statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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dense_range.subset_closure_image_preimage_of_is_open (hf : dense_range f) {s : set β}
(hs : is_open s) : s ⊆ closure (f '' (f ⁻¹' s)) | by { rw image_preimage_eq_inter_range, exact hf.open_subset_closure_inter hs } | lemma | dense_range.subset_closure_image_preimage_of_is_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"dense_range",
"is_open"
] | If `f` has dense range and `s` is an open set in the codomain of `f`, then the image of the
preimage of `s` under `f` is dense in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range.dense_of_maps_to {f : α → β} (hf' : dense_range f) (hf : continuous f)
{s : set α} (hs : dense s) {t : set β} (ht : maps_to f s t) :
dense t | (hf'.dense_image hf hs).mono ht.image_subset | lemma | dense_range.dense_of_maps_to | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"dense",
"dense_range"
] | If a continuous map with dense range maps a dense set to a subset of `t`, then `t` is a dense
set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range.comp {g : β → γ} {f : κ → β} (hg : dense_range g) (hf : dense_range f)
(cg : continuous g) :
dense_range (g ∘ f) | by { rw [dense_range, range_comp], exact hg.dense_image cg hf } | lemma | dense_range.comp | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"continuous",
"dense_range"
] | Composition of a continuous map with dense range and a function with dense range has dense
range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range.nonempty_iff (hf : dense_range f) : nonempty κ ↔ nonempty β | range_nonempty_iff_nonempty.symm.trans hf.nonempty_iff | lemma | dense_range.nonempty_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.nonempty [h : nonempty β] (hf : dense_range f) : nonempty κ | hf.nonempty_iff.mpr h | lemma | dense_range.nonempty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.some (hf : dense_range f) (b : β) : κ | classical.choice $ hf.nonempty_iff.mpr ⟨b⟩ | def | dense_range.some | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_range"
] | Given a function `f : α → β` with dense range and `b : β`, returns some `a : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range.exists_mem_open (hf : dense_range f) {s : set β} (ho : is_open s)
(hs : s.nonempty) :
∃ a, f a ∈ s | exists_range_iff.1 $ hf.exists_mem_open ho hs | lemma | dense_range.exists_mem_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_range",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.mem_nhds {f : κ → β} (h : dense_range f) {b : β} {U : set β}
(U_in : U ∈ 𝓝 b) : ∃ a, f a ∈ U | let ⟨a, ha⟩ := h.exists_mem_open is_open_interior ⟨b, mem_interior_iff_mem_nhds.2 U_in⟩
in ⟨a, interior_subset ha⟩ | lemma | dense_range.mem_nhds | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"dense_range",
"interior_subset",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_open.gen (s : set α) (u : set β) : set C(α,β) | {f | f '' s ⊆ u} | def | continuous_map.compact_open.gen | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [] | A generating set for the compact-open topology (when `s` is compact and `u` is open). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gen_empty (u : set β) : compact_open.gen (∅ : set α) u = set.univ | set.ext (λ f, iff_true_intro ((congr_arg (⊆ u) (image_empty f)).mpr u.empty_subset)) | lemma | continuous_map.gen_empty | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gen_univ (s : set α) : compact_open.gen s (set.univ : set β) = set.univ | set.ext (λ f, iff_true_intro (f '' s).subset_univ) | lemma | continuous_map.gen_univ | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gen_inter (s : set α) (u v : set β) :
compact_open.gen s (u ∩ v) = compact_open.gen s u ∩ compact_open.gen s v | set.ext (λ f, subset_inter_iff) | lemma | continuous_map.gen_inter | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gen_union (s t : set α) (u : set β) :
compact_open.gen (s ∪ t) u = compact_open.gen s u ∩ compact_open.gen t u | set.ext (λ f, (iff_of_eq (congr_arg (⊆ u) (image_union f s t))).trans union_subset_iff) | lemma | continuous_map.gen_union | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"iff_of_eq",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gen_empty_right {s : set α} (h : s.nonempty) : compact_open.gen s (∅ : set β) = ∅ | eq_empty_of_forall_not_mem $ λ f, (h.image _).not_subset_empty | lemma | continuous_map.gen_empty_right | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_open : topological_space C(α, β) | topological_space.generate_from
{m | ∃ (s : set α) (hs : is_compact s) (u : set β) (hu : is_open u), m = compact_open.gen s u} | instance | continuous_map.compact_open | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"is_compact",
"is_open",
"topological_space",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_gen {s : set α} (hs : is_compact s) {u : set β} (hu : is_open u) :
is_open (compact_open.gen s u) | topological_space.generate_open.basic _ (by dsimp [mem_set_of_eq]; tauto) | lemma | continuous_map.is_open_gen | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"is_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_gen {s : set α} (hs : is_compact s) {u : set γ} (hu : is_open u) :
continuous_map.comp g ⁻¹' (compact_open.gen s u) = compact_open.gen s (g ⁻¹' u) | begin
ext ⟨f, _⟩,
change g ∘ f '' s ⊆ u ↔ f '' s ⊆ g ⁻¹' u,
rw [image_comp, image_subset_iff]
end | lemma | continuous_map.preimage_gen | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous_map.comp",
"is_compact",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_comp : continuous (continuous_map.comp g : C(α, β) → C(α, γ)) | continuous_generated_from $ assume m ⟨s, hs, u, hu, hm⟩,
by rw [hm, preimage_gen g hs hu]; exact continuous_map.is_open_gen hs (hu.preimage g.2) | lemma | continuous_map.continuous_comp | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_generated_from",
"continuous_map.comp",
"continuous_map.is_open_gen"
] | C(α, -) is a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_gen {s : set α} (hs : is_compact s) {u : set γ} (hu : is_open u) :
(λ g : C(β, γ), g.comp f) ⁻¹' compact_open.gen s u = compact_open.gen (f '' s) u | begin
ext ⟨g, _⟩,
change g ∘ f '' s ⊆ u ↔ g '' (f '' s) ⊆ u,
rw set.image_comp,
end | lemma | continuous_map.image_gen | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"is_compact",
"is_open",
"set.image_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_comp_left : continuous (λ g, g.comp f : C(β, γ) → C(α, γ)) | continuous_generated_from $ assume m ⟨s, hs, u, hu, hm⟩,
by { rw [hm, image_gen f hs hu], exact continuous_map.is_open_gen (hs.image f.2) hu } | lemma | continuous_map.continuous_comp_left | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_generated_from",
"continuous_map.is_open_gen"
] | C(-, γ) is a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_comp' [locally_compact_space β] :
continuous (λ x : C(α, β) × C(β, γ), x.2.comp x.1) | continuous_generated_from begin
rintros M ⟨K, hK, U, hU, rfl⟩,
conv { congr, rw [compact_open.gen, preimage_set_of_eq],
congr, funext, rw [coe_comp, image_comp, image_subset_iff] },
rw is_open_iff_forall_mem_open,
rintros ⟨φ₀, ψ₀⟩ H,
obtain ⟨L, hL, hKL, hLU⟩ := exists_compact_between (hK.image φ₀.2) (hU.p... | lemma | continuous_map.continuous_comp' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_generated_from",
"continuous_map.is_open_gen",
"exists_compact_between",
"interior",
"is_open_iff_forall_mem_open",
"is_open_interior",
"locally_compact_space",
"subset_trans"
] | Composition is a continuous map from `C(α, β) × C(β, γ)` to `C(α, γ)`, provided that `β` is
locally compact. This is Prop. 9 of Chap. X, §3, №. 4 of Bourbaki's *Topologie Générale*. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.comp' {X : Type*} [topological_space X] [locally_compact_space β]
{f : X → C(α, β)} {g : X → C(β, γ)} (hf : continuous f) (hg : continuous g) :
continuous (λ x, (g x).comp (f x)) | continuous_comp'.comp (hf.prod_mk hg : continuous $ λ x, (f x, g x)) | lemma | continuous_map.continuous.comp' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"locally_compact_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_eval' [locally_compact_space α] : continuous (λ p : C(α, β) × α, p.1 p.2) | continuous_iff_continuous_at.mpr $ assume ⟨f, x⟩ n hn,
let ⟨v, vn, vo, fxv⟩ := mem_nhds_iff.mp hn in
have v ∈ 𝓝 (f x), from is_open.mem_nhds vo fxv,
let ⟨s, hs, sv, sc⟩ :=
locally_compact_space.local_compact_nhds x (f ⁻¹' v)
(f.continuous.tendsto x this) in
let ⟨u, us, uo, xu⟩ := mem_nhds_iff.mp hs i... | lemma | continuous_map.continuous_eval' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_map.is_open_gen",
"is_open",
"is_open.mem_nhds",
"locally_compact_space"
] | The evaluation map `C(α, β) × α → β` is continuous if `α` is locally compact.
See also `continuous_map.continuous_eval` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_eval_const' [locally_compact_space α] (a : α) : continuous (λ f : C(α, β), f a) | continuous_eval'.comp (continuous_id.prod_mk continuous_const) | lemma | continuous_map.continuous_eval_const' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_const",
"locally_compact_space"
] | See also `continuous_map.continuous_eval_const` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_coe' [locally_compact_space α] : @continuous (C(α, β)) (α → β) _ _ coe_fn | continuous_pi continuous_eval_const' | lemma | continuous_map.continuous_coe' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_pi",
"locally_compact_space"
] | See also `continuous_map.continuous_coe` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_open_le_induced (s : set α) :
(continuous_map.compact_open : topological_space C(α, β))
≤ topological_space.induced (continuous_map.restrict s) continuous_map.compact_open | begin
simp only [induced_generate_from_eq, continuous_map.compact_open],
apply topological_space.generate_from_anti,
rintros b ⟨a, ⟨c, hc, u, hu, rfl⟩, rfl⟩,
refine ⟨coe '' c, hc.image continuous_subtype_coe, u, hu, _⟩,
ext f,
simp only [compact_open.gen, mem_set_of_eq, mem_preimage, continuous_map.coe_rest... | lemma | continuous_map.compact_open_le_induced | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous_map.coe_restrict",
"continuous_map.compact_open",
"continuous_map.restrict",
"continuous_subtype_coe",
"induced_generate_from_eq",
"topological_space",
"topological_space.generate_from_anti",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_open_eq_Inf_induced :
(continuous_map.compact_open : topological_space C(α, β))
= ⨅ (s : set α) (hs : is_compact s),
topological_space.induced (continuous_map.restrict s) continuous_map.compact_open | begin
refine le_antisymm _ _,
{ refine le_infi₂ _,
exact λ s hs, compact_open_le_induced s },
simp only [← generate_from_Union, induced_generate_from_eq, continuous_map.compact_open],
apply topological_space.generate_from_anti,
rintros _ ⟨s, hs, u, hu, rfl⟩,
rw mem_Union₂,
refine ⟨s, hs, _, ⟨univ, is_... | lemma | continuous_map.compact_open_eq_Inf_induced | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous_map.coe_restrict",
"continuous_map.compact_open",
"continuous_map.restrict",
"generate_from_Union",
"induced_generate_from_eq",
"is_compact",
"le_infi₂",
"topological_space",
"topological_space.generate_from_anti",
"topological_space.induced"
] | The compact-open topology on `C(α, β)` is equal to the infimum of the compact-open topologies
on `C(s, β)` for `s` a compact subset of `α`. The key point of the proof is that the union of the
compact subsets of `α` is equal to the union of compact subsets of the compact subsets of `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_restrict (s : set α) : continuous (λ F : C(α, β), F.restrict s) | by { rw continuous_iff_le_induced, exact compact_open_le_induced s } | lemma | continuous_map.continuous_restrict | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_iff_le_induced"
] | For any subset `s` of `α`, the restriction of continuous functions to `s` is continuous as a
function from `C(α, β)` to `C(s, β)` with their respective compact-open topologies. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_compact_open_eq_Inf_nhds_induced (f : C(α, β)) :
𝓝 f = ⨅ s (hs : is_compact s), (𝓝 (f.restrict s)).comap (continuous_map.restrict s) | by { rw [compact_open_eq_Inf_induced], simp [nhds_infi, nhds_induced] } | lemma | continuous_map.nhds_compact_open_eq_Inf_nhds_induced | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous_map.restrict",
"is_compact",
"nhds_induced",
"nhds_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_compact_open_restrict {ι : Type*} {l : filter ι} {F : ι → C(α, β)} {f : C(α, β)}
(hFf : filter.tendsto F l (𝓝 f)) (s : set α) :
filter.tendsto (λ i, (F i).restrict s) l (𝓝 (f.restrict s)) | (continuous_restrict s).continuous_at.tendsto.comp hFf | lemma | continuous_map.tendsto_compact_open_restrict | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"filter",
"filter.tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_compact_open_iff_forall {ι : Type*} {l : filter ι} (F : ι → C(α, β)) (f : C(α, β)) :
filter.tendsto F l (𝓝 f)
↔ ∀ s (hs : is_compact s), filter.tendsto (λ i, (F i).restrict s) l (𝓝 (f.restrict s)) | by { rw [compact_open_eq_Inf_induced], simp [nhds_infi, nhds_induced, filter.tendsto_comap_iff] } | lemma | continuous_map.tendsto_compact_open_iff_forall | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"filter",
"filter.tendsto",
"filter.tendsto_comap_iff",
"is_compact",
"nhds_induced",
"nhds_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_tendsto_compact_open_iff_forall [locally_compact_space α] [t2_space α] [t2_space β]
{ι : Type*} {l : filter ι} [filter.ne_bot l] (F : ι → C(α, β)) :
(∃ f, filter.tendsto F l (𝓝 f))
↔ ∀ (s : set α) (hs : is_compact s), ∃ f, filter.tendsto (λ i, (F i).restrict s) l (𝓝 f) | begin
split,
{ rintros ⟨f, hf⟩ s hs,
exact ⟨f.restrict s, tendsto_compact_open_restrict hf s⟩ },
{ intros h,
choose f hf using h,
-- By uniqueness of limits in a `t2_space`, since `λ i, F i x` tends to both `f s₁ hs₁ x` and
-- `f s₂ hs₂ x`, we have `f s₁ hs₁ x = f s₂ hs₂ x`
have h : ∀ s₁ (hs₁ ... | lemma | continuous_map.exists_tendsto_compact_open_iff_forall | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"exists_compact_mem_nhds",
"filter",
"filter.ne_bot",
"filter.tendsto",
"is_compact",
"locally_compact_space",
"t2_space",
"tendsto_nhds_unique"
] | A family `F` of functions in `C(α, β)` converges in the compact-open topology, if and only if
it converges in the compact-open topology on each compact subset of `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coev (b : β) : C(α, β × α) | ⟨prod.mk b, continuous_const.prod_mk continuous_id⟩ | def | continuous_map.coev | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [] | The coevaluation map `β → C(α, β × α)` sending a point `x : β` to the continuous function
on `α` sending `y` to `(x, y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_coev {y : β} (s : set α) : (coev α β y) '' s = ({y} : set β) ×ˢ s | by tidy | lemma | continuous_map.image_coev | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coev : continuous (coev α β) | continuous_generated_from $ begin
rintros _ ⟨s, sc, u, uo, rfl⟩,
rw is_open_iff_forall_mem_open,
intros y hy,
change (coev α β y) '' s ⊆ u at hy,
rw image_coev s at hy,
rcases generalized_tube_lemma is_compact_singleton sc uo hy
with ⟨v, w, vo, wo, yv, sw, vwu⟩,
refine ⟨v, _, vo, singleton_subset_iff.... | lemma | continuous_map.continuous_coev | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_generated_from",
"generalized_tube_lemma",
"is_compact_singleton",
"is_open_iff_forall_mem_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry' (f : C(α × β, γ)) (a : α) : C(β, γ) | ⟨function.curry f a⟩ | def | continuous_map.curry' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [] | Auxiliary definition, see `continuous_map.curry` and `homeomorph.curry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_curry' (f : C(α × β, γ)) : continuous (curry' f) | have hf : curry' f = continuous_map.comp f ∘ coev _ _, by { ext, refl },
hf ▸ continuous.comp (continuous_comp f) continuous_coev | lemma | continuous_map.continuous_curry' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous.comp",
"continuous_map.comp"
] | If a map `α × β → γ` is continuous, then its curried form `α → C(β, γ)` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_of_continuous_uncurry (f : α → C(β, γ))
(h : continuous (function.uncurry (λ x y, f x y))) : continuous f | by { convert continuous_curry' ⟨_, h⟩, ext, refl } | lemma | continuous_map.continuous_of_continuous_uncurry | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous"
] | To show continuity of a map `α → C(β, γ)`, it suffices to show that its uncurried form
`α × β → γ` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry (f : C(α × β, γ)) : C(α, C(β, γ)) | ⟨_, continuous_curry' f⟩ | def | continuous_map.curry | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [] | The curried form of a continuous map `α × β → γ` as a continuous map `α → C(β, γ)`.
If `a × β` is locally compact, this is continuous. If `α` and `β` are both locally
compact, then this is a homeomorphism, see `homeomorph.curry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_curry [locally_compact_space (α × β)] :
continuous (curry : C(α × β, γ) → C(α, C(β, γ))) | begin
apply continuous_of_continuous_uncurry,
apply continuous_of_continuous_uncurry,
rw ←homeomorph.comp_continuous_iff' (homeomorph.prod_assoc _ _ _).symm,
convert continuous_eval';
tidy
end | lemma | continuous_map.continuous_curry | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_curry",
"homeomorph.prod_assoc",
"locally_compact_space"
] | The currying process is a continuous map between function spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry_apply (f : C(α × β, γ)) (a : α) (b : β) : f.curry a b = f (a, b) | rfl | lemma | continuous_map.curry_apply | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"curry_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_uncurry_of_continuous [locally_compact_space β] (f : C(α, C(β, γ))) :
continuous (function.uncurry (λ x y, f x y)) | continuous_eval'.comp $ f.continuous.prod_map continuous_id | lemma | continuous_map.continuous_uncurry_of_continuous | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_id",
"locally_compact_space"
] | The uncurried form of a continuous map `α → C(β, γ)` is a continuous map `α × β → γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uncurry [locally_compact_space β] (f : C(α, C(β, γ))) : C(α × β, γ) | ⟨_, continuous_uncurry_of_continuous f⟩ | def | continuous_map.uncurry | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"locally_compact_space"
] | The uncurried form of a continuous map `α → C(β, γ)` as a continuous map `α × β → γ` (if `β` is
locally compact). If `α` is also locally compact, then this is a homeomorphism between the two
function spaces, see `homeomorph.curry`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_uncurry [locally_compact_space α] [locally_compact_space β] :
continuous (uncurry : C(α, C(β, γ)) → C(α × β, γ)) | begin
apply continuous_of_continuous_uncurry,
rw ←homeomorph.comp_continuous_iff' (homeomorph.prod_assoc _ _ _),
apply continuous.comp continuous_eval' (continuous.prod_map continuous_eval' continuous_id);
apply_instance
end | lemma | continuous_map.continuous_uncurry | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous.comp",
"continuous.prod_map",
"continuous_id",
"homeomorph.prod_assoc",
"locally_compact_space"
] | The uncurrying process is a continuous map between function spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const' : C(β, C(α, β)) | curry ⟨prod.fst, continuous_fst⟩ | def | continuous_map.const' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [] | The family of constant maps: `β → C(α, β)` as a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_const' : (const' : β → C(α, β)) = const α | rfl | lemma | continuous_map.coe_const' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_const' : continuous (const α : β → C(α, β)) | const'.continuous | lemma | continuous_map.continuous_const' | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry [locally_compact_space α] [locally_compact_space β] : C(α × β, γ) ≃ₜ C(α, C(β, γ)) | ⟨⟨curry, uncurry, by tidy, by tidy⟩, continuous_curry, continuous_uncurry⟩ | def | homeomorph.curry | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous_curry",
"locally_compact_space"
] | Currying as a homeomorphism between the function spaces `C(α × β, γ)` and `C(α, C(β, γ))`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map_of_unique [unique α] : β ≃ₜ C(α, β) | { to_fun := const α,
inv_fun := λ f, f default,
left_inv := λ a, rfl,
right_inv := λ f, by { ext, rw unique.eq_default a, refl },
continuous_to_fun := continuous_const',
continuous_inv_fun := continuous_eval'.comp (continuous_id.prod_mk continuous_const) } | def | homeomorph.continuous_map_of_unique | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous_const",
"inv_fun",
"unique",
"unique.eq_default"
] | If `α` has a single element, then `β` is homeomorphic to `C(α, β)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map_of_unique_apply [unique α] (b : β) (a : α) :
continuous_map_of_unique b a = b | rfl | lemma | homeomorph.continuous_map_of_unique_apply | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map_of_unique_symm_apply [unique α] (f : C(α, β)) :
continuous_map_of_unique.symm f = f default | rfl | lemma | homeomorph.continuous_map_of_unique_symm_apply | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map.continuous_lift_prod_left (hf : quotient_map f) {g : X × Y → Z}
(hg : continuous (λ p : X₀ × Y, g (f p.1, p.2))) : continuous g | begin
let Gf : C(X₀, C(Y, Z)) := continuous_map.curry ⟨_, hg⟩,
have h : ∀ x : X, continuous (λ y, g (x, y)),
{ intros x,
obtain ⟨x₀, rfl⟩ := hf.surjective x,
exact (Gf x₀).continuous },
let G : X → C(Y, Z) := λ x, ⟨_, h x⟩,
have : continuous G,
{ rw hf.continuous_iff,
exact Gf.continuous },
co... | lemma | quotient_map.continuous_lift_prod_left | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_map.continuous_uncurry_of_continuous",
"continuous_map.curry",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map.continuous_lift_prod_right (hf : quotient_map f) {g : Y × X → Z}
(hg : continuous (λ p : Y × X₀, g (p.1, f p.2))) : continuous g | begin
have : continuous (λ p : X₀ × Y, g ((prod.swap p).1, f (prod.swap p).2)),
{ exact hg.comp continuous_swap },
have : continuous (λ p : X₀ × Y, (g ∘ prod.swap) (f p.1, p.2)) := this,
convert (hf.continuous_lift_prod_left this).comp continuous_swap,
ext x,
simp,
end | lemma | quotient_map.continuous_lift_prod_right | topology | src/topology/compact_open.lean | [
"tactic.tidy",
"topology.continuous_function.basic",
"topology.homeomorph",
"topology.subset_properties",
"topology.maps"
] | [
"continuous",
"continuous_swap",
"prod.swap",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected (s : set α) : Prop | ∀ (u v : set α), is_open u → is_open v → s ⊆ u ∪ v →
(s ∩ u).nonempty → (s ∩ v).nonempty → (s ∩ (u ∩ v)).nonempty | def | is_preconnected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_open"
] | A preconnected set is one where there is no non-trivial open partition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected (s : set α) : Prop | s.nonempty ∧ is_preconnected s | def | is_connected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected"
] | A connected set is one that is nonempty and where there is no non-trivial open partition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.nonempty {s : set α} (h : is_connected s) :
s.nonempty | h.1 | lemma | is_connected.nonempty | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected.is_preconnected {s : set α} (h : is_connected s) :
is_preconnected s | h.2 | lemma | is_connected.is_preconnected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preirreducible.is_preconnected {s : set α} (H : is_preirreducible s) :
is_preconnected s | λ _ _ hu hv _, H _ _ hu hv | theorem | is_preirreducible.is_preconnected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible.is_connected {s : set α} (H : is_irreducible s) : is_connected s | ⟨H.nonempty, H.is_preirreducible.is_preconnected⟩ | theorem | is_irreducible.is_connected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_empty : is_preconnected (∅ : set α) | is_preirreducible_empty.is_preconnected | theorem | is_preconnected_empty | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_singleton {x} : is_connected ({x} : set α) | is_irreducible_singleton.is_connected | theorem | is_connected_singleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_singleton {x} : is_preconnected ({x} : set α) | is_connected_singleton.is_preconnected | theorem | is_preconnected_singleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.is_preconnected {s : set α} (hs : s.subsingleton) :
is_preconnected s | hs.induction_on is_preconnected_empty (λ x, is_preconnected_singleton) | theorem | set.subsingleton.is_preconnected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected_empty",
"is_preconnected_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_of_forall {s : set α} (x : α)
(H : ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ is_preconnected t) :
is_preconnected s | begin
rintros u v hu hv hs ⟨z, zs, zu⟩ ⟨y, ys, yv⟩,
have xs : x ∈ s, by { rcases H y ys with ⟨t, ts, xt, yt, ht⟩, exact ts xt },
wlog xu : x ∈ u,
{ rw inter_comm u v, rw union_comm at hs,
exact this x H v u hv hu hs y ys yv z zs zu xs ((hs xs).resolve_right xu), },
rcases H y ys with ⟨t, ts, xt, yt, ht⟩,
... | theorem | is_preconnected_of_forall | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected"
] | If any point of a set is joined to a fixed point by a preconnected subset,
then the original set is preconnected as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_of_forall_pair {s : set α}
(H : ∀ x y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ is_preconnected t) :
is_preconnected s | begin
rcases eq_empty_or_nonempty s with (rfl|⟨x, hx⟩),
exacts [is_preconnected_empty, is_preconnected_of_forall x $ λ y, H x hx y],
end | theorem | is_preconnected_of_forall_pair | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected_empty",
"is_preconnected_of_forall"
] | If any two points of a set are contained in a preconnected subset,
then the original set is preconnected as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_sUnion (x : α) (c : set (set α)) (H1 : ∀ s ∈ c, x ∈ s)
(H2 : ∀ s ∈ c, is_preconnected s) : is_preconnected (⋃₀ c) | begin
apply is_preconnected_of_forall x,
rintros y ⟨s, sc, ys⟩,
exact ⟨s, subset_sUnion_of_mem sc, H1 s sc, ys, H2 s sc⟩
end | theorem | is_preconnected_sUnion | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected_of_forall"
] | A union of a family of preconnected sets with a common point is preconnected as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_Union {ι : Sort*} {s : ι → set α} (h₁ : (⋂ i, s i).nonempty)
(h₂ : ∀ i, is_preconnected (s i)) :
is_preconnected (⋃ i, s i) | exists.elim h₁ $ λ f hf, is_preconnected_sUnion f _ hf (forall_range_iff.2 h₂) | theorem | is_preconnected_Union | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected_sUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.union (x : α) {s t : set α} (H1 : x ∈ s) (H2 : x ∈ t)
(H3 : is_preconnected s) (H4 : is_preconnected t) : is_preconnected (s ∪ t) | sUnion_pair s t ▸ is_preconnected_sUnion x {s, t}
(by rintro r (rfl | rfl | h); assumption)
(by rintro r (rfl | rfl | h); assumption) | theorem | is_preconnected.union | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected_sUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.union' {s t : set α} (H : (s ∩ t).nonempty)
(hs : is_preconnected s) (ht : is_preconnected t) : is_preconnected (s ∪ t) | by { rcases H with ⟨x, hxs, hxt⟩, exact hs.union x hxs hxt ht } | theorem | is_preconnected.union' | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected.union {s t : set α} (H : (s ∩ t).nonempty)
(Hs : is_connected s) (Ht : is_connected t) : is_connected (s ∪ t) | begin
rcases H with ⟨x, hx⟩,
refine ⟨⟨x, mem_union_left t (mem_of_mem_inter_left hx)⟩, _⟩,
exact is_preconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx)
Hs.is_preconnected Ht.is_preconnected
end | theorem | is_connected.union | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_preconnected.union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.sUnion_directed {S : set (set α)}
(K : directed_on (⊆) S)
(H : ∀ s ∈ S, is_preconnected s) : is_preconnected (⋃₀ S) | begin
rintros u v hu hv Huv ⟨a, ⟨s, hsS, has⟩, hau⟩ ⟨b, ⟨t, htS, hbt⟩, hbv⟩,
obtain ⟨r, hrS, hsr, htr⟩ : ∃ r ∈ S, s ⊆ r ∧ t ⊆ r := K s hsS t htS,
have Hnuv : (r ∩ (u ∩ v)).nonempty,
from H _ hrS u v hu hv ((subset_sUnion_of_mem hrS).trans Huv)
⟨a, hsr has, hau⟩ ⟨b, htr hbt, hbv⟩,
have Kruv : r ∩ (u ∩ v) ⊆... | theorem | is_preconnected.sUnion_directed | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"directed_on",
"is_preconnected"
] | The directed sUnion of a set S of preconnected subsets is preconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.bUnion_of_refl_trans_gen {ι : Type*} {t : set ι} {s : ι → set α}
(H : ∀ i ∈ t, is_preconnected (s i))
(K : ∀ i j ∈ t, refl_trans_gen (λ i j : ι, (s i ∩ s j).nonempty ∧ i ∈ t) i j) :
is_preconnected (⋃ n ∈ t, s n) | begin
let R := λ i j : ι, (s i ∩ s j).nonempty ∧ i ∈ t,
have P : ∀ (i j ∈ t), refl_trans_gen R i j →
∃ (p ⊆ t), i ∈ p ∧ j ∈ p ∧ is_preconnected (⋃ j ∈ p, s j),
{ intros i hi j hj h,
induction h,
case refl
{ refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, _⟩,
rw [b... | theorem | is_preconnected.bUnion_of_refl_trans_gen | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"ih",
"is_preconnected",
"is_preconnected_of_forall_pair"
] | The bUnion of a family of preconnected sets is preconnected if the graph determined by
whether two sets intersect is preconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.bUnion_of_refl_trans_gen {ι : Type*} {t : set ι} {s : ι → set α}
(ht : t.nonempty)
(H : ∀ i ∈ t, is_connected (s i))
(K : ∀ i j ∈ t, refl_trans_gen (λ i j : ι, (s i ∩ s j).nonempty ∧ i ∈ t) i j) :
is_connected (⋃ n ∈ t, s n) | ⟨nonempty_bUnion.2 $ ⟨ht.some, ht.some_mem, (H _ ht.some_mem).nonempty⟩,
is_preconnected.bUnion_of_refl_trans_gen (λ i hi, (H i hi).is_preconnected) K⟩ | theorem | is_connected.bUnion_of_refl_trans_gen | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_preconnected",
"is_preconnected.bUnion_of_refl_trans_gen"
] | The bUnion of a family of preconnected sets is preconnected if the graph determined by
whether two sets intersect is preconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.Union_of_refl_trans_gen {ι : Type*} {s : ι → set α}
(H : ∀ i, is_preconnected (s i))
(K : ∀ i j, refl_trans_gen (λ i j : ι, (s i ∩ s j).nonempty) i j) :
is_preconnected (⋃ n, s n) | by { rw [← bUnion_univ], exact is_preconnected.bUnion_of_refl_trans_gen (λ i _, H i)
(λ i _ j _, by simpa [mem_univ] using K i j) } | theorem | is_preconnected.Union_of_refl_trans_gen | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected.bUnion_of_refl_trans_gen"
] | Preconnectedness of the Union of a family of preconnected sets
indexed by the vertices of a preconnected graph,
where two vertices are joined when the corresponding sets intersect. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.Union_of_refl_trans_gen {ι : Type*} [nonempty ι] {s : ι → set α}
(H : ∀ i, is_connected (s i))
(K : ∀ i j, refl_trans_gen (λ i j : ι, (s i ∩ s j).nonempty) i j) :
is_connected (⋃ n, s n) | ⟨nonempty_Union.2 $ nonempty.elim ‹_› $ λ i : ι, ⟨i, (H _).nonempty⟩,
is_preconnected.Union_of_refl_trans_gen (λ i, (H i).is_preconnected) K⟩ | theorem | is_connected.Union_of_refl_trans_gen | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_preconnected",
"is_preconnected.Union_of_refl_trans_gen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.Union_of_chain {s : β → set α}
(H : ∀ n, is_preconnected (s n))
(K : ∀ n, (s n ∩ s (succ n)).nonempty) :
is_preconnected (⋃ n, s n) | is_preconnected.Union_of_refl_trans_gen H $
λ i j, refl_trans_gen_of_succ _ (λ i _, K i) $ λ i _, by { rw inter_comm, exact K i } | theorem | is_preconnected.Union_of_chain | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected.Union_of_refl_trans_gen",
"refl_trans_gen_of_succ"
] | The Union of connected sets indexed by a type with an archimedean successor (like `ℕ` or `ℤ`)
such that any two neighboring sets meet is preconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.Union_of_chain [nonempty β] {s : β → set α}
(H : ∀ n, is_connected (s n))
(K : ∀ n, (s n ∩ s (succ n)).nonempty) :
is_connected (⋃ n, s n) | is_connected.Union_of_refl_trans_gen H $
λ i j, refl_trans_gen_of_succ _ (λ i _, K i) $ λ i _, by { rw inter_comm, exact K i } | theorem | is_connected.Union_of_chain | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_connected.Union_of_refl_trans_gen",
"refl_trans_gen_of_succ"
] | The Union of connected sets indexed by a type with an archimedean successor (like `ℕ` or `ℤ`)
such that any two neighboring sets meet is connected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.bUnion_of_chain
{s : β → set α} {t : set β} (ht : ord_connected t)
(H : ∀ n ∈ t, is_preconnected (s n))
(K : ∀ n : β, n ∈ t → succ n ∈ t → (s n ∩ s (succ n)).nonempty) :
is_preconnected (⋃ n ∈ t, s n) | begin
have h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t :=
λ i j k hi hj hk, ht.out hi hj (Ico_subset_Icc_self hk),
have h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t := λ i j k hi hj hk,
ht.out hi hj ⟨hk.1.trans $ le_succ k, succ_le_of_lt hk.2⟩,
have h3 : ∀ {i j k : β}, i ∈ t → ... | theorem | is_preconnected.bUnion_of_chain | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected.bUnion_of_refl_trans_gen",
"refl_trans_gen_of_succ"
] | The Union of preconnected sets indexed by a subset of a type with an archimedean successor
(like `ℕ` or `ℤ`) such that any two neighboring sets meet is preconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.bUnion_of_chain
{s : β → set α} {t : set β} (hnt : t.nonempty) (ht : ord_connected t)
(H : ∀ n ∈ t, is_connected (s n))
(K : ∀ n : β, n ∈ t → succ n ∈ t → (s n ∩ s (succ n)).nonempty) :
is_connected (⋃ n ∈ t, s n) | ⟨nonempty_bUnion.2 $ ⟨hnt.some, hnt.some_mem, (H _ hnt.some_mem).nonempty⟩,
is_preconnected.bUnion_of_chain ht (λ i hi, (H i hi).is_preconnected) K⟩ | theorem | is_connected.bUnion_of_chain | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_preconnected",
"is_preconnected.bUnion_of_chain"
] | The Union of connected sets indexed by a subset of a type with an archimedean successor
(like `ℕ` or `ℤ`) such that any two neighboring sets meet is preconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.subset_closure {s : set α} {t : set α}
(H : is_preconnected s) (Kst : s ⊆ t) (Ktcs : t ⊆ closure s) :
is_preconnected t | λ u v hu hv htuv ⟨y, hyt, hyu⟩ ⟨z, hzt, hzv⟩,
let ⟨p, hpu, hps⟩ := mem_closure_iff.1 (Ktcs hyt) u hu hyu,
⟨q, hqv, hqs⟩ := mem_closure_iff.1 (Ktcs hzt) v hv hzv,
⟨r, hrs, hruv⟩ := H u v hu hv (subset.trans Kst htuv) ⟨p, hps, hpu⟩ ⟨q, hqs, hqv⟩ in
⟨r, Kst hrs, hruv⟩ | theorem | is_preconnected.subset_closure | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"closure",
"is_preconnected"
] | Theorem of bark and tree :
if a set is within a (pre)connected set and its closure,
then it is (pre)connected as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.subset_closure {s : set α} {t : set α}
(H : is_connected s) (Kst : s ⊆ t) (Ktcs : t ⊆ closure s): is_connected t | let hsne := H.left,
ht := Kst,
htne := nonempty.mono ht hsne in
⟨nonempty.mono Kst H.left, is_preconnected.subset_closure H.right Kst Ktcs ⟩ | theorem | is_connected.subset_closure | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"closure",
"is_connected",
"is_preconnected.subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.closure {s : set α} (H : is_preconnected s) :
is_preconnected (closure s) | is_preconnected.subset_closure H subset_closure $ subset.refl $ closure s | theorem | is_preconnected.closure | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"closure",
"is_preconnected",
"is_preconnected.subset_closure",
"subset_closure"
] | The closure of a (pre)connected set is (pre)connected as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.closure {s : set α} (H : is_connected s) :
is_connected (closure s) | is_connected.subset_closure H subset_closure $ subset.refl $ closure s | theorem | is_connected.closure | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"closure",
"is_connected",
"is_connected.subset_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.image [topological_space β] {s : set α} (H : is_preconnected s)
(f : α → β) (hf : continuous_on f s) : is_preconnected (f '' s) | begin
-- Unfold/destruct definitions in hypotheses
rintros u v hu hv huv ⟨_, ⟨x, xs, rfl⟩, xu⟩ ⟨_, ⟨y, ys, rfl⟩, yv⟩,
rcases continuous_on_iff'.1 hf u hu with ⟨u', hu', u'_eq⟩,
rcases continuous_on_iff'.1 hf v hv with ⟨v', hv', v'_eq⟩,
-- Reformulate `huv : f '' s ⊆ u ∪ v` in terms of `u'` and `v'`
replace ... | theorem | is_preconnected.image | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous_on",
"is_preconnected",
"topological_space"
] | The image of a (pre)connected set is (pre)connected as well. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected.image [topological_space β] {s : set α} (H : is_connected s)
(f : α → β) (hf : continuous_on f s) : is_connected (f '' s) | ⟨nonempty_image_iff.mpr H.nonempty, H.is_preconnected.image f hf⟩ | theorem | is_connected.image | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous_on",
"is_connected",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_closed_iff {s : set α} :
is_preconnected s ↔ ∀ t t', is_closed t → is_closed t' → s ⊆ t ∪ t' →
(s ∩ t).nonempty → (s ∩ t').nonempty → (s ∩ (t ∩ t')).nonempty | ⟨begin
rintros h t t' ht ht' htt' ⟨x, xs, xt⟩ ⟨y, ys, yt'⟩,
rw [←not_disjoint_iff_nonempty_inter, ←subset_compl_iff_disjoint_right, compl_inter],
intros h',
have xt' : x ∉ t', from (h' xs).resolve_left (absurd xt),
have yt : y ∉ t, from (h' ys).resolve_right (absurd yt'),
have := h _ _ ht.is_open_compl ht'.... | theorem | is_preconnected_closed_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_closed",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.is_preconnected_image [topological_space β] {s : set α} {f : α → β}
(hf : inducing f) : is_preconnected (f '' s) ↔ is_preconnected s | begin
refine ⟨λ h, _, λ h, h.image _ hf.continuous.continuous_on⟩,
rintro u v hu' hv' huv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩,
rcases hf.is_open_iff.1 hu' with ⟨u, hu, rfl⟩,
rcases hf.is_open_iff.1 hv' with ⟨v, hv, rfl⟩,
replace huv : f '' s ⊆ u ∪ v, by rwa image_subset_iff,
rcases h u v hu hv huv ⟨f x, mem_image_o... | lemma | inducing.is_preconnected_image | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"inducing",
"is_preconnected",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.preimage_of_open_map [topological_space β] {s : set β}
(hs : is_preconnected s) {f : α → β} (hinj : function.injective f) (hf : is_open_map f)
(hsf : s ⊆ range f) :
is_preconnected (f ⁻¹' s) | λ u v hu hv hsuv hsu hsv,
begin
obtain ⟨b, hbs, hbu, hbv⟩ := hs (f '' u) (f '' v) (hf u hu) (hf v hv) _ _ _,
obtain ⟨a, rfl⟩ := hsf hbs,
rw hinj.mem_set_image at hbu hbv,
exact ⟨a, hbs, hbu, hbv⟩,
{ have := image_subset f hsuv,
rwa [set.image_preimage_eq_of_subset hsf, image_union] at this },
{ obtain ⟨... | lemma | is_preconnected.preimage_of_open_map | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_open_map",
"is_preconnected",
"set.image_preimage_eq_of_subset",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.preimage_of_closed_map [topological_space β] {s : set β}
(hs : is_preconnected s) {f : α → β} (hinj : function.injective f) (hf : is_closed_map f)
(hsf : s ⊆ range f) :
is_preconnected (f ⁻¹' s) | is_preconnected_closed_iff.2 $ λ u v hu hv hsuv hsu hsv,
begin
obtain ⟨b, hbs, hbu, hbv⟩ :=
is_preconnected_closed_iff.1 hs (f '' u) (f '' v) (hf u hu) (hf v hv) _ _ _,
obtain ⟨a, rfl⟩ := hsf hbs,
rw hinj.mem_set_image at hbu hbv,
exact ⟨a, hbs, hbu, hbv⟩,
{ have := image_subset f hsuv,
rwa [set.image... | lemma | is_preconnected.preimage_of_closed_map | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_closed_map",
"is_preconnected",
"set.image_preimage_eq_of_subset",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected.preimage_of_open_map [topological_space β] {s : set β} (hs : is_connected s)
{f : α → β} (hinj : function.injective f) (hf : is_open_map f) (hsf : s ⊆ range f) :
is_connected (f ⁻¹' s) | ⟨hs.nonempty.preimage' hsf, hs.is_preconnected.preimage_of_open_map hinj hf hsf⟩ | lemma | is_connected.preimage_of_open_map | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_open_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected.preimage_of_closed_map [topological_space β] {s : set β} (hs : is_connected s)
{f : α → β} (hinj : function.injective f) (hf : is_closed_map f) (hsf : s ⊆ range f) :
is_connected (f ⁻¹' s) | ⟨hs.nonempty.preimage' hsf, hs.is_preconnected.preimage_of_closed_map hinj hf hsf⟩ | lemma | is_connected.preimage_of_closed_map | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_closed_map",
"is_connected",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.subset_or_subset (hu : is_open u) (hv : is_open v) (huv : disjoint u v)
(hsuv : s ⊆ u ∪ v) (hs : is_preconnected s) :
s ⊆ u ∨ s ⊆ v | begin
specialize hs u v hu hv hsuv,
obtain hsu | hsu := (s ∩ u).eq_empty_or_nonempty,
{ exact or.inr ((set.disjoint_iff_inter_eq_empty.2 hsu).subset_right_of_subset_union hsuv) },
{ replace hs := mt (hs hsu),
simp_rw [set.not_nonempty_iff_eq_empty, ←set.disjoint_iff_inter_eq_empty,
disjoint_iff_inter_... | lemma | is_preconnected.subset_or_subset | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"disjoint",
"is_open",
"is_preconnected",
"set.not_nonempty_iff_eq_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.subset_left_of_subset_union (hu : is_open u) (hv : is_open v)
(huv : disjoint u v) (hsuv : s ⊆ u ∪ v) (hsu : (s ∩ u).nonempty) (hs : is_preconnected s) :
s ⊆ u | disjoint.subset_left_of_subset_union hsuv
begin
by_contra hsv,
rw not_disjoint_iff_nonempty_inter at hsv,
obtain ⟨x, _, hx⟩ := hs u v hu hv hsuv hsu hsv,
exact set.disjoint_iff.1 huv hx,
end | lemma | is_preconnected.subset_left_of_subset_union | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"by_contra",
"disjoint",
"disjoint.subset_left_of_subset_union",
"is_open",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.subset_right_of_subset_union (hu : is_open u) (hv : is_open v)
(huv : disjoint u v) (hsuv : s ⊆ u ∪ v) (hsv : (s ∩ v).nonempty) (hs : is_preconnected s) :
s ⊆ v | hs.subset_left_of_subset_union hv hu huv.symm (union_comm u v ▸ hsuv) hsv | lemma | is_preconnected.subset_right_of_subset_union | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"disjoint",
"is_open",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.subset_of_closure_inter_subset (hs : is_preconnected s)
(hu : is_open u) (h'u : (s ∩ u).nonempty) (h : closure u ∩ s ⊆ u) : s ⊆ u | begin
have A : s ⊆ u ∪ (closure u)ᶜ,
{ assume x hx,
by_cases xu : x ∈ u,
{ exact or.inl xu },
{ right,
assume h'x,
exact xu (h (mem_inter h'x hx)) } },
apply hs.subset_left_of_subset_union hu is_closed_closure.is_open_compl _ A h'u,
exact disjoint_compl_right.mono_right (compl_subset_com... | lemma | is_preconnected.subset_of_closure_inter_subset | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"closure",
"is_open",
"is_preconnected",
"subset_closure"
] | If a preconnected set `s` intersects an open set `u`, and limit points of `u` inside `s` are
contained in `u`, then the whole set `s` is contained in `u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.prod [topological_space β] {s : set α} {t : set β}
(hs : is_preconnected s) (ht : is_preconnected t) :
is_preconnected (s ×ˢ t) | begin
apply is_preconnected_of_forall_pair,
rintro ⟨a₁, b₁⟩ ⟨ha₁, hb₁⟩ ⟨a₂, b₂⟩ ⟨ha₂, hb₂⟩,
refine ⟨prod.mk a₁ '' t ∪ flip prod.mk b₂ '' s, _,
or.inl ⟨b₁, hb₁, rfl⟩, or.inr ⟨a₂, ha₂, rfl⟩, _⟩,
{ rintro _ (⟨y, hy, rfl⟩|⟨x, hx, rfl⟩),
exacts [⟨ha₁, hy⟩, ⟨hx, hb₂⟩] },
{ exact (ht.image _ (continuous.prod... | theorem | is_preconnected.prod | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous.prod.mk",
"continuous_const",
"continuous_on",
"is_preconnected",
"is_preconnected_of_forall_pair",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected.prod [topological_space β] {s : set α} {t : set β}
(hs : is_connected s) (ht : is_connected t) : is_connected (s ×ˢ t) | ⟨hs.1.prod ht.1, hs.2.prod ht.2⟩ | theorem | is_connected.prod | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_univ_pi [Π i, topological_space (π i)] {s : Π i, set (π i)}
(hs : ∀ i, is_preconnected (s i)) :
is_preconnected (pi univ s) | begin
rintros u v uo vo hsuv ⟨f, hfs, hfu⟩ ⟨g, hgs, hgv⟩,
rcases exists_finset_piecewise_mem_of_mem_nhds (uo.mem_nhds hfu) g with ⟨I, hI⟩,
induction I using finset.induction_on with i I hi ihI,
{ refine ⟨g, hgs, ⟨_, hgv⟩⟩, simpa using hI },
{ rw [finset.piecewise_insert] at hI,
have := I.piecewise_mem_set... | theorem | is_preconnected_univ_pi | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous_id",
"continuous_on",
"exists_finset_piecewise_mem_of_mem_nhds",
"finset.induction_on",
"finset.piecewise_insert",
"is_preconnected",
"topological_space",
"update",
"update_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_univ_pi [Π i, topological_space (π i)] {s : Π i, set (π i)} :
is_connected (pi univ s) ↔ ∀ i, is_connected (s i) | begin
simp only [is_connected, ← univ_pi_nonempty_iff, forall_and_distrib, and.congr_right_iff],
refine λ hne, ⟨λ hc i, _, is_preconnected_univ_pi⟩,
rw [← eval_image_univ_pi hne],
exact hc.image _ (continuous_apply _).continuous_on
end | theorem | is_connected_univ_pi | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"and.congr_right_iff",
"continuous_apply",
"continuous_on",
"forall_and_distrib",
"is_connected",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma.is_connected_iff [Π i, topological_space (π i)] {s : set (Σ i, π i)} :
is_connected s ↔ ∃ i t, is_connected t ∧ s = sigma.mk i '' t | begin
refine ⟨λ hs, _, _⟩,
{ obtain ⟨⟨i, x⟩, hx⟩ := hs.nonempty,
have : s ⊆ range (sigma.mk i),
{ have h : range (sigma.mk i) = sigma.fst ⁻¹' {i}, by { ext, simp },
rw h,
exact is_preconnected.subset_left_of_subset_union
(is_open_sigma_fst_preimage _) (is_open_sigma_fst_preimage {x | x ≠... | lemma | sigma.is_connected_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_open_map_sigma_mk",
"is_open_sigma_fst_preimage",
"is_preconnected.subset_left_of_subset_union",
"set.image_preimage_eq_of_subset",
"sigma_mk_injective",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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