Loogle!

Result

Found 24 declarations mentioning TopologicalSpace.Compacts.map.

• TopologicalSpace.Compacts.map 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (hf : Continuous f) (K : TopologicalSpace.Compacts α) : TopologicalSpace.Compacts β
• TopologicalSpace.Compacts.map_id 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} [TopologicalSpace α] (K : TopologicalSpace.Compacts α) : TopologicalSpace.Compacts.map id ⋯ K = K
• TopologicalSpace.Compacts.coe_map 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] {f : α → β} (hf : Continuous f) (s : TopologicalSpace.Compacts α) : ↑(TopologicalSpace.Compacts.map f hf s) = f '' ↑s
• TopologicalSpace.CompactOpens.map_toCompacts 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (hf : Continuous f) (hf' : IsOpenMap f) (s : TopologicalSpace.CompactOpens α) : (TopologicalSpace.CompactOpens.map f hf hf' s).toCompacts = TopologicalSpace.Compacts.map f hf s.toCompacts
• TopologicalSpace.Compacts.map_comp 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (f : β → γ) (g : α → β) (hf : Continuous f) (hg : Continuous g) (K : TopologicalSpace.Compacts α) : TopologicalSpace.Compacts.map (f ∘ g) ⋯ K = TopologicalSpace.Compacts.map f hf (TopologicalSpace.Compacts.map g hg K)
• TopologicalSpace.Compacts.equiv_apply 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) (K : TopologicalSpace.Compacts α) : (TopologicalSpace.Compacts.equiv f) K = TopologicalSpace.Compacts.map ⇑f ⋯ K
• TopologicalSpace.Compacts.equiv_symm_apply 📋 Mathlib.Topology.Sets.Compacts
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) (K : TopologicalSpace.Compacts β) : (TopologicalSpace.Compacts.equiv f).symm K = TopologicalSpace.Compacts.map ⇑f.symm ⋯ K
• TopologicalSpace.Compacts.equiv.eq_1 📋 Mathlib.MeasureTheory.Measure.Content
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β) : TopologicalSpace.Compacts.equiv f = { toFun := TopologicalSpace.Compacts.map ⇑f ⋯, invFun := TopologicalSpace.Compacts.map ⇑f.symm ⋯, left_inv := ⋯, right_inv := ⋯ }
• MeasureTheory.Content.outerMeasure_preimage 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] (f : G ≃ₜ G) (h : ∀ ⦃K : TopologicalSpace.Compacts G⦄, μ (TopologicalSpace.Compacts.map ⇑f ⋯ K) = μ K) (A : Set G) : μ.outerMeasure (⇑f ⁻¹' A) = μ.outerMeasure A
• MeasureTheory.Content.innerContent_pos_of_is_add_left_invariant 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [AddGroup G] [IsTopologicalAddGroup G] (h3 : ∀ (g : G) {K : TopologicalSpace.Compacts G}, μ (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = μ K) (K : TopologicalSpace.Compacts G) (hK : μ K ≠ 0) (U : TopologicalSpace.Opens G) (hU : (↑U).Nonempty) : 0 < μ.innerContent U
• MeasureTheory.Content.innerContent_pos_of_is_mul_left_invariant 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [Group G] [IsTopologicalGroup G] (h3 : ∀ (g : G) {K : TopologicalSpace.Compacts G}, μ (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = μ K) (K : TopologicalSpace.Compacts G) (hK : μ K ≠ 0) (U : TopologicalSpace.Opens G) (hU : (↑U).Nonempty) : 0 < μ.innerContent U
• MeasureTheory.Content.is_add_left_invariant_outerMeasure 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [AddGroup G] [ContinuousAdd G] (h : ∀ (g : G) {K : TopologicalSpace.Compacts G}, μ (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = μ K) (g : G) (A : Set G) : μ.outerMeasure ((fun x => g + x) ⁻¹' A) = μ.outerMeasure A
• MeasureTheory.Content.is_mul_left_invariant_outerMeasure 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [Group G] [ContinuousMul G] (h : ∀ (g : G) {K : TopologicalSpace.Compacts G}, μ (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = μ K) (g : G) (A : Set G) : μ.outerMeasure ((fun x => g * x) ⁻¹' A) = μ.outerMeasure A
• MeasureTheory.Content.outerMeasure_pos_of_is_add_left_invariant 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [AddGroup G] [IsTopologicalAddGroup G] (h3 : ∀ (g : G) {K : TopologicalSpace.Compacts G}, μ (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = μ K) (K : TopologicalSpace.Compacts G) (hK : μ K ≠ 0) {U : Set G} (h1U : IsOpen U) (h2U : U.Nonempty) : 0 < μ.outerMeasure U
• MeasureTheory.Content.outerMeasure_pos_of_is_mul_left_invariant 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [R1Space G] [Group G] [IsTopologicalGroup G] (h3 : ∀ (g : G) {K : TopologicalSpace.Compacts G}, μ (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = μ K) (K : TopologicalSpace.Compacts G) (hK : μ K ≠ 0) {U : Set G} (h1U : IsOpen U) (h2U : U.Nonempty) : 0 < μ.outerMeasure U
• MeasureTheory.Content.innerContent_comap 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) (f : G ≃ₜ G) (h : ∀ ⦃K : TopologicalSpace.Compacts G⦄, μ (TopologicalSpace.Compacts.map ⇑f ⋯ K) = μ K) (U : TopologicalSpace.Opens G) : μ.innerContent ((TopologicalSpace.Opens.comap ↑f) U) = μ.innerContent U
• MeasureTheory.Content.is_add_left_invariant_innerContent 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [AddGroup G] [ContinuousAdd G] (h : ∀ (g : G) {K : TopologicalSpace.Compacts G}, μ (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = μ K) (g : G) (U : TopologicalSpace.Opens G) : μ.innerContent ((TopologicalSpace.Opens.comap ↑(Homeomorph.addLeft g)) U) = μ.innerContent U
• MeasureTheory.Content.is_mul_left_invariant_innerContent 📋 Mathlib.MeasureTheory.Measure.Content
  {G : Type w} [TopologicalSpace G] (μ : MeasureTheory.Content G) [Group G] [ContinuousMul G] (h : ∀ (g : G) {K : TopologicalSpace.Compacts G}, μ (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = μ K) (g : G) (U : TopologicalSpace.Opens G) : μ.innerContent ((TopologicalSpace.Opens.comap ↑(Homeomorph.mulLeft g)) U) = μ.innerContent U
• MeasureTheory.Measure.haar.is_left_invariant_addCHaar 📋 Mathlib.MeasureTheory.Measure.Haar.Basic
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (g : G) (K : TopologicalSpace.Compacts G) : MeasureTheory.Measure.haar.addCHaar K₀ (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = MeasureTheory.Measure.haar.addCHaar K₀ K
• MeasureTheory.Measure.haar.is_left_invariant_chaar 📋 Mathlib.MeasureTheory.Measure.Haar.Basic
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (g : G) (K : TopologicalSpace.Compacts G) : MeasureTheory.Measure.haar.chaar K₀ (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = MeasureTheory.Measure.haar.chaar K₀ K
• MeasureTheory.Measure.haar.is_left_invariant_addPrehaar 📋 Mathlib.MeasureTheory.Measure.Haar.Basic
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty) (g : G) (K : TopologicalSpace.Compacts G) : MeasureTheory.Measure.haar.addPrehaar (↑K₀) U (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K
• MeasureTheory.Measure.haar.is_left_invariant_prehaar 📋 Mathlib.MeasureTheory.Measure.Haar.Basic
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty) (g : G) (K : TopologicalSpace.Compacts G) : MeasureTheory.Measure.haar.prehaar (↑K₀) U (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = MeasureTheory.Measure.haar.prehaar (↑K₀) U K
• MeasureTheory.Measure.haar.is_left_invariant_addHaarContent 📋 Mathlib.MeasureTheory.Measure.Haar.Basic
  {G : Type u_1} [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (g : G) (K : TopologicalSpace.Compacts G) : (MeasureTheory.Measure.haar.addHaarContent K₀) (TopologicalSpace.Compacts.map (fun b => g + b) ⋯ K) = (MeasureTheory.Measure.haar.addHaarContent K₀) K
• MeasureTheory.Measure.haar.is_left_invariant_haarContent 📋 Mathlib.MeasureTheory.Measure.Haar.Basic
  {G : Type u_1} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] {K₀ : TopologicalSpace.PositiveCompacts G} (g : G) (K : TopologicalSpace.Compacts G) : (MeasureTheory.Measure.haar.haarContent K₀) (TopologicalSpace.Compacts.map (fun b => g * b) ⋯ K) = (MeasureTheory.Measure.haar.haarContent K₀) K

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 19971e9 serving mathlib revision 40fea08