedComponentsMap_continuous {β : Type*} [TopologicalSpace β] {f : α → β}
    (h : Continuous f) : Continuous h.connectedComponentsMap 
Arguments: {α : Type u} [TopologicalSpace α] {β : Type u_3} [TopologicalSpace β] {f : α → β} (h : Continuous f)
Goal: Continuous h.connectedComponentsMap
-- IsPreconnected.constant
theorem IsPreconnected.constant {Y : Type*} [TopologicalSpace Y] [DiscreteTopology Y] {s : Set α}
    (hs : IsPreconnected s) {f : α → Y} (hf : ContinuousOn f s) {x y : α} (hx : x ∈ s)
    (hy : y ∈ s) : f x = f y 
Arguments: {α : Type u} [TopologicalSpace α] {Y : Type u_3} [TopologicalSpace Y] [DiscreteTopology Y] {s : Set α} (hs : IsPreconnected s) {f : α → Y} (hf : ContinuousOn f s) {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: f x = f y
-- PreconnectedSpace.constant
theorem PreconnectedSpace.constant {Y : Type*} [TopologicalSpace Y] [DiscreteTopology Y]
    (hp : PreconnectedSpace α) {f : α → Y} (hf : Continuous f) {x y : α} : f x = f y 
Arguments: {α : Type u} [TopologicalSpace α] {Y : Type u_3} [TopologicalSpace Y] [DiscreteTopology Y] (hp : PreconnectedSpace α) {f : α → Y} (hf : Continuous f) {x y : α}
Goal: f x = f y
-- IsPreconnected.constant_of_mapsTo
theorem IsPreconnected.constant_of_mapsTo {S : Set α} (hS : IsPreconnected S)
    {T : Set β} [DiscreteTopology T] {f : α → β} (hc : ContinuousOn f S) (hTm : MapsTo f S T)
    {x y : α} (hx : x ∈ S) (hy : y ∈ S) : f x = f y 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {S : Set α} (hS : IsPreconnected S) {T : Set β} [DiscreteTopology ↑T] {f : α → β} (hc : ContinuousOn f S) (hTm : Set.MapsTo f S T) {x y : α} (hx : x ∈ S) (hy : y ∈ S)
Goal: f x = f y
-- IsPreconnected.eqOn_const_of_mapsTo
theorem IsPreconnected.eqOn_const_of_mapsTo {S : Set α} (hS : IsPreconnected S)
    {T : Set β} [DiscreteTopology T] {f : α → β} (hc : ContinuousOn f S) (hTm : MapsTo f S T)
    (hne : T.Nonempty) : ∃ y ∈ T, EqOn f (const α y) S 
Arguments: {α : Type u} {β : Type v} [TopologicalSpace α] [TopologicalSpace β] {S : Set α} (hS : IsPreconnected S) {T : Set β} [DiscreteTopology ↑T] {f : α → β} (hc : ContinuousOn f S) (hTm : Set.MapsTo f S T) (hne : T.Nonempty)
Goal: ∃ y ∈ T, Set.EqOn f (Function.const α y) S
-- Homeomorph.continuous_toFun
structure Homeomorph (X : Type*) (Y : Type*) [TopologicalSpace X] [TopologicalSpace Y]
    extends X ≃ Y where
    continuous_toFun : Continuous toFun := by continuity
    continuous_invFun : Continuous invFun := by continuity
Arguments: {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] (self : X ≃ₜ Y)
Goal: Continuous self.toFun
-- Homeomorph.continuous_invFun
structure Homeomorph (X : Type*) (Y : Type*) [TopologicalSpace X] [TopologicalSpace Y]
    extends X ≃ Y where
    continuous_toFun : Continuous toFun := by continuity
    continuous_invFun : Continuous invFun := by continuity
Arguments: {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] (self : X ≃ₜ Y)
Goal: Continuous self.invFun
-- Homeomorph.toEquiv_injective
theorem toEquiv_injective : Function.Injective (toEquiv : X ≃ₜ Y → X ≃ Y)
  
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Function.Injective Homeomorph.toEquiv
-- Homeomorph.homeomorph_mk_coe
@[simp] theorem homeomorph_mk_coe (a : X ≃ Y) (b c) : (Homeomorph.mk a b c : X → Y) = a 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (a : X ≃ Y) (b : Continuous a.toFun) (c : Continuous a.invFun)
Goal: ⇑{ toEquiv := a, continuous_toFun := b, continuous_invFun := c } = ⇑a
-- Homeomorph.symm_symm
@[simp] theorem symm_symm (h : X ≃ₜ Y) : h.symm.symm = h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: h.symm.symm = h
-- Homeomorph.symm_bijective
theorem symm_bijective : Function.Bijective (Homeomorph.symm : (X ≃ₜ Y) → Y ≃ₜ X) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Function.Bijective Homeomorph.symm
-- Homeomorph.coe_toEquiv
@[simp]
theorem coe_toEquiv (h : X ≃ₜ Y) : ⇑h.toEquiv = h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: ⇑h.toEquiv = ⇑h
-- Homeomorph.coe_symm_toEquiv
@[simp]
theorem coe_symm_toEquiv (h : X ≃ₜ Y) : ⇑h.toEquiv.symm = h.symm 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: ⇑h.symm = ⇑h.symm
-- Homeomorph.ext
@[ext]
theorem ext {h h' : X ≃ₜ Y} (H : ∀ x, h x = h' x) : h = h' 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {h h' : X ≃ₜ Y} (H : ∀ (x : X), h x = h' x)
Goal: h = h'
-- Homeomorph.refl_apply
@[simps! (config := .asFn) apply]
protected def refl (X : Type*) [TopologicalSpace X] : X ≃ₜ X 
Arguments: (X : Type u_6) [TopologicalSpace X]
Goal: ⇑(Homeomorph.refl X) = id
-- Homeomorph.trans_apply
@[simp]
theorem trans_apply (h₁ : X ≃ₜ Y) (h₂ : Y ≃ₜ Z) (x : X) : h₁.trans h₂ x = h₂ (h₁ x) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h₁ : X ≃ₜ Y) (h₂ : Y ≃ₜ Z) (x : X)
Goal: (h₁.trans h₂) x = h₂ (h₁ x)
-- Homeomorph.symm_trans_apply
@[simp]
theorem symm_trans_apply (f : X ≃ₜ Y) (g : Y ≃ₜ Z) (z : Z) :
    (f.trans g).symm z = f.symm (g.symm z) 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (f : X ≃ₜ Y) (g : Y ≃ₜ Z) (z : Z)
Goal: (f.trans g).symm z = f.symm (g.symm z)
-- Homeomorph.homeomorph_mk_coe_symm
@[simp]
theorem homeomorph_mk_coe_symm (a : X ≃ Y) (b c) :
    ((Homeomorph.mk a b c).symm : Y → X) = a.symm 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (a : X ≃ Y) (b : Continuous a.toFun) (c : Continuous a.invFun)
Goal: ⇑{ toEquiv := a, continuous_toFun := b, continuous_invFun := c }.symm = ⇑a.symm
-- Homeomorph.refl_symm
@[simp]
theorem refl_symm : (Homeomorph.refl X).symm = Homeomorph.refl X 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: (Homeomorph.refl X).symm = Homeomorph.refl X
-- Homeomorph.continuous
@[continuity, fun_prop]
protected theorem continuous (h : X ≃ₜ Y) : Continuous h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Continuous ⇑h
-- Homeomorph.continuous_symm
@[continuity]
protected theorem continuous_symm (h : X ≃ₜ Y) : Continuous h.symm 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Continuous ⇑h.symm
-- Homeomorph.apply_symm_apply
@[simp]
theorem apply_symm_apply (h : X ≃ₜ Y) (y : Y) : h (h.symm y) = y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (y : Y)
Goal: h (h.symm y) = y
-- Homeomorph.symm_apply_apply
@[simp]
theorem symm_apply_apply (h : X ≃ₜ Y) (x : X) : h.symm (h x) = x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X)
Goal: h.symm (h x) = x
-- Homeomorph.self_trans_symm
@[simp]
theorem self_trans_symm (h : X ≃ₜ Y) : h.trans h.symm = Homeomorph.refl X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: h.trans h.symm = Homeomorph.refl X
-- Homeomorph.symm_trans_self
@[simp]
theorem symm_trans_self (h : X ≃ₜ Y) : h.symm.trans h = Homeomorph.refl Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: h.symm.trans h = Homeomorph.refl Y
-- Homeomorph.bijective
protected theorem bijective (h : X ≃ₜ Y) : Function.Bijective h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Function.Bijective ⇑h
-- Homeomorph.injective
protected theorem injective (h : X ≃ₜ Y) : Function.Injective h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Function.Injective ⇑h
-- Homeomorph.surjective
protected theorem surjective (h : X ≃ₜ Y) : Function.Surjective h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Function.Surjective ⇑h
-- Homeomorph.symm_comp_self
@[simp]
theorem symm_comp_self (h : X ≃ₜ Y) : h.symm ∘ h = id 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: ⇑h.symm ∘ ⇑h = id
-- Homeomorph.self_comp_symm
@[simp]
theorem self_comp_symm (h : X ≃ₜ Y) : h ∘ h.symm = id 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: ⇑h ∘ ⇑h.symm = id
-- Homeomorph.range_coe
@[simp]
theorem range_coe (h : X ≃ₜ Y) : range h = univ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Set.range ⇑h = Set.univ
-- Homeomorph.image_symm
theorem image_symm (h : X ≃ₜ Y) : image h.symm = preimage h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Set.image ⇑h.symm = Set.preimage ⇑h
-- Homeomorph.preimage_symm
theorem preimage_symm (h : X ≃ₜ Y) : preimage h.symm = image h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Set.preimage ⇑h.symm = Set.image ⇑h
-- Homeomorph.image_preimage
@[simp]
theorem image_preimage (h : X ≃ₜ Y) (s : Set Y) : h '' (h ⁻¹' s) = s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set Y)
Goal: ⇑h '' (⇑h ⁻¹' s) = s
-- Homeomorph.preimage_image
@[simp]
theorem preimage_image (h : X ≃ₜ Y) (s : Set X) : h ⁻¹' (h '' s) = s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set X)
Goal: ⇑h ⁻¹' (⇑h '' s) = s
-- Homeomorph.image_compl
lemma image_compl (h : X ≃ₜ Y) (s : Set X) : h '' (sᶜ) = (h '' s)ᶜ 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set X)
Goal: ⇑h '' sᶜ = (⇑h '' s)ᶜ
-- Homeomorph.inducing
protected theorem inducing (h : X ≃ₜ Y) : Inducing h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Inducing ⇑h
-- Homeomorph.induced_eq
theorem induced_eq (h : X ≃ₜ Y) : TopologicalSpace.induced h ‹_› = ‹_› 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: TopologicalSpace.induced (⇑h) inst✝ = inst✝¹
-- Homeomorph.quotientMap
protected theorem quotientMap (h : X ≃ₜ Y) : QuotientMap h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: QuotientMap ⇑h
-- Homeomorph.coinduced_eq
theorem coinduced_eq (h : X ≃ₜ Y) : TopologicalSpace.coinduced h ‹_› = ‹_› 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: TopologicalSpace.coinduced (⇑h) inst✝ = inst✝¹
-- Homeomorph.embedding
protected theorem embedding (h : X ≃ₜ Y) : Embedding h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Embedding ⇑h
-- Homeomorph.secondCountableTopology
protected theorem secondCountableTopology [SecondCountableTopology Y]
    (h : X ≃ₜ Y) : SecondCountableTopology X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [SecondCountableTopology Y] (h : X ≃ₜ Y)
Goal: SecondCountableTopology X
-- Homeomorph.isCompact_image
@[simp]
theorem isCompact_image {s : Set X} (h : X ≃ₜ Y) : IsCompact (h '' s) ↔ IsCompact s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (h : X ≃ₜ Y)
Goal: IsCompact (⇑h '' s) ↔ IsCompact s
-- Homeomorph.isCompact_preimage
@[simp]
theorem isCompact_preimage {s : Set Y} (h : X ≃ₜ Y) : IsCompact (h ⁻¹' s) ↔ IsCompact s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set Y} (h : X ≃ₜ Y)
Goal: IsCompact (⇑h ⁻¹' s) ↔ IsCompact s
-- Homeomorph.isSigmaCompact_image
@[simp]
theorem isSigmaCompact_image {s : Set X} (h : X ≃ₜ Y) :
    IsSigmaCompact (h '' s) ↔ IsSigmaCompact s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (h : X ≃ₜ Y)
Goal: IsSigmaCompact (⇑h '' s) ↔ IsSigmaCompact s
-- Homeomorph.isSigmaCompact_preimage
@[simp]
theorem isSigmaCompact_preimage {s : Set Y} (h : X ≃ₜ Y) :
    IsSigmaCompact (h ⁻¹' s) ↔ IsSigmaCompact s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set Y} (h : X ≃ₜ Y)
Goal: IsSigmaCompact (⇑h ⁻¹' s) ↔ IsSigmaCompact s
-- Homeomorph.isPreconnected_image
@[simp]
theorem isPreconnected_image {s : Set X} (h : X ≃ₜ Y) :
    IsPreconnected (h '' s) ↔ IsPreconnected s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (h : X ≃ₜ Y)
Goal: IsPreconnected (⇑h '' s) ↔ IsPreconnected s
-- Homeomorph.isPreconnected_preimage
@[simp]
theorem isPreconnected_preimage {s : Set Y} (h : X ≃ₜ Y) :
    IsPreconnected (h ⁻¹' s) ↔ IsPreconnected s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set Y} (h : X ≃ₜ Y)
Goal: IsPreconnected (⇑h ⁻¹' s) ↔ IsPreconnected s
-- Homeomorph.isConnected_image
@[simp]
theorem isConnected_image {s : Set X} (h : X ≃ₜ Y) :
    IsConnected (h '' s) ↔ IsConnected s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (h : X ≃ₜ Y)
Goal: IsConnected (⇑h '' s) ↔ IsConnected s
-- Homeomorph.isConnected_preimage
@[simp]
theorem isConnected_preimage {s : Set Y} (h : X ≃ₜ Y) :
    IsConnected (h ⁻¹' s) ↔ IsConnected s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set Y} (h : X ≃ₜ Y)
Goal: IsConnected (⇑h ⁻¹' s) ↔ IsConnected s
-- Homeomorph.image_connectedComponentIn
theorem image_connectedComponentIn {s : Set X} (h : X ≃ₜ Y) {x : X} (hx : x ∈ s) :
    h '' connectedComponentIn s x = connectedComponentIn (h '' s) (h x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} (h : X ≃ₜ Y) {x : X} (hx : x ∈ s)
Goal: ⇑h '' connectedComponentIn s x = connectedComponentIn (⇑h '' s) (h x)
-- Homeomorph.comap_cocompact
@[simp]
theorem comap_cocompact (h : X ≃ₜ Y) : comap h (cocompact Y) = cocompact X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Filter.comap (⇑h) (Filter.cocompact Y) = Filter.cocompact X
-- Homeomorph.map_cocompact
@[simp]
theorem map_cocompact (h : X ≃ₜ Y) : map h (cocompact X) = cocompact Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: Filter.map (⇑h) (Filter.cocompact X) = Filter.cocompact Y
-- Homeomorph.compactSpace
protected theorem compactSpace [CompactSpace X] (h : X ≃ₜ Y) : CompactSpace Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [CompactSpace X] (h : X ≃ₜ Y)
Goal: CompactSpace Y
-- Homeomorph.t0Space
protected theorem t0Space [T0Space X] (h : X ≃ₜ Y) : T0Space Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T0Space X] (h : X ≃ₜ Y)
Goal: T0Space Y
-- Homeomorph.t1Space
protected theorem t1Space [T1Space X] (h : X ≃ₜ Y) : T1Space Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T1Space X] (h : X ≃ₜ Y)
Goal: T1Space Y
-- Homeomorph.t2Space
protected theorem t2Space [T2Space X] (h : X ≃ₜ Y) : T2Space Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T2Space X] (h : X ≃ₜ Y)
Goal: T2Space Y
-- Homeomorph.t3Space
protected theorem t3Space [T3Space X] (h : X ≃ₜ Y) : T3Space Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T3Space X] (h : X ≃ₜ Y)
Goal: T3Space Y
-- Homeomorph.denseEmbedding
protected theorem denseEmbedding (h : X ≃ₜ Y) : DenseEmbedding h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: DenseEmbedding ⇑h
-- Homeomorph.isOpen_preimage
@[simp]
theorem isOpen_preimage (h : X ≃ₜ Y) {s : Set Y} : IsOpen (h ⁻¹' s) ↔ IsOpen s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) {s : Set Y}
Goal: IsOpen (⇑h ⁻¹' s) ↔ IsOpen s
-- Homeomorph.isOpen_image
@[simp]
theorem isOpen_image (h : X ≃ₜ Y) {s : Set X} : IsOpen (h '' s) ↔ IsOpen s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) {s : Set X}
Goal: IsOpen (⇑h '' s) ↔ IsOpen s
-- Homeomorph.isOpenMap
protected theorem isOpenMap (h : X ≃ₜ Y) : IsOpenMap h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: IsOpenMap ⇑h
-- Homeomorph.isClosed_preimage
@[simp]
theorem isClosed_preimage (h : X ≃ₜ Y) {s : Set Y} : IsClosed (h ⁻¹' s) ↔ IsClosed s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) {s : Set Y}
Goal: IsClosed (⇑h ⁻¹' s) ↔ IsClosed s
-- Homeomorph.isClosed_image
@[simp]
theorem isClosed_image (h : X ≃ₜ Y) {s : Set X} : IsClosed (h '' s) ↔ IsClosed s 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) {s : Set X}
Goal: IsClosed (⇑h '' s) ↔ IsClosed s
-- Homeomorph.isClosedMap
protected theorem isClosedMap (h : X ≃ₜ Y) : IsClosedMap h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: IsClosedMap ⇑h
-- Homeomorph.openEmbedding
protected theorem openEmbedding (h : X ≃ₜ Y) : OpenEmbedding h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: OpenEmbedding ⇑h
-- Homeomorph.closedEmbedding
protected theorem closedEmbedding (h : X ≃ₜ Y) : ClosedEmbedding h 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: ClosedEmbedding ⇑h
-- Homeomorph.normalSpace
protected theorem normalSpace [NormalSpace X] (h : X ≃ₜ Y) : NormalSpace Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [NormalSpace X] (h : X ≃ₜ Y)
Goal: NormalSpace Y
-- Homeomorph.t4Space
protected theorem t4Space [T4Space X] (h : X ≃ₜ Y) : T4Space Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [T4Space X] (h : X ≃ₜ Y)
Goal: T4Space Y
-- Homeomorph.preimage_closure
theorem preimage_closure (h : X ≃ₜ Y) (s : Set Y) : h ⁻¹' closure s = closure (h ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set Y)
Goal: ⇑h ⁻¹' closure s = closure (⇑h ⁻¹' s)
-- Homeomorph.image_closure
theorem image_closure (h : X ≃ₜ Y) (s : Set X) : h '' closure s = closure (h '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set X)
Goal: ⇑h '' closure s = closure (⇑h '' s)
-- Homeomorph.preimage_interior
theorem preimage_interior (h : X ≃ₜ Y) (s : Set Y) : h ⁻¹' interior s = interior (h ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set Y)
Goal: ⇑h ⁻¹' interior s = interior (⇑h ⁻¹' s)
-- Homeomorph.image_interior
theorem image_interior (h : X ≃ₜ Y) (s : Set X) : h '' interior s = interior (h '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set X)
Goal: ⇑h '' interior s = interior (⇑h '' s)
-- Homeomorph.preimage_frontier
theorem preimage_frontier (h : X ≃ₜ Y) (s : Set Y) : h ⁻¹' frontier s = frontier (h ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set Y)
Goal: ⇑h ⁻¹' frontier s = frontier (⇑h ⁻¹' s)
-- Homeomorph.image_frontier
theorem image_frontier (h : X ≃ₜ Y) (s : Set X) : h '' frontier s = frontier (h '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (s : Set X)
Goal: ⇑h '' frontier s = frontier (⇑h '' s)
-- HasCompactSupport.comp_homeomorph
@[to_additive]
theorem _root_.HasCompactMulSupport.comp_homeomorph {M} [One M] {f : Y → M}
    (hf : HasCompactMulSupport f) (φ : X ≃ₜ Y) : HasCompactMulSupport (f ∘ φ) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {M : Type u_6} [Zero M] {f : Y → M} (hf : HasCompactSupport f) (φ : X ≃ₜ Y)
Goal: HasCompactSupport (f ∘ ⇑φ)
-- HasCompactMulSupport.comp_homeomorph
@[to_additive]
theorem _root_.HasCompactMulSupport.comp_homeomorph {M} [One M] {f : Y → M}
    (hf : HasCompactMulSupport f) (φ : X ≃ₜ Y) : HasCompactMulSupport (f ∘ φ) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {M : Type u_6} [One M] {f : Y → M} (hf : HasCompactMulSupport f) (φ : X ≃ₜ Y)
Goal: HasCompactMulSupport (f ∘ ⇑φ)
-- Homeomorph.map_nhds_eq
@[simp]
theorem map_nhds_eq (h : X ≃ₜ Y) (x : X) : map h (𝓝 x) = 𝓝 (h x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X)
Goal: Filter.map (⇑h) (𝓝 x) = 𝓝 (h x)
-- Homeomorph.map_punctured_nhds_eq
@[simp]
theorem map_punctured_nhds_eq (h : X ≃ₜ Y) (x : X) : map h (𝓝[≠] x) = 𝓝[≠] (h x) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X)
Goal: Filter.map (⇑h) (𝓝[≠] x) = 𝓝[≠] h x
-- Homeomorph.symm_map_nhds_eq
theorem symm_map_nhds_eq (h : X ≃ₜ Y) (x : X) : map h.symm (𝓝 (h x)) = 𝓝 x 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X)
Goal: Filter.map (⇑h.symm) (𝓝 (h x)) = 𝓝 x
-- Homeomorph.nhds_eq_comap
theorem nhds_eq_comap (h : X ≃ₜ Y) (x : X) : 𝓝 x = comap h (𝓝 (h x)) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (x : X)
Goal: 𝓝 x = Filter.comap (⇑h) (𝓝 (h x))
-- Homeomorph.comap_nhds_eq
@[simp]
theorem comap_nhds_eq (h : X ≃ₜ Y) (y : Y) : comap h (𝓝 y) = 𝓝 (h.symm y) 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y) (y : Y)
Goal: Filter.comap (⇑h) (𝓝 y) = 𝓝 (h.symm y)
-- Homeomorph.locallyConnectedSpace
theorem locallyConnectedSpace [i : LocallyConnectedSpace Y] (h : X ≃ₜ Y) :
    LocallyConnectedSpace X 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [i : LocallyConnectedSpace Y] (h : X ≃ₜ Y)
Goal: LocallyConnectedSpace X
-- Homeomorph.locallyCompactSpace_iff
theorem locallyCompactSpace_iff (h : X ≃ₜ Y) :
    LocallyCompactSpace X ↔ LocallyCompactSpace Y 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] (h : X ≃ₜ Y)
Goal: LocallyCompactSpace X ↔ LocallyCompactSpace Y
-- Homeomorph.comp_continuousOn_iff
@[simp]
theorem comp_continuousOn_iff (h : X ≃ₜ Y) (f : Z → X) (s : Set Z) :
    ContinuousOn (h ∘ f) s ↔ ContinuousOn f s 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h : X ≃ₜ Y) (f : Z → X) (s : Set Z)
Goal: ContinuousOn (⇑h ∘ f) s ↔ ContinuousOn f s
-- Homeomorph.comp_continuous_iff
@[simp]
theorem comp_continuous_iff (h : X ≃ₜ Y) {f : Z → X} : Continuous (h ∘ f) ↔ Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h : X ≃ₜ Y) {f : Z → X}
Goal: Continuous (⇑h ∘ f) ↔ Continuous f
-- Homeomorph.comp_continuous_iff'
@[simp]
theorem comp_continuous_iff' (h : X ≃ₜ Y) {f : Y → Z} : Continuous (f ∘ h) ↔ Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h : X ≃ₜ Y) {f : Y → Z}
Goal: Continuous (f ∘ ⇑h) ↔ Continuous f
-- Homeomorph.comp_continuousAt_iff
theorem comp_continuousAt_iff (h : X ≃ₜ Y) (f : Z → X) (z : Z) :
    ContinuousAt (h ∘ f) z ↔ ContinuousAt f z 
Arguments: {X : Type u_1} {Y : Type u_2} {Z : Type u_3} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (h : X ≃ₜ Y) (f : Z → X) (z : Z)
Goal: ContinuousAt (⇑h ∘ f) z ↔ ContinuousAt f z