fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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flip_unflip {X α β : Type*} [Finite α] [TopologicalSpace X]
(f : α → LocallyConstant X β) : (unflip f).flip = f := rfl | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | flip_unflip | null |
comap (f : C(X, Y)) (g : LocallyConstant Y Z) : LocallyConstant X Z :=
⟨g ∘ f, g.isLocallyConstant.comp_continuous f.continuous⟩
@[simp] | def | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | comap | Pull back of locally constant maps under a continuous map, by pre-composition. |
coe_comap (f : C(X, Y)) (g : LocallyConstant Y Z) :
(comap f g) = g ∘ f := rfl | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | coe_comap | null |
coe_comap_apply (f : C(X, Y)) (g : LocallyConstant Y Z) (x : X) :
comap f g x = g (f x) := rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | coe_comap_apply | null |
comap_id : comap (@ContinuousMap.id X _) = @id (LocallyConstant X Z) := rfl | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | comap_id | null |
comap_comp {W : Type*} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y)) :
comap (Z := Z) (g.comp f) = comap f ∘ comap g := rfl | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | comap_comp | null |
comap_comap {W : Type*} [TopologicalSpace W] (f : C(W, X)) (g : C(X, Y))
(x : LocallyConstant Y Z) : comap f (comap g x) = comap (g.comp f) x := rfl | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | comap_comap | null |
comap_const (f : C(X, Y)) (y : Y) (h : ∀ x, f x = y) :
(comap f : LocallyConstant Y Z → LocallyConstant X Z) = fun g => const X (g y) := by
ext; simp [h] | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | comap_const | null |
comap_injective (f : C(X, Y)) (hfs : f.1.Surjective) :
(comap (Z := Z) f).Injective := by
intro a b h
ext y
obtain ⟨x, hx⟩ := hfs y
simpa [← hx] using LocallyConstant.congr_fun h x | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | comap_injective | null |
desc {X α β : Type*} [TopologicalSpace X] {g : α → β} (f : X → α) (h : LocallyConstant X β)
(cond : g ∘ f = h) (inj : Function.Injective g) : LocallyConstant X α where
toFun := f
isLocallyConstant := IsLocallyConstant.desc _ g (cond.symm ▸ h.isLocallyConstant) inj
@[simp] | def | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | desc | If a locally constant function factors through an injection, then it factors through a locally
constant function. |
coe_desc {X α β : Type*} [TopologicalSpace X] (f : X → α) (g : α → β)
(h : LocallyConstant X β) (cond : g ∘ f = h) (inj : Function.Injective g) :
⇑(desc f h cond inj) = f :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | coe_desc | null |
@[to_additive (attr := simps) /-- Given a clopen set `U` and a locally constant function `f`,
`LocallyConstant.indicator` returns the locally constant function that is `f` on `U` and `0`
otherwise. -/]
noncomputable mulIndicator (hU : IsClopen U) : LocallyConstant X R where
toFun := Set.mulIndicator U f
isLocallyConstant := fun s => by
rw [mulIndicator_preimage, Set.ite, Set.diff_eq]
exact ((f.2 s).inter hU.isOpen).union ((IsLocallyConstant.const 1 s).inter hU.compl.isOpen)
variable (a : X)
open Classical in
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | mulIndicator | Given a clopen set `U` and a locally constant function `f`, `LocallyConstant.mulIndicator`
returns the locally constant function that is `f` on `U` and `1` otherwise. |
mulIndicator_apply_eq_if (hU : IsClopen U) :
mulIndicator f hU a = if a ∈ U then f a else 1 :=
Set.mulIndicator_apply U f a
variable {a}
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | mulIndicator_apply_eq_if | null |
mulIndicator_of_mem (hU : IsClopen U) (h : a ∈ U) : f.mulIndicator hU a = f a :=
Set.mulIndicator_of_mem h _
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | mulIndicator_of_mem | null |
mulIndicator_of_notMem (hU : IsClopen U) (h : a ∉ U) : f.mulIndicator hU a = 1 :=
Set.mulIndicator_of_notMem h _
@[deprecated (since := "2025-05-23")] alias indicator_of_not_mem := indicator_of_notMem
@[to_additive existing, deprecated (since := "2025-05-23")]
alias mulIndicator_of_not_mem := mulIndicator_of_notMem | theorem | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | mulIndicator_of_notMem | null |
@[simps]
congrLeft [TopologicalSpace Y] (e : X ≃ₜ Y) : LocallyConstant X Z ≃ LocallyConstant Y Z where
toFun := comap e.symm
invFun := comap e
left_inv := by
intro
simp [comap_comap]
right_inv := by
intro
simp [comap_comap] | def | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | congrLeft | The equivalence between `LocallyConstant X Z` and `LocallyConstant Y Z` given a
homeomorphism `X ≃ₜ Y` |
@[simps]
congrRight (e : Y ≃ Z) : LocallyConstant X Y ≃ LocallyConstant X Z where
toFun := map e
invFun := map e.symm
left_inv := by intro; ext; simp
right_inv := by intro; ext; simp
variable (X) in | def | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | congrRight | The equivalence between `LocallyConstant X Y` and `LocallyConstant X Z` given an
equivalence `Y ≃ Z` |
equivClopens [∀ (s : Set X) x, Decidable (x ∈ s)] :
LocallyConstant X (Fin 2) ≃ TopologicalSpace.Clopens X where
toFun f := ⟨f ⁻¹' {0}, f.2.isClopen_fiber _⟩
invFun s := ofIsClopen s.2
left_inv _ := locallyConstant_eq_of_fiber_zero_eq _ _ (by simp)
right_inv _ := by simp | def | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | equivClopens | The set of clopen subsets of a topological space is equivalent to the locally constant maps to
a two-element set |
piecewise {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂) (h : C₁ ∪ C₂ = Set.univ)
(f : LocallyConstant C₁ Z) (g : LocallyConstant C₂ Z)
(hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, hx.1⟩ = g ⟨x, hx.2⟩)
[DecidablePred (· ∈ C₁)] : LocallyConstant X Z where
toFun i := if hi : i ∈ C₁ then f ⟨i, hi⟩ else g ⟨i, (Set.compl_subset_iff_union.mpr h) hi⟩
isLocallyConstant := by
let dZ : TopologicalSpace Z := ⊥
haveI : DiscreteTopology Z := discreteTopology_bot Z
obtain ⟨f, hf⟩ := f
obtain ⟨g, hg⟩ := g
rw [IsLocallyConstant.iff_continuous] at hf hg ⊢
dsimp only [coe_mk]
rw [Set.union_eq_iUnion] at h
refine (locallyFinite_of_finite _).continuous h (fun i ↦ ?_) (fun i ↦ ?_)
· cases i <;> [exact h₂; exact h₁]
· cases i <;> rw [continuousOn_iff_continuous_restrict]
· convert hg
ext x
simp only [cond_false, restrict_apply, Subtype.coe_eta, dite_eq_right_iff]
exact fun hx ↦ hfg x ⟨hx, x.prop⟩
· simp only [cond_true, restrict_dite, Subtype.coe_eta]
exact hf
@[simp] | def | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | piecewise | Given two closed sets covering a topological space, and locally constant maps on these two sets,
then if these two locally constant maps agree on the intersection, we get a piecewise defined
locally constant map on the whole space.
TODO: Generalise this construction to `ContinuousMap`. |
piecewise_apply_left {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂)
(h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant C₁ Z) (g : LocallyConstant C₂ Z)
(hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, hx.1⟩ = g ⟨x, hx.2⟩)
[DecidablePred (· ∈ C₁)] (x : X) (hx : x ∈ C₁) :
piecewise h₁ h₂ h f g hfg x = f ⟨x, hx⟩ := by
simp only [piecewise,
coe_mk]
rw [dif_pos hx]
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | piecewise_apply_left | null |
piecewise_apply_right {C₁ C₂ : Set X} (h₁ : IsClosed C₁) (h₂ : IsClosed C₂)
(h : C₁ ∪ C₂ = Set.univ) (f : LocallyConstant C₁ Z) (g : LocallyConstant C₂ Z)
(hfg : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f ⟨x, hx.1⟩ = g ⟨x, hx.2⟩)
[DecidablePred (· ∈ C₁)] (x : X) (hx : x ∈ C₂) :
piecewise h₁ h₂ h f g hfg x = g ⟨x, hx⟩ := by
simp only [piecewise,
coe_mk]
split_ifs with h
· exact hfg x ⟨h, hx⟩
· rfl | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | piecewise_apply_right | null |
piecewise' {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁)
(h₂ : IsClosed C₂) (f₁ : LocallyConstant C₁ Z) (f₂ : LocallyConstant C₂ Z)
[DecidablePred (· ∈ C₁)] (hf : ∀ x (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, hx.1⟩ = f₂ ⟨x, hx.2⟩) :
LocallyConstant C₀ Z :=
letI : ∀ j : C₀, Decidable (j ∈ Subtype.val ⁻¹' C₁) := fun j ↦ decidable_of_iff (↑j ∈ C₁) Iff.rfl
piecewise (h₁.preimage continuous_subtype_val) (h₂.preimage continuous_subtype_val)
(by simpa [eq_univ_iff_forall] using h₀)
(f₁.comap ⟨(restrictPreimage C₁ ((↑) : C₀ → X)), continuous_subtype_val.restrictPreimage⟩)
(f₂.comap ⟨(restrictPreimage C₂ ((↑) : C₀ → X)), continuous_subtype_val.restrictPreimage⟩) <| by
rintro ⟨x, hx₀⟩ ⟨hx₁ : x ∈ C₁, hx₂ : x ∈ C₂⟩
simpa using hf x ⟨hx₁, hx₂⟩
@[simp] | def | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | piecewise' | A variant of `LocallyConstant.piecewise` where the two closed sets cover a subset.
TODO: Generalise this construction to `ContinuousMap`. |
piecewise'_apply_left {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁)
(h₂ : IsClosed C₂) (f₁ : LocallyConstant C₁ Z) (f₂ : LocallyConstant C₂ Z)
[DecidablePred (· ∈ C₁)] (hf : ∀ x (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, hx.1⟩ = f₂ ⟨x, hx.2⟩)
(x : C₀) (hx : x.val ∈ C₁) :
piecewise' h₀ h₁ h₂ f₁ f₂ hf x = f₁ ⟨x.val, hx⟩ := by
letI : ∀ j : C₀, Decidable (j ∈ Subtype.val ⁻¹' C₁) := fun j ↦ decidable_of_iff (↑j ∈ C₁) Iff.rfl
rw [piecewise', piecewise_apply_left (f := (f₁.comap
⟨(restrictPreimage C₁ ((↑) : C₀ → X)), continuous_subtype_val.restrictPreimage⟩))
(hx := hx)]
rfl
@[simp] | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | piecewise'_apply_left | null |
piecewise'_apply_right {C₀ C₁ C₂ : Set X} (h₀ : C₀ ⊆ C₁ ∪ C₂) (h₁ : IsClosed C₁)
(h₂ : IsClosed C₂) (f₁ : LocallyConstant C₁ Z) (f₂ : LocallyConstant C₂ Z)
[DecidablePred (· ∈ C₁)] (hf : ∀ x (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, hx.1⟩ = f₂ ⟨x, hx.2⟩)
(x : C₀) (hx : x.val ∈ C₂) :
piecewise' h₀ h₁ h₂ f₁ f₂ hf x = f₂ ⟨x.val, hx⟩ := by
letI : ∀ j : C₀, Decidable (j ∈ Subtype.val ⁻¹' C₁) := fun j ↦ decidable_of_iff (↑j ∈ C₁) Iff.rfl
rw [piecewise', piecewise_apply_right (f := (f₁.comap
⟨(restrictPreimage C₁ ((↑) : C₀ → X)), continuous_subtype_val.restrictPreimage⟩))
(hx := hx)]
rfl | lemma | Topology | [
"Mathlib.Algebra.Notation.Indicator",
"Mathlib.Tactic.FinCases",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/LocallyConstant/Basic.lean | piecewise'_apply_right | null |
protected IsInducing.induced (f : X → Y) : @IsInducing X Y (induced f ‹_›) _ f :=
@IsInducing.mk _ _ (TopologicalSpace.induced f ‹_›) _ _ rfl
variable [TopologicalSpace X] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsInducing.induced | null |
protected IsInducing.id : IsInducing (@id X) := ⟨induced_id.symm⟩
variable [TopologicalSpace Z] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsInducing.id | null |
protected IsInducing.comp (hg : IsInducing g) (hf : IsInducing f) :
IsInducing (g ∘ f) :=
⟨by rw [hf.eq_induced, hg.eq_induced, induced_compose]⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsInducing.comp | null |
IsInducing.of_comp_iff (hg : IsInducing g) : IsInducing (g ∘ f) ↔ IsInducing f := by
refine ⟨fun h ↦ ?_, hg.comp⟩
rw [isInducing_iff, hg.eq_induced, induced_compose, h.eq_induced] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsInducing.of_comp_iff | null |
IsInducing.of_comp (hf : Continuous f) (hg : Continuous g) (hgf : IsInducing (g ∘ f)) :
IsInducing f :=
⟨le_antisymm hf.le_induced (by grw [hgf.eq_induced, ← induced_compose, ← hg.le_induced])⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | IsInducing.of_comp | null |
isInducing_iff_nhds : IsInducing f ↔ ∀ x, 𝓝 x = comap f (𝓝 (f x)) :=
(isInducing_iff _).trans (induced_iff_nhds_eq f) | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isInducing_iff_nhds | null |
nhds_eq_comap (hf : IsInducing f) : ∀ x : X, 𝓝 x = comap f (𝓝 <| f x) :=
isInducing_iff_nhds.1 hf | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | nhds_eq_comap | null |
basis_nhds {p : ι → Prop} {s : ι → Set Y} (hf : IsInducing f) {x : X}
(h_basis : (𝓝 (f x)).HasBasis p s) : (𝓝 x).HasBasis p (preimage f ∘ s) :=
hf.nhds_eq_comap x ▸ h_basis.comap f | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | basis_nhds | null |
nhdsSet_eq_comap (hf : IsInducing f) (s : Set X) :
𝓝ˢ s = comap f (𝓝ˢ (f '' s)) := by
simp only [nhdsSet, sSup_image, comap_iSup, hf.nhds_eq_comap, iSup_image] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | nhdsSet_eq_comap | null |
map_nhds_eq (hf : IsInducing f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x :=
hf.eq_induced ▸ map_nhds_induced_eq x | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | map_nhds_eq | null |
map_nhds_of_mem (hf : IsInducing f) (x : X) (h : range f ∈ 𝓝 (f x)) :
(𝓝 x).map f = 𝓝 (f x) := hf.eq_induced ▸ map_nhds_induced_of_mem h | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | map_nhds_of_mem | null |
mapClusterPt_iff (hf : IsInducing f) {x : X} {l : Filter X} :
MapClusterPt (f x) l f ↔ ClusterPt x l := by
delta MapClusterPt ClusterPt
rw [← Filter.push_pull', ← hf.nhds_eq_comap, map_neBot_iff] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | mapClusterPt_iff | null |
image_mem_nhdsWithin (hf : IsInducing f) {x : X} {s : Set X} (hs : s ∈ 𝓝 x) :
f '' s ∈ 𝓝[range f] f x :=
hf.map_nhds_eq x ▸ image_mem_map hs | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | image_mem_nhdsWithin | null |
tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : IsInducing g) :
Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := by
rw [hg.nhds_eq_comap, tendsto_comap_iff] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | tendsto_nhds_iff | null |
continuousAt_iff (hg : IsInducing g) {x : X} :
ContinuousAt f x ↔ ContinuousAt (g ∘ f) x :=
hg.tendsto_nhds_iff | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuousAt_iff | null |
continuous_iff (hg : IsInducing g) :
Continuous f ↔ Continuous (g ∘ f) := by
simp_rw [continuous_iff_continuousAt, hg.continuousAt_iff] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuous_iff | null |
continuousAt_iff' (hf : IsInducing f) {x : X} (h : range f ∈ 𝓝 (f x)) :
ContinuousAt (g ∘ f) x ↔ ContinuousAt g (f x) := by
simp_rw [ContinuousAt, Filter.Tendsto, ← hf.map_nhds_of_mem _ h, Filter.map_map, comp] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuousAt_iff' | null |
protected continuous (hf : IsInducing f) : Continuous f :=
hf.continuous_iff.mp continuous_id | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuous | null |
closure_eq_preimage_closure_image (hf : IsInducing f) (s : Set X) :
closure s = f ⁻¹' closure (f '' s) := by
ext x
rw [Set.mem_preimage, ← closure_induced, hf.eq_induced] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | closure_eq_preimage_closure_image | null |
isClosed_iff (hf : IsInducing f) {s : Set X} :
IsClosed s ↔ ∃ t, IsClosed t ∧ f ⁻¹' t = s := by rw [hf.eq_induced, isClosed_induced_iff] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosed_iff | null |
isClosed_iff' (hf : IsInducing f) {s : Set X} :
IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by rw [hf.eq_induced, isClosed_induced_iff'] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosed_iff' | null |
isClosed_preimage (h : IsInducing f) (s : Set Y) (hs : IsClosed s) :
IsClosed (f ⁻¹' s) :=
(isClosed_iff h).mpr ⟨s, hs, rfl⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosed_preimage | null |
isOpen_iff (hf : IsInducing f) {s : Set X} :
IsOpen s ↔ ∃ t, IsOpen t ∧ f ⁻¹' t = s := by rw [hf.eq_induced, isOpen_induced_iff] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpen_iff | null |
setOf_isOpen (hf : IsInducing f) :
{s : Set X | IsOpen s} = preimage f '' {t | IsOpen t} :=
Set.ext fun _ ↦ hf.isOpen_iff | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | setOf_isOpen | null |
dense_iff (hf : IsInducing f) {s : Set X} :
Dense s ↔ ∀ x, f x ∈ closure (f '' s) := by
simp only [Dense, hf.closure_eq_preimage_closure_image, mem_preimage] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | dense_iff | null |
of_subsingleton [Subsingleton X] (f : X → Y) : IsInducing f :=
⟨Subsingleton.elim _ _⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_subsingleton | null |
induced [t : TopologicalSpace Y] (hf : Injective f) :
@IsEmbedding X Y (t.induced f) t f :=
@IsEmbedding.mk X Y (t.induced f) t _ (.induced f) hf
alias _root_.Function.Injective.isEmbedding_induced := IsEmbedding.induced
variable [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | induced | null |
isInducing (hf : IsEmbedding f) : IsInducing f := hf.toIsInducing | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isInducing | null |
mk' (f : X → Y) (inj : Injective f) (induced : ∀ x, comap f (𝓝 (f x)) = 𝓝 x) :
IsEmbedding f :=
⟨isInducing_iff_nhds.2 fun x => (induced x).symm, inj⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | mk' | null |
protected id : IsEmbedding (@id X) := ⟨.id, fun _ _ h => h⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | id | null |
protected comp (hg : IsEmbedding g) (hf : IsEmbedding f) : IsEmbedding (g ∘ f) :=
{ hg.isInducing.comp hf.isInducing with injective := fun _ _ h => hf.injective <| hg.injective h } | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | comp | null |
of_comp_iff (hg : IsEmbedding g) : IsEmbedding (g ∘ f) ↔ IsEmbedding f := by
simp_rw [isEmbedding_iff, hg.isInducing.of_comp_iff, hg.injective.of_comp_iff f] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_comp_iff | null |
protected of_comp (hf : Continuous f) (hg : Continuous g) (hgf : IsEmbedding (g ∘ f)) :
IsEmbedding f where
toIsInducing := hgf.isInducing.of_comp hf hg
injective := hgf.injective.of_comp | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_comp | null |
of_leftInverse {f : X → Y} {g : Y → X} (h : LeftInverse f g) (hf : Continuous f)
(hg : Continuous g) : IsEmbedding g := .of_comp hg hf <| h.comp_eq_id.symm ▸ .id
alias _root_.Function.LeftInverse.isEmbedding := of_leftInverse | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_leftInverse | null |
map_nhds_eq (hf : IsEmbedding f) (x : X) : (𝓝 x).map f = 𝓝[range f] f x :=
hf.1.map_nhds_eq x | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | map_nhds_eq | null |
map_nhds_of_mem (hf : IsEmbedding f) (x : X) (h : range f ∈ 𝓝 (f x)) :
(𝓝 x).map f = 𝓝 (f x) :=
hf.1.map_nhds_of_mem x h | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | map_nhds_of_mem | null |
tendsto_nhds_iff {f : ι → Y} {l : Filter ι} {y : Y} (hg : IsEmbedding g) :
Tendsto f l (𝓝 y) ↔ Tendsto (g ∘ f) l (𝓝 (g y)) := hg.isInducing.tendsto_nhds_iff | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | tendsto_nhds_iff | null |
continuous_iff (hg : IsEmbedding g) : Continuous f ↔ Continuous (g ∘ f) :=
hg.isInducing.continuous_iff | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuous_iff | null |
continuous (hf : IsEmbedding f) : Continuous f := hf.isInducing.continuous | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuous | null |
closure_eq_preimage_closure_image (hf : IsEmbedding f) (s : Set X) :
closure s = f ⁻¹' closure (f '' s) :=
hf.1.closure_eq_preimage_closure_image s | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | closure_eq_preimage_closure_image | null |
discreteTopology [DiscreteTopology Y] (hf : IsEmbedding f) : DiscreteTopology X :=
.of_continuous_injective hf.continuous hf.injective | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | discreteTopology | 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`. |
of_subsingleton [Subsingleton X] (f : X → Y) : IsEmbedding f :=
⟨.of_subsingleton f, f.injective_of_subsingleton⟩ | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_subsingleton | null |
isQuotientMap_iff : IsQuotientMap f ↔ Surjective f ∧ ∀ s, IsOpen s ↔ IsOpen (f ⁻¹' s) :=
(isQuotientMap_iff' _).trans <| and_congr Iff.rfl TopologicalSpace.ext_iff | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isQuotientMap_iff | null |
isQuotientMap_iff_isClosed :
IsQuotientMap f ↔ Surjective f ∧ ∀ s : Set Y, IsClosed s ↔ IsClosed (f ⁻¹' s) :=
isQuotientMap_iff.trans <| Iff.rfl.and <| compl_surjective.forall.trans <| by
simp only [isOpen_compl_iff, preimage_compl] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isQuotientMap_iff_isClosed | null |
protected id : IsQuotientMap (@id X) :=
⟨fun x => ⟨x, rfl⟩, coinduced_id.symm⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | id | null |
protected comp (hg : IsQuotientMap g) (hf : IsQuotientMap f) : IsQuotientMap (g ∘ f) :=
⟨hg.surjective.comp hf.surjective, by rw [hg.eq_coinduced, hf.eq_coinduced, coinduced_compose]⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | comp | null |
protected of_comp (hf : Continuous f) (hg : Continuous g)
(hgf : IsQuotientMap (g ∘ f)) : IsQuotientMap g :=
⟨hgf.1.of_comp,
le_antisymm (by grw [hgf.eq_coinduced, ← coinduced_compose, hf.coinduced_le]) hg.coinduced_le⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_comp | null |
of_inverse {g : Y → X} (hf : Continuous f) (hg : Continuous g) (h : LeftInverse g f) :
IsQuotientMap g := .of_comp hf hg <| h.comp_eq_id.symm ▸ IsQuotientMap.id | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_inverse | null |
protected continuous_iff (hf : IsQuotientMap f) : Continuous g ↔ Continuous (g ∘ f) := by
rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.eq_coinduced, coinduced_compose] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuous_iff | null |
protected continuous (hf : IsQuotientMap f) : Continuous f :=
hf.continuous_iff.mp continuous_id | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | continuous | null |
protected isOpen_preimage (hf : IsQuotientMap f) {s : Set Y} : IsOpen (f ⁻¹' s) ↔ IsOpen s :=
((isQuotientMap_iff.1 hf).2 s).symm | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpen_preimage | null |
protected isClosed_preimage (hf : IsQuotientMap f) {s : Set Y} :
IsClosed (f ⁻¹' s) ↔ IsClosed s :=
((isQuotientMap_iff_isClosed.1 hf).2 s).symm | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isClosed_preimage | null |
protected id : IsOpenMap (@id X) := fun s hs => by rwa [image_id] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | id | null |
protected comp (hg : IsOpenMap g) (hf : IsOpenMap f) :
IsOpenMap (g ∘ f) := fun s hs => by rw [image_comp]; exact hg _ (hf _ hs) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | comp | null |
isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by
rw [← image_univ]
exact hf _ isOpen_univ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpen_range | null |
image_mem_nhds (hf : IsOpenMap f) {x : X} {s : Set X} (hx : s ∈ 𝓝 x) : f '' s ∈ 𝓝 (f x) :=
let ⟨t, hts, ht, hxt⟩ := mem_nhds_iff.1 hx
mem_of_superset (IsOpen.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_mono hts) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | image_mem_nhds | null |
range_mem_nhds (hf : IsOpenMap f) (x : X) : range f ∈ 𝓝 (f x) :=
hf.isOpen_range.mem_nhds <| mem_range_self _ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | range_mem_nhds | null |
mapsTo_interior (hf : IsOpenMap f) {s : Set X} {t : Set Y} (h : MapsTo f s t) :
MapsTo f (interior s) (interior t) :=
mapsTo_iff_image_subset.2 <|
interior_maximal (h.mono interior_subset Subset.rfl).image_subset (hf _ isOpen_interior) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | mapsTo_interior | null |
image_interior_subset (hf : IsOpenMap f) (s : Set X) :
f '' interior s ⊆ interior (f '' s) :=
(hf.mapsTo_interior (mapsTo_image f s)).image_subset | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | image_interior_subset | null |
nhds_le (hf : IsOpenMap f) (x : X) : 𝓝 (f x) ≤ map f (𝓝 x) :=
le_map fun _ => hf.image_mem_nhds | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | nhds_le | null |
map_nhds_eq (hf : IsOpenMap f) {x : X} (hf' : ContinuousAt f x) : map f (𝓝 x) = 𝓝 (f x) :=
le_antisymm hf' (hf.nhds_le x) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | map_nhds_eq | null |
map_nhdsSet_eq (hf : IsOpenMap f) (hf' : Continuous f) (s : Set X) :
map f (𝓝ˢ s) = 𝓝ˢ (f '' s) := by
rw [← biUnion_of_singleton s]
simp_rw [image_iUnion, nhdsSet_iUnion, map_iSup, image_singleton, nhdsSet_singleton,
hf.map_nhds_eq hf'.continuousAt] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | map_nhdsSet_eq | null |
of_nhds_le (hf : ∀ x, 𝓝 (f x) ≤ map f (𝓝 x)) : IsOpenMap f := fun _s hs =>
isOpen_iff_mem_nhds.2 fun _y ⟨_x, hxs, hxy⟩ => hxy ▸ hf _ (image_mem_map <| hs.mem_nhds hxs) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_nhds_le | null |
of_sections
(h : ∀ x, ∃ g : Y → X, ContinuousAt g (f x) ∧ g (f x) = x ∧ RightInverse g f) : IsOpenMap f :=
of_nhds_le fun x =>
let ⟨g, hgc, hgx, hgf⟩ := h x
calc
𝓝 (f x) = map f (map g (𝓝 (f x))) := by rw [map_map, hgf.comp_eq_id, map_id]
_ ≤ map f (𝓝 (g (f x))) := map_mono hgc
_ = map f (𝓝 x) := by rw [hgx] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_sections | null |
of_inverse {f' : Y → X} (h : Continuous f') (l_inv : LeftInverse f f')
(r_inv : RightInverse f f') : IsOpenMap f :=
of_sections fun _ => ⟨f', h.continuousAt, r_inv _, l_inv⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_inverse | null |
isQuotientMap (open_map : IsOpenMap f) (cont : Continuous f) (surj : Surjective f) :
IsQuotientMap f :=
isQuotientMap_iff.2
⟨surj, fun s => ⟨fun h => h.preimage cont, fun h => surj.image_preimage s ▸ open_map _ h⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isQuotientMap | A continuous surjective open map is a quotient map. |
interior_preimage_subset_preimage_interior (hf : IsOpenMap f) {s : Set Y} :
interior (f ⁻¹' s) ⊆ f ⁻¹' interior s :=
hf.mapsTo_interior (mapsTo_preimage _ _) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | interior_preimage_subset_preimage_interior | null |
preimage_interior_eq_interior_preimage (hf₁ : IsOpenMap f) (hf₂ : Continuous f)
(s : Set Y) : f ⁻¹' interior s = interior (f ⁻¹' s) :=
Subset.antisymm (preimage_interior_subset_interior_preimage hf₂)
(interior_preimage_subset_preimage_interior hf₁) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | preimage_interior_eq_interior_preimage | null |
preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set Y} :
f ⁻¹' closure s ⊆ closure (f ⁻¹' s) := by
rw [← compl_subset_compl]
simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | preimage_closure_subset_closure_preimage | null |
preimage_closure_eq_closure_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) :
f ⁻¹' closure s = closure (f ⁻¹' s) :=
hf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s) | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | preimage_closure_eq_closure_preimage | null |
preimage_frontier_subset_frontier_preimage (hf : IsOpenMap f) {s : Set Y} :
f ⁻¹' frontier s ⊆ frontier (f ⁻¹' s) := by
simpa only [frontier_eq_closure_inter_closure, preimage_inter] using
inter_subset_inter hf.preimage_closure_subset_closure_preimage
hf.preimage_closure_subset_closure_preimage | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | preimage_frontier_subset_frontier_preimage | null |
preimage_frontier_eq_frontier_preimage (hf : IsOpenMap f) (hfc : Continuous f) (s : Set Y) :
f ⁻¹' frontier s = frontier (f ⁻¹' s) := by
simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl,
hf.preimage_closure_eq_closure_preimage hfc] | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | preimage_frontier_eq_frontier_preimage | null |
of_isEmpty [h : IsEmpty X] (f : X → Y) : IsOpenMap f := of_nhds_le h.elim | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | of_isEmpty | null |
clusterPt_comap (hf : IsOpenMap f) {x : X} {l : Filter Y} (h : ClusterPt (f x) l) :
ClusterPt x (comap f l) := by
rw [ClusterPt, ← map_neBot_iff, Filter.push_pull]
exact h.neBot.mono <| inf_le_inf_right _ <| hf.nhds_le _ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | clusterPt_comap | null |
isOpenMap_iff_kernImage :
IsOpenMap f ↔ ∀ {u : Set X}, IsClosed u → IsClosed (kernImage f u) := by
rw [IsOpenMap, compl_surjective.forall]
simp [kernImage_eq_compl] | lemma | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpenMap_iff_kernImage | null |
isOpenMap_iff_nhds_le : IsOpenMap f ↔ ∀ x : X, 𝓝 (f x) ≤ (𝓝 x).map f :=
⟨fun hf => hf.nhds_le, IsOpenMap.of_nhds_le⟩ | theorem | Topology | [
"Mathlib.Topology.Order",
"Mathlib.Topology.NhdsSet"
] | Mathlib/Topology/Maps/Basic.lean | isOpenMap_iff_nhds_le | null |
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