ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- exists_perfect_nonempty_of_isClosed_of_not_countable
theorem exists_perfect_nonempty_of_isClosed_of_not_countable [SecondCountableTopology α]
    (hclosed : IsClosed C) (hunc : ¬C.Countable) : ∃ D : Set α, Perfect D ∧ D.Nonempty ∧ D ⊆ C 
Arguments: {α : Type u_1} [TopologicalSpace α] {C : Set α} [SecondCountableTopology α] (hclosed : IsClosed C) (hunc : ¬C.Countable)
Goal: ∃ D, Perfect D ∧ D.Nonempty ∧ D ⊆ C
-- perfectSpace_iff_forall_not_isolated
theorem perfectSpace_iff_forall_not_isolated : PerfectSpace X ↔ ∀ x : X, Filter.NeBot (𝓝[≠] x) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: PerfectSpace X ↔ ∀ (x : X), (𝓝[≠] x).NeBot
-- inducing_induced
theorem inducing_induced (f : X → Y) : @Inducing X Y (TopologicalSpace.induced f ‹_›) _ f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace Y] (f : X → Y)
Goal: Inducing f
-- inducing_id
theorem inducing_id : Inducing (@id X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Inducing id
-- Inducing.comp
protected theorem Inducing.comp (hg : Inducing g) (hf : Inducing f) :
    Inducing (g ∘ f) 
Arguments: {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 : Inducing g) (hf : Inducing f)
Goal: Inducing (g ∘ f)
-- Inducing.of_comp_iff
theorem Inducing.of_comp_iff (hg : Inducing g) :
    Inducing (g ∘ f) ↔ Inducing f 
Arguments: {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 : Inducing g)
Goal: Inducing (g ∘ f) ↔ Inducing f
-- inducing_of_inducing_compose
theorem inducing_of_inducing_compose
    (hf : Continuous f) (hg : Continuous g) (hgf : Inducing (g ∘ f)) : Inducing f 
Arguments: {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 : Inducing (g ∘ f))
Goal: Inducing f
-- inducing_iff_nhds
theorem inducing_iff_nhds : Inducing f ↔ ∀ x, 𝓝 x = comap f (𝓝 (f x)) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: Inducing f ↔ ∀ (x : X), 𝓝 x = Filter.comap f (𝓝 (f x))
-- Inducing.nhds_eq_comap
theorem nhds_eq_comap (hf : Inducing f) : ∀ x : X, 𝓝 x = comap f (𝓝 <| f x) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) (x : X)
Goal: 𝓝 x = Filter.comap f (𝓝 (f x))
-- Inducing.basis_nhds
theorem basis_nhds {p : ι → Prop} {s : ι → Set Y} (hf : Inducing f) {x : X}
    (h_basis : (𝓝 (f x)).HasBasis p s) : (𝓝 x).HasBasis p (preimage f ∘ s) 
Arguments: {X : Type u_1} {Y : Type u_2} {ι : Type u_4} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] {p : ι → Prop} {s : ι → Set Y} (hf : Inducing f) {x : X} (h_basis : (𝓝 (f x)).HasBasis p s)
Goal: (𝓝 x).HasBasis p (Set.preimage f ∘ s)
-- Inducing.nhdsSet_eq_comap
theorem nhdsSet_eq_comap (hf : Inducing f) (s : Set X) :
    𝓝ˢ s = comap f (𝓝ˢ (f '' s)) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) (s : Set X)
Goal: 𝓝ˢ s = Filter.comap f (𝓝ˢ (f '' s))
-- Inducing.map_nhds_eq
theorem map_nhds_eq (hf : Inducing f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) (x : X)
Goal: Filter.map f (𝓝 x) = 𝓝[Set.range f] f x
-- Inducing.map_nhds_of_mem
theorem map_nhds_of_mem (hf : Inducing f) (x : X) (h : range f ∈ 𝓝 (f x)) :
    (𝓝 x).map f = 𝓝 (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) (x : X) (h : Set.range f ∈ 𝓝 (f x))
Goal: Filter.map f (𝓝 x) = 𝓝 (f x)
-- Inducing.mapClusterPt_iff
theorem mapClusterPt_iff (hf : Inducing f) {x : X} {l : Filter X} :
    MapClusterPt (f x) l f ↔ ClusterPt x l 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) {x : X} {l : Filter X}
Goal: MapClusterPt (f x) l f ↔ ClusterPt x l
-- Inducing.image_mem_nhdsWithin
theorem image_mem_nhdsWithin (hf : Inducing f) {x : X} {s : Set X} (hs : s ∈ 𝓝 x) :
    f '' s ∈ 𝓝[range f] f x 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) {x : X} {s : Set X} (hs : s ∈ 𝓝 x)
Goal: f '' s ∈ 𝓝[Set.range f] f x
-- Inducing.tendsto_nhds_iff
theorem tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : Inducing g) :
    Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) 
Arguments: {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 : Inducing g)
Goal: Filter.Tendsto f l (𝓝 y) ↔ Filter.Tendsto (g ∘ f) l (𝓝 (g y))
-- Inducing.continuousAt_iff
theorem continuousAt_iff (hg : Inducing g) {x : X} :
    ContinuousAt f x ↔ ContinuousAt (g ∘ f) x 
Arguments: {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 : Inducing g) {x : X}
Goal: ContinuousAt f x ↔ ContinuousAt (g ∘ f) x
-- Inducing.continuous_iff
theorem continuous_iff (hg : Inducing g) :
    Continuous f ↔ Continuous (g ∘ f) 
Arguments: {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 : Inducing g)
Goal: Continuous f ↔ Continuous (g ∘ f)
-- Inducing.continuousAt_iff'
theorem continuousAt_iff' (hf : Inducing f) {x : X} (h : range f ∈ 𝓝 (f x)) :
    ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) 
Arguments: {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 : Inducing f) {x : X} (h : Set.range f ∈ 𝓝 (f x))
Goal: ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x)
-- Inducing.continuous
protected theorem continuous (hf : Inducing f) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f)
Goal: Continuous f
-- Inducing.closure_eq_preimage_closure_image
theorem closure_eq_preimage_closure_image (hf : Inducing f) (s : Set X) :
    closure s = f ⁻¹' closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) (s : Set X)
Goal: closure s = f ⁻¹' closure (f '' s)
-- Inducing.isClosed_iff
theorem isClosed_iff (hf : Inducing f) {s : Set X} :
    IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) {s : Set X}
Goal: IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s
-- Inducing.isClosed_iff'
theorem isClosed_iff' (hf : Inducing f) {s : Set X} :
    IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) {s : Set X}
Goal: IsClosed s ↔ ∀ (x : X), f x ∈ closure (f '' s) → x ∈ s
-- Inducing.isClosed_preimage
theorem isClosed_preimage (h : Inducing f) (s : Set Y) (hs : IsClosed s) :
    IsClosed (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h : Inducing f) (s : Set Y) (hs : IsClosed s)
Goal: IsClosed (f ⁻¹' s)
-- Inducing.isOpen_iff
theorem isOpen_iff (hf : Inducing f) {s : Set X} :
    IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) {s : Set X}
Goal: IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s
-- Inducing.setOf_isOpen
theorem setOf_isOpen (hf : Inducing f) :
    {s : Set X | IsOpen s} = preimage f '' {t | IsOpen t} 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f)
Goal: {s | IsOpen s} = Set.preimage f '' {t | IsOpen t}
-- Inducing.dense_iff
theorem dense_iff (hf : Inducing f) {s : Set X} :
    Dense s ↔ ∀ x, f x ∈ closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Inducing f) {s : Set X}
Goal: Dense s ↔ ∀ (x : X), f x ∈ closure (f '' s)
-- Inducing.of_subsingleton
theorem of_subsingleton [Subsingleton X] (f : X → Y) : Inducing f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Subsingleton X] (f : X → Y)
Goal: Inducing f
-- Function.Injective.embedding_induced
theorem Function.Injective.embedding_induced [t : TopologicalSpace Y] (hf : Injective f) :
    @_root_.Embedding X Y (t.induced f) t f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [t : TopologicalSpace Y] (hf : Function.Injective f)
Goal: Embedding f
-- Embedding.mk'
theorem Embedding.mk' (f : X → Y) (inj : Injective f) (induced : ∀ x, comap f (𝓝 (f x)) = 𝓝 x) :
    Embedding f 
Arguments: {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 (𝓝 (f x)) = 𝓝 x)
Goal: Embedding f
-- embedding_id
theorem embedding_id : Embedding (@id X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: Embedding id
-- Embedding.comp
protected theorem Embedding.comp (hg : Embedding g) (hf : Embedding f) :
    Embedding (g ∘ f) 
Arguments: {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 : Embedding g) (hf : Embedding f)
Goal: Embedding (g ∘ f)
-- Embedding.of_comp_iff
theorem Embedding.of_comp_iff (hg : Embedding g) : Embedding (g ∘ f) ↔ Embedding f 
Arguments: {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 : Embedding g)
Goal: Embedding (g ∘ f) ↔ Embedding f
-- embedding_of_embedding_compose
theorem embedding_of_embedding_compose
    (hf : Continuous f) (hg : Continuous g) (hgf : Embedding (g ∘ f)) : Embedding f 
Arguments: {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 : Embedding (g ∘ f))
Goal: Embedding f
-- Function.LeftInverse.embedding
protected theorem Function.LeftInverse.embedding {f : X → Y} {g : Y → X} (h : LeftInverse f g)
    (hf : Continuous f) (hg : Continuous g) : _root_.Embedding g 
Arguments: {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)
Goal: Embedding g
-- Embedding.map_nhds_eq
theorem Embedding.map_nhds_eq (hf : Embedding f) (x : X) :
    (𝓝 x).map f = 𝓝[range f] f x 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Embedding f) (x : X)
Goal: Filter.map f (𝓝 x) = 𝓝[Set.range f] f x
-- Embedding.map_nhds_of_mem
theorem Embedding.map_nhds_of_mem (hf : Embedding f) (x : X) (h : range f ∈ 𝓝 (f x)) :
    (𝓝 x).map f = 𝓝 (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Embedding f) (x : X) (h : Set.range f ∈ 𝓝 (f x))
Goal: Filter.map f (𝓝 x) = 𝓝 (f x)
-- Embedding.tendsto_nhds_iff
theorem Embedding.tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y}
    (hg : Embedding g) : Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) 
Arguments: {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 : Embedding g)
Goal: Filter.Tendsto f l (𝓝 y) ↔ Filter.Tendsto (g ∘ f) l (𝓝 (g y))
-- Embedding.continuous_iff
theorem Embedding.continuous_iff (hg : Embedding g) :
    Continuous f ↔ Continuous (g ∘ f) 
Arguments: {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 : Embedding g)
Goal: Continuous f ↔ Continuous (g ∘ f)
-- Embedding.continuous
theorem Embedding.continuous (hf : Embedding f) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Embedding f)
Goal: Continuous f
-- Embedding.closure_eq_preimage_closure_image
theorem Embedding.closure_eq_preimage_closure_image (hf : Embedding f) (s : Set X) :
    closure s = f ⁻¹' closure (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : Embedding f) (s : Set X)
Goal: closure s = f ⁻¹' closure (f '' s)
-- Embedding.discreteTopology
theorem Embedding.discreteTopology [DiscreteTopology Y] (hf : Embedding f) : DiscreteTopology X 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] [DiscreteTopology Y] (hf : Embedding f)
Goal: DiscreteTopology X
-- Embedding.of_subsingleton
theorem Embedding.of_subsingleton [Subsingleton X] (f : X → Y) : Embedding f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [Subsingleton X] (f : X → Y)
Goal: Embedding f
-- quotientMap_iff
theorem quotientMap_iff : QuotientMap f ↔ Surjective f ∧ ∀ s : Set Y, IsOpen s ↔ IsOpen (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: QuotientMap f ↔ Function.Surjective f ∧ ∀ (s : Set Y), IsOpen s ↔ IsOpen (f ⁻¹' s)
-- quotientMap_iff_closed
theorem quotientMap_iff_closed :
    QuotientMap f ↔ Surjective f ∧ ∀ s : Set Y, IsClosed s ↔ IsClosed (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: QuotientMap f ↔ Function.Surjective f ∧ ∀ (s : Set Y), IsClosed s ↔ IsClosed (f ⁻¹' s)
-- QuotientMap.id
protected theorem id : QuotientMap (@id X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: QuotientMap id
-- QuotientMap.comp
protected theorem comp (hg : QuotientMap g) (hf : QuotientMap f) : QuotientMap (g ∘ f) 
Arguments: {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 : QuotientMap g) (hf : QuotientMap f)
Goal: QuotientMap (g ∘ f)
-- QuotientMap.of_quotientMap_compose
protected theorem of_quotientMap_compose (hf : Continuous f) (hg : Continuous g)
    (hgf : QuotientMap (g ∘ f)) : QuotientMap g 
Arguments: {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 : QuotientMap (g ∘ f))
Goal: QuotientMap g
-- QuotientMap.of_inverse
theorem of_inverse {g : Y → X} (hf : Continuous f) (hg : Continuous g) (h : LeftInverse g f) :
    QuotientMap g 
Arguments: {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)
Goal: QuotientMap g
-- QuotientMap.continuous_iff
protected theorem continuous_iff (hf : QuotientMap f) : Continuous g ↔ Continuous (g ∘ f) 
Arguments: {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 : QuotientMap f)
Goal: Continuous g ↔ Continuous (g ∘ f)
-- QuotientMap.continuous
protected theorem continuous (hf : QuotientMap f) : Continuous f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : QuotientMap f)
Goal: Continuous f
-- QuotientMap.surjective
protected theorem surjective (hf : QuotientMap f) : Surjective f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : QuotientMap f)
Goal: Function.Surjective f
-- QuotientMap.isOpen_preimage
protected theorem isOpen_preimage (hf : QuotientMap f) {s : Set Y} : IsOpen (f ⁻¹' s) ↔ IsOpen s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : QuotientMap f) {s : Set Y}
Goal: IsOpen (f ⁻¹' s) ↔ IsOpen s
-- QuotientMap.isClosed_preimage
protected theorem isClosed_preimage (hf : QuotientMap f) {s : Set Y} :
    IsClosed (f ⁻¹' s) ↔ IsClosed s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : QuotientMap f) {s : Set Y}
Goal: IsClosed (f ⁻¹' s) ↔ IsClosed s
-- IsOpenMap.id
protected theorem id : IsOpenMap (@id X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsOpenMap id
-- IsOpenMap.comp
protected theorem comp (hg : IsOpenMap g) (hf : IsOpenMap f) :
    IsOpenMap (g ∘ f) 
Arguments: {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)
Goal: IsOpenMap (g ∘ f)
-- IsOpenMap.isOpen_range
theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f)
Goal: IsOpen (Set.range f)
-- IsOpenMap.image_mem_nhds
theorem image_mem_nhds (hf : IsOpenMap f) {x : X} {s : Set X} (hx : s ∈ 𝓝 x) : f '' s ∈ 𝓝 (f x) 
Arguments: {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 ∈ 𝓝 x)
Goal: f '' s ∈ 𝓝 (f x)
-- IsOpenMap.range_mem_nhds
theorem range_mem_nhds (hf : IsOpenMap f) (x : X) : range f ∈ 𝓝 (f x) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (x : X)
Goal: Set.range f ∈ 𝓝 (f x)
-- IsOpenMap.mapsTo_interior
theorem mapsTo_interior (hf : IsOpenMap f) {s : Set X} {t : Set Y} (h : MapsTo f s t) :
    MapsTo f (interior s) (interior t) 
Arguments: {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)
Goal: Set.MapsTo f (interior s) (interior t)
-- IsOpenMap.image_interior_subset
theorem image_interior_subset (hf : IsOpenMap f) (s : Set X) :
    f '' interior s ⊆ interior (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (s : Set X)
Goal: f '' interior s ⊆ interior (f '' s)
-- IsOpenMap.nhds_le
theorem nhds_le (hf : IsOpenMap f) (x : X) : 𝓝 (f x) ≤ (𝓝 x).map f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (x : X)
Goal: 𝓝 (f x) ≤ Filter.map f (𝓝 x)
-- IsOpenMap.of_nhds_le
theorem of_nhds_le (hf : ∀ x, 𝓝 (f x) ≤ map f (𝓝 x)) : IsOpenMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : ∀ (x : X), 𝓝 (f x) ≤ Filter.map f (𝓝 x))
Goal: IsOpenMap f
-- IsOpenMap.of_sections
theorem of_sections
    (h : ∀ x, ∃ g : Y → X, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (h : ∀ (x : X), ∃ g, ContinuousAt g (f x) ∧ g (f x) = x ∧ Function.RightInverse g f)
Goal: IsOpenMap f
-- IsOpenMap.of_inverse
theorem of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f')
    (r_inv : RightInverse f f') : IsOpenMap f 
Arguments: {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')
Goal: IsOpenMap f
-- IsOpenMap.to_quotientMap
theorem to_quotientMap (open_map : IsOpenMap f) (cont : Continuous f) (surj : Surjective f) :
    QuotientMap f 
Arguments: {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)
Goal: QuotientMap f
-- IsOpenMap.interior_preimage_subset_preimage_interior
theorem interior_preimage_subset_preimage_interior (hf : IsOpenMap f) {s : Set Y} :
    interior (f ⁻¹' s) ⊆ f ⁻¹' interior s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {s : Set Y}
Goal: interior (f ⁻¹' s) ⊆ f ⁻¹' interior s
-- IsOpenMap.preimage_interior_eq_interior_preimage
theorem preimage_interior_eq_interior_preimage (hf₁ : IsOpenMap f) (hf₂ : Continuous f)
    (s : Set Y) : f ⁻¹' interior s = interior (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf₁ : IsOpenMap f) (hf₂ : Continuous f) (s : Set Y)
Goal: f ⁻¹' interior s = interior (f ⁻¹' s)
-- IsOpenMap.preimage_closure_subset_closure_preimage
theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set Y} :
    f ⁻¹' closure s ⊆ closure (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {s : Set Y}
Goal: f ⁻¹' closure s ⊆ closure (f ⁻¹' s)
-- IsOpenMap.preimage_closure_eq_closure_preimage
theorem preimage_closure_eq_closure_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) :
    f ⁻¹' closure s = closure (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y)
Goal: f ⁻¹' closure s = closure (f ⁻¹' s)
-- IsOpenMap.preimage_frontier_subset_frontier_preimage
theorem preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set Y} :
    f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) {s : Set Y}
Goal: f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s)
-- IsOpenMap.preimage_frontier_eq_frontier_preimage
theorem preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) :
    f ⁻¹' frontier s = frontier (f ⁻¹' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y)
Goal: f ⁻¹' frontier s = frontier (f ⁻¹' s)
-- IsOpenMap.of_isEmpty
theorem of_isEmpty [h : IsEmpty X] (f : X → Y) : IsOpenMap f 
Arguments: {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [h : IsEmpty X] (f : X → Y)
Goal: IsOpenMap f
-- isOpenMap_iff_nhds_le
theorem isOpenMap_iff_nhds_le : IsOpenMap f ↔ ∀ x : X, 𝓝 (f x) ≤ (𝓝 x).map f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: IsOpenMap f ↔ ∀ (x : X), 𝓝 (f x) ≤ Filter.map f (𝓝 x)
-- isOpenMap_iff_interior
theorem isOpenMap_iff_interior : IsOpenMap f ↔ ∀ s, f '' interior s ⊆ interior (f '' s) 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y]
Goal: IsOpenMap f ↔ ∀ (s : Set X), f '' interior s ⊆ interior (f '' s)
-- Inducing.isOpenMap
protected theorem Inducing.isOpenMap (hi : Inducing f) (ho : IsOpen (range f)) : IsOpenMap f 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hi : Inducing f) (ho : IsOpen (Set.range f))
Goal: IsOpenMap f
-- IsClosedMap.id
protected theorem id : IsClosedMap (@id X) 
Arguments: {X : Type u_1} [TopologicalSpace X]
Goal: IsClosedMap id
-- IsClosedMap.comp
protected theorem comp (hg : IsClosedMap g) (hf : IsClosedMap f) : IsClosedMap (g ∘ f) 
Arguments: {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)
Goal: IsClosedMap (g ∘ f)
-- IsClosedMap.closure_image_subset
theorem closure_image_subset (hf : IsClosedMap f) (s : Set X) :
    closure (f '' s) ⊆ f '' closure s 
Arguments: {X : Type u_1} {Y : Type u_2} {f : X → Y} [TopologicalSpace X] [TopologicalSpace Y] (hf : IsClosedMap f) (s : Set X)
Goal: closure (f '' s) ⊆ f '' closure s
-- IsClosedMap.of_inverse
theorem of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f')
    (r_inv : RightInverse f f') : IsClosedMap f 
Arguments: {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')
Goal: IsClosedMap f
-- IsClosedMap.of_nonempty
theorem of_nonempty (h : ∀ s, IsClosed s → s.Nonempty → IsClosed (f '' s)) :
    IsClosedMap f 
Arguments: {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))
Goal: IsClosedMap f