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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|---|---|---|---|---|---|---|---|---|---|---|
function.locally_compact_space_of_finite [finite ι] [locally_compact_space β] :
locally_compact_space (ι → β) | pi.locally_compact_space_of_finite | instance | function.locally_compact_space_of_finite | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"finite",
"locally_compact_space",
"pi.locally_compact_space_of_finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.locally_compact_space [locally_compact_space β] [compact_space β] :
locally_compact_space (ι → β) | pi.locally_compact_space | instance | function.locally_compact_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compact_space",
"locally_compact_space",
"pi.locally_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_compact_subset [locally_compact_space α] {x : α} {U : set α}
(hU : is_open U) (hx : x ∈ U) : ∃ (K : set α), is_compact K ∧ x ∈ interior K ∧ K ⊆ U | begin
rcases locally_compact_space.local_compact_nhds x U (hU.mem_nhds hx) with ⟨K, h1K, h2K, h3K⟩,
exact ⟨K, h3K, mem_interior_iff_mem_nhds.2 h1K, h2K⟩,
end | lemma | exists_compact_subset | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"interior",
"is_compact",
"is_open",
"locally_compact_space"
] | A reformulation of the definition of locally compact space: In a locally compact space,
every open set containing `x` has a compact subset containing `x` in its interior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_compact_mem_nhds [locally_compact_space α] (x : α) :
∃ K, is_compact K ∧ K ∈ 𝓝 x | let ⟨K, hKc, hx, H⟩ := exists_compact_subset is_open_univ (mem_univ x)
in ⟨K, hKc, mem_interior_iff_mem_nhds.1 hx⟩ | lemma | exists_compact_mem_nhds | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"exists_compact_subset",
"is_compact",
"is_open_univ",
"locally_compact_space"
] | In a locally compact space every point has a compact neighborhood. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_compact_between [hα : locally_compact_space α] {K U : set α} (hK : is_compact K)
(hU : is_open U) (h_KU : K ⊆ U) : ∃ L, is_compact L ∧ K ⊆ interior L ∧ L ⊆ U | begin
choose V hVc hxV hKV using λ x : K, exists_compact_subset hU (h_KU x.2),
have : K ⊆ ⋃ x, interior (V x), from λ x hx, mem_Union.2 ⟨⟨x, hx⟩, hxV _⟩,
rcases hK.elim_finite_subcover _ (λ x, @is_open_interior α _ (V x)) this with ⟨t, ht⟩,
refine ⟨_, t.is_compact_bUnion (λ x _, hVc x), λ x hx, _, set.Union₂_su... | lemma | exists_compact_between | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"exists_compact_subset",
"interior",
"interior_mono",
"is_compact",
"is_open",
"is_open_interior",
"locally_compact_space",
"set.Union₂_subset"
] | In a locally compact space, for every containement `K ⊆ U` of a compact set `K` in an open
set `U`, there is a compact neighborhood `L` such that `K ⊆ L ⊆ U`: equivalently, there is a
compact `L` such that `K ⊆ interior L` and `L ⊆ U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_compact_superset [locally_compact_space α] {K : set α} (hK : is_compact K) :
∃ K', is_compact K' ∧ K ⊆ interior K' | let ⟨L, hLc, hKL, _⟩ := exists_compact_between hK is_open_univ K.subset_univ in ⟨L, hLc, hKL⟩ | lemma | exists_compact_superset | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"exists_compact_between",
"interior",
"is_compact",
"is_open_univ",
"locally_compact_space"
] | In a locally compact space, every compact set is contained in the interior of a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding.locally_compact_space [locally_compact_space β] {f : α → β}
(hf : closed_embedding f) : locally_compact_space α | begin
have : ∀ x : α, (𝓝 x).has_basis (λ s, s ∈ 𝓝 (f x) ∧ is_compact s) (λ s, f ⁻¹' s),
{ intro x,
rw hf.to_embedding.to_inducing.nhds_eq_comap,
exact (compact_basis_nhds _).comap _ },
exact locally_compact_space_of_has_basis this (λ x s hs, hf.is_compact_preimage hs.2)
end | lemma | closed_embedding.locally_compact_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"closed_embedding",
"compact_basis_nhds",
"is_compact",
"locally_compact_space",
"locally_compact_space_of_has_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.locally_compact_space [locally_compact_space α] {s : set α}
(hs : is_closed s) : locally_compact_space s | (closed_embedding_subtype_coe hs).locally_compact_space | lemma | is_closed.locally_compact_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"closed_embedding_subtype_coe",
"is_closed",
"locally_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.locally_compact_space [locally_compact_space β] {f : α → β}
(hf : open_embedding f) : locally_compact_space α | begin
have : ∀ x : α, (𝓝 x).has_basis (λ s, (s ∈ 𝓝 (f x) ∧ is_compact s) ∧ s ⊆ range f) (λ s, f ⁻¹' s),
{ intro x,
rw hf.to_embedding.to_inducing.nhds_eq_comap,
exact ((compact_basis_nhds _).restrict_subset $
hf.open_range.mem_nhds $ mem_range_self _).comap _ },
refine locally_compact_space_of_has... | lemma | open_embedding.locally_compact_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compact_basis_nhds",
"is_compact",
"locally_compact_space",
"locally_compact_space_of_has_basis",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.locally_compact_space [locally_compact_space α] {s : set α}
(hs : is_open s) : locally_compact_space s | hs.open_embedding_subtype_coe.locally_compact_space | lemma | is_open.locally_compact_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_open",
"locally_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter.le_nhds_Lim [compact_space α] (F : ultrafilter α) :
↑F ≤ 𝓝 (@Lim _ _ (F : filter α).nonempty_of_ne_bot F) | begin
rcases is_compact_univ.ultrafilter_le_nhds F (by simp) with ⟨x, -, h⟩,
exact le_nhds_Lim ⟨x,h⟩,
end | lemma | ultrafilter.le_nhds_Lim | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"Lim",
"compact_space",
"filter",
"le_nhds_Lim",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.exists_minimal_nonempty_closed_subset [compact_space α]
{S : set α} (hS : is_closed S) (hne : S.nonempty) :
∃ (V : set α),
V ⊆ S ∧ V.nonempty ∧ is_closed V ∧
(∀ (V' : set α), V' ⊆ V → V'.nonempty → is_closed V' → V' = V) | begin
let opens := {U : set α | Sᶜ ⊆ U ∧ is_open U ∧ Uᶜ.nonempty},
obtain ⟨U, ⟨Uc, Uo, Ucne⟩, h⟩ := zorn_subset opens (λ c hc hz, begin
by_cases hcne : c.nonempty,
{ obtain ⟨U₀, hU₀⟩ := hcne,
haveI : nonempty {U // U ∈ c} := ⟨⟨U₀, hU₀⟩⟩,
obtain ⟨U₀compl, U₀opn, U₀ne⟩ := hc hU₀,
use ⋃₀ c,
... | theorem | is_closed.exists_minimal_nonempty_closed_subset | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compact_space",
"compl_compl",
"exists_prop",
"is_closed",
"is_compact",
"is_compact.nonempty_Inter_of_directed_nonempty_compact_closed",
"is_open",
"is_open_sUnion",
"not_and",
"not_exists",
"set.mem_Inter",
"set.subset.trans",
"zorn_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_compact_space (α : Type*) [topological_space α] : Prop | (exists_compact_covering : ∃ K : ℕ → set α, (∀ n, is_compact (K n)) ∧ (⋃ n, K n) = univ) | class | sigma_compact_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_compact",
"topological_space"
] | A σ-compact space is a space that is the union of a countable collection of compact subspaces.
Note that a locally compact separable T₂ space need not be σ-compact.
The sequence can be extracted using `topological_space.compact_covering`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space.sigma_compact [compact_space α] : sigma_compact_space α | ⟨⟨λ _, univ, λ _, is_compact_univ, Union_const _⟩⟩ | instance | compact_space.sigma_compact | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compact_space",
"is_compact_univ",
"sigma_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_compact_space.of_countable (S : set (set α)) (Hc : S.countable)
(Hcomp : ∀ s ∈ S, is_compact s) (HU : ⋃₀ S = univ) : sigma_compact_space α | ⟨(exists_seq_cover_iff_countable ⟨_, is_compact_empty⟩).2 ⟨S, Hc, Hcomp, HU⟩⟩ | lemma | sigma_compact_space.of_countable | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_compact",
"sigma_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_compact_space_of_locally_compact_second_countable [locally_compact_space α]
[second_countable_topology α] : sigma_compact_space α | begin
choose K hKc hxK using λ x : α, exists_compact_mem_nhds x,
rcases countable_cover_nhds hxK with ⟨s, hsc, hsU⟩,
refine sigma_compact_space.of_countable _ (hsc.image K) (ball_image_iff.2 $ λ x _, hKc x) _,
rwa sUnion_image
end | instance | sigma_compact_space_of_locally_compact_second_countable | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"exists_compact_mem_nhds",
"locally_compact_space",
"sigma_compact_space",
"sigma_compact_space.of_countable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_covering : ℕ → set α | accumulate exists_compact_covering.some | def | compact_covering | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [] | A choice of compact covering for a `σ`-compact space, chosen to be monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_compact_covering (n : ℕ) : is_compact (compact_covering α n) | is_compact_accumulate (classical.some_spec sigma_compact_space.exists_compact_covering).1 n | lemma | is_compact_compact_covering | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compact_covering",
"is_compact",
"is_compact_accumulate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_compact_covering : (⋃ n, compact_covering α n) = univ | begin
rw [compact_covering, Union_accumulate],
exact (classical.some_spec sigma_compact_space.exists_compact_covering).2
end | lemma | Union_compact_covering | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compact_covering"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_covering_subset ⦃m n : ℕ⦄ (h : m ≤ n) :
compact_covering α m ⊆ compact_covering α n | monotone_accumulate h | lemma | compact_covering_subset | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compact_covering"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem_compact_covering (x : α) : ∃ n, x ∈ compact_covering α n | Union_eq_univ_iff.mp (Union_compact_covering α) x | lemma | exists_mem_compact_covering | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"Union_compact_covering",
"compact_covering"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.sigma_compact_space {e : β → α} (he : closed_embedding e) :
sigma_compact_space β | ⟨⟨λ n, e ⁻¹' compact_covering α n, λ n, he.is_compact_preimage (is_compact_compact_covering _ _),
by rw [← preimage_Union, Union_compact_covering, preimage_univ]⟩⟩ | theorem | closed_embedding.sigma_compact_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"Union_compact_covering",
"closed_embedding",
"compact_covering",
"is_compact_compact_covering",
"sigma_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite.countable_univ {ι : Type*} {f : ι → set α} (hf : locally_finite f)
(hne : ∀ i, (f i).nonempty) :
(univ : set ι).countable | begin
have := λ n, hf.finite_nonempty_inter_compact (is_compact_compact_covering α n),
refine (countable_Union (λ n, (this n).countable)).mono (λ i hi, _),
rcases hne i with ⟨x, hx⟩,
rcases Union_eq_univ_iff.1 (Union_compact_covering α) x with ⟨n, hn⟩,
exact mem_Union.2 ⟨n, x, hx, hn⟩
end | lemma | locally_finite.countable_univ | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"Union_compact_covering",
"countable",
"is_compact_compact_covering",
"locally_finite"
] | If `α` is a `σ`-compact space, then a locally finite family of nonempty sets of `α` can have
only countably many elements, `set.countable` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_finite.encodable {ι : Type*} {f : ι → set α}
(hf : locally_finite f) (hne : ∀ i, (f i).nonempty) : encodable ι | @encodable.of_equiv _ _ (hf.countable_univ hne).to_encodable (equiv.set.univ _).symm | def | locally_finite.encodable | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"encodable",
"encodable.of_equiv",
"equiv.set.univ",
"locally_finite"
] | If `f : ι → set α` is a locally finite covering of a σ-compact topological space by nonempty
sets, then the index type `ι` is encodable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_cover_nhds_within_of_sigma_compact {f : α → set α} {s : set α} (hs : is_closed s)
(hf : ∀ x ∈ s, f x ∈ 𝓝[s] x) : ∃ t ⊆ s, t.countable ∧ s ⊆ ⋃ x ∈ t, f x | begin
simp only [nhds_within, mem_inf_principal] at hf,
choose t ht hsub using λ n, ((is_compact_compact_covering α n).inter_right hs).elim_nhds_subcover
_ (λ x hx, hf x hx.right),
refine ⟨⋃ n, (t n : set α), Union_subset $ λ n x hx, (ht n x hx).2,
countable_Union $ λ n, (t n).countable_to_set, λ x hx, me... | lemma | countable_cover_nhds_within_of_sigma_compact | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"exists_mem_compact_covering",
"is_closed",
"is_compact_compact_covering",
"nhds_within"
] | In a topological space with sigma compact topology, if `f` is a function that sends each point
`x` of a closed set `s` to a neighborhood of `x` within `s`, then for some countable set `t ⊆ s`,
the neighborhoods `f x`, `x ∈ t`, cover the whole set `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
countable_cover_nhds_of_sigma_compact {f : α → set α}
(hf : ∀ x, f x ∈ 𝓝 x) : ∃ s : set α, s.countable ∧ (⋃ x ∈ s, f x) = univ | begin
simp only [← nhds_within_univ] at hf,
rcases countable_cover_nhds_within_of_sigma_compact is_closed_univ (λ x _, hf x)
with ⟨s, -, hsc, hsU⟩,
exact ⟨s, hsc, univ_subset_iff.1 hsU⟩
end | lemma | countable_cover_nhds_of_sigma_compact | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"countable_cover_nhds_within_of_sigma_compact",
"is_closed_univ",
"nhds_within_univ"
] | In a topological space with sigma compact topology, if `f` is a function that sends each
point `x` to a neighborhood of `x`, then for some countable set `s`, the neighborhoods `f x`,
`x ∈ s`, cover the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_exhaustion (X : Type*) [topological_space X] | (to_fun : ℕ → set X)
(is_compact' : ∀ n, is_compact (to_fun n))
(subset_interior_succ' : ∀ n, to_fun n ⊆ interior (to_fun (n + 1)))
(Union_eq' : (⋃ n, to_fun n) = univ) | structure | compact_exhaustion | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"interior",
"is_compact",
"topological_space"
] | An [exhaustion by compact sets](https://en.wikipedia.org/wiki/Exhaustion_by_compact_sets) of a
topological space is a sequence of compact sets `K n` such that `K n ⊆ interior (K (n + 1))` and
`(⋃ n, K n) = univ`.
If `X` is a locally compact sigma compact space, then `compact_exhaustion.choice X` provides
a choice of a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact (n : ℕ) : is_compact (K n) | K.is_compact' n | lemma | compact_exhaustion.is_compact | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_succ (n : ℕ) : K n ⊆ interior (K (n + 1)) | K.subset_interior_succ' n | lemma | compact_exhaustion.subset_interior_succ | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_succ (n : ℕ) : K n ⊆ K (n + 1) | subset.trans (K.subset_interior_succ n) interior_subset | lemma | compact_exhaustion.subset_succ | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"interior_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset ⦃m n : ℕ⦄ (h : m ≤ n) : K m ⊆ K n | show K m ≤ K n, from monotone_nat_of_le_succ K.subset_succ h | lemma | compact_exhaustion.subset | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"monotone_nat_of_le_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior ⦃m n : ℕ⦄ (h : m < n) : K m ⊆ interior (K n) | subset.trans (K.subset_interior_succ m) $ interior_mono $ K.subset h | lemma | compact_exhaustion.subset_interior | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"interior",
"interior_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Union_eq : (⋃ n, K n) = univ | K.Union_eq' | lemma | compact_exhaustion.Union_eq | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_mem (x : α) : ∃ n, x ∈ K n | Union_eq_univ_iff.1 K.Union_eq x | lemma | compact_exhaustion.exists_mem | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
find (x : α) : ℕ | nat.find (K.exists_mem x) | def | compact_exhaustion.find | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [] | The minimal `n` such that `x ∈ K n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_find (x : α) : x ∈ K (K.find x) | nat.find_spec (K.exists_mem x) | lemma | compact_exhaustion.mem_find | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_iff_find_le {x : α} {n : ℕ} : x ∈ K n ↔ K.find x ≤ n | ⟨λ h, nat.find_min' (K.exists_mem x) h, λ h, K.subset h $ K.mem_find x⟩ | lemma | compact_exhaustion.mem_iff_find_le | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
shiftr : compact_exhaustion α | { to_fun := λ n, nat.cases_on n ∅ K,
is_compact' := λ n, nat.cases_on n is_compact_empty K.is_compact,
subset_interior_succ' := λ n, nat.cases_on n (empty_subset _) K.subset_interior_succ,
Union_eq' := Union_eq_univ_iff.2 $ λ x, ⟨K.find x + 1, K.mem_find x⟩ } | def | compact_exhaustion.shiftr | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compact_exhaustion",
"is_compact_empty"
] | Prepend the empty set to a compact exhaustion `K n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
find_shiftr (x : α) : K.shiftr.find x = K.find x + 1 | nat.find_comp_succ _ _ (not_mem_empty _) | lemma | compact_exhaustion.find_shiftr | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"nat.find_comp_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_diff_shiftr_find (x : α) : x ∈ K.shiftr (K.find x + 1) \ K.shiftr (K.find x) | ⟨K.mem_find _, mt K.shiftr.mem_iff_find_le.1 $
by simp only [find_shiftr, not_le, nat.lt_succ_self]⟩ | lemma | compact_exhaustion.mem_diff_shiftr_find | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
choice (X : Type*) [topological_space X] [locally_compact_space X]
[sigma_compact_space X] : compact_exhaustion X | begin
apply classical.choice,
let K : ℕ → {s : set X // is_compact s} :=
λ n, nat.rec_on n ⟨∅, is_compact_empty⟩
(λ n s, ⟨(exists_compact_superset s.2).some ∪ compact_covering X n,
(exists_compact_superset s.2).some_spec.1.union (is_compact_compact_covering _ _)⟩),
refine ⟨⟨λ n, K n, λ n, (K n).... | def | compact_exhaustion.choice | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"Union_compact_covering",
"compact_covering",
"compact_exhaustion",
"exists_compact_superset",
"interior_mono",
"is_compact",
"is_compact_compact_covering",
"locally_compact_space",
"sigma_compact_space",
"topological_space"
] | A choice of an
[exhaustion by compact sets](https://en.wikipedia.org/wiki/Exhaustion_by_compact_sets)
of a locally compact sigma compact space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_clopen (s : set α) : Prop | is_open s ∧ is_closed s | def | is_clopen | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_closed",
"is_open"
] | A set is clopen if it is both open and closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_clopen.is_open (hs : is_clopen s) : is_open s | hs.1 | lemma | is_clopen.is_open | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.is_closed (hs : is_clopen s) : is_closed s | hs.2 | lemma | is_clopen.is_closed | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_iff_frontier_eq_empty {s : set α} : is_clopen s ↔ frontier s = ∅ | begin
rw [is_clopen, ← closure_eq_iff_is_closed, ← interior_eq_iff_is_open, frontier, diff_eq_empty],
refine ⟨λ h, (h.2.trans h.1.symm).subset, λ h, _⟩,
exact ⟨interior_subset.antisymm (subset_closure.trans h),
(h.trans interior_subset).antisymm subset_closure⟩
end | lemma | is_clopen_iff_frontier_eq_empty | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"closure_eq_iff_is_closed",
"frontier",
"interior_eq_iff_is_open",
"interior_subset",
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.union {s t : set α} (hs : is_clopen s) (ht : is_clopen t) : is_clopen (s ∪ t) | ⟨hs.1.union ht.1, hs.2.union ht.2⟩ | theorem | is_clopen.union | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.inter {s t : set α} (hs : is_clopen s) (ht : is_clopen t) : is_clopen (s ∩ t) | ⟨hs.1.inter ht.1, hs.2.inter ht.2⟩ | theorem | is_clopen.inter | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_empty : is_clopen (∅ : set α) | ⟨is_open_empty, is_closed_empty⟩ | theorem | is_clopen_empty | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_univ : is_clopen (univ : set α) | ⟨is_open_univ, is_closed_univ⟩ | theorem | is_clopen_univ | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.compl {s : set α} (hs : is_clopen s) : is_clopen sᶜ | ⟨hs.2.is_open_compl, hs.1.is_closed_compl⟩ | theorem | is_clopen.compl | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_compl_iff {s : set α} : is_clopen sᶜ ↔ is_clopen s | ⟨λ h, compl_compl s ▸ is_clopen.compl h, is_clopen.compl⟩ | theorem | is_clopen_compl_iff | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"compl_compl",
"is_clopen",
"is_clopen.compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.diff {s t : set α} (hs : is_clopen s) (ht : is_clopen t) : is_clopen (s \ t) | hs.inter ht.compl | theorem | is_clopen.diff | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.prod {s : set α} {t : set β} (hs : is_clopen s) (ht : is_clopen t) :
is_clopen (s ×ˢ t) | ⟨hs.1.prod ht.1, hs.2.prod ht.2⟩ | lemma | is_clopen.prod | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_Union {β : Type*} [finite β] {s : β → set α} (h : ∀ i, is_clopen (s i)) :
is_clopen (⋃ i, s i) | ⟨is_open_Union (forall_and_distrib.1 h).1, is_closed_Union (forall_and_distrib.1 h).2⟩ | lemma | is_clopen_Union | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"finite",
"is_clopen",
"is_closed_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_bUnion {β : Type*} {s : set β} {f : β → set α} (hs : s.finite)
(h : ∀ i ∈ s, is_clopen $ f i) :
is_clopen (⋃ i ∈ s, f i) | ⟨is_open_bUnion (λ i hi, (h i hi).1), is_closed_bUnion hs (λ i hi, (h i hi).2)⟩ | lemma | is_clopen_bUnion | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen",
"is_closed_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_bUnion_finset {β : Type*} {s : finset β} {f : β → set α}
(h : ∀ i ∈ s, is_clopen $ f i) :
is_clopen (⋃ i ∈ s, f i) | is_clopen_bUnion s.finite_to_set h | lemma | is_clopen_bUnion_finset | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"finset",
"is_clopen",
"is_clopen_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_Inter {β : Type*} [finite β] {s : β → set α} (h : ∀ i, is_clopen (s i)) :
is_clopen (⋂ i, s i) | ⟨(is_open_Inter (forall_and_distrib.1 h).1), (is_closed_Inter (forall_and_distrib.1 h).2)⟩ | lemma | is_clopen_Inter | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"finite",
"is_clopen",
"is_closed_Inter",
"is_open_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_bInter {β : Type*} {s : set β} (hs : s.finite) {f : β → set α}
(h : ∀ i ∈ s, is_clopen (f i)) :
is_clopen (⋂ i ∈ s, f i) | ⟨is_open_bInter hs (λ i hi, (h i hi).1), is_closed_bInter (λ i hi, (h i hi).2)⟩ | lemma | is_clopen_bInter | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen",
"is_closed_bInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_bInter_finset {β : Type*} {s : finset β} {f : β → set α}
(h : ∀ i ∈ s, is_clopen (f i)) :
is_clopen (⋂ i ∈ s, f i) | is_clopen_bInter s.finite_to_set h | lemma | is_clopen_bInter_finset | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"finset",
"is_clopen",
"is_clopen_bInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.preimage {s : set β} (h : is_clopen s) {f : α → β} (hf : continuous f) :
is_clopen (f ⁻¹' s) | ⟨h.1.preimage hf, h.2.preimage hf⟩ | lemma | is_clopen.preimage | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"continuous",
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.preimage_clopen_of_clopen
{f : α → β} {s : set α} {t : set β} (hf : continuous_on f s) (hs : is_clopen s)
(ht : is_clopen t) : is_clopen (s ∩ f⁻¹' t) | ⟨continuous_on.preimage_open_of_open hf hs.1 ht.1,
continuous_on.preimage_closed_of_closed hf hs.2 ht.2⟩ | lemma | continuous_on.preimage_clopen_of_clopen | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"continuous_on",
"continuous_on.preimage_closed_of_closed",
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_inter_of_disjoint_cover_clopen {Z a b : set α} (h : is_clopen Z)
(cover : Z ⊆ a ∪ b) (ha : is_open a) (hb : is_open b) (hab : disjoint a b) : is_clopen (Z ∩ a) | begin
refine ⟨is_open.inter h.1 ha, _⟩,
have : is_closed (Z ∩ bᶜ) := is_closed.inter h.2 (is_closed_compl_iff.2 hb),
convert this using 1,
refine (inter_subset_inter_right Z hab.subset_compl_right).antisymm _,
rintro x ⟨hx₁, hx₂⟩,
exact ⟨hx₁, by simpa [not_mem_of_mem_compl hx₂] using cover hx₁⟩,
end | theorem | is_clopen_inter_of_disjoint_cover_clopen | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"disjoint",
"is_clopen",
"is_closed",
"is_closed.inter",
"is_open"
] | The intersection of a disjoint covering by two open sets of a clopen set will be clopen. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_clopen_discrete [discrete_topology α] (x : set α) : is_clopen x | ⟨is_open_discrete _, is_closed_discrete _⟩ | lemma | is_clopen_discrete | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"discrete_topology",
"is_clopen",
"is_closed_discrete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
clopen_range_sigma_mk {ι : Type*} {σ : ι → Type*} [Π i, topological_space (σ i)] {i : ι} :
is_clopen (set.range (@sigma.mk ι σ i)) | ⟨open_embedding_sigma_mk.open_range, closed_embedding_sigma_mk.closed_range⟩ | lemma | clopen_range_sigma_mk | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen",
"set.range",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map.is_clopen_preimage {f : α → β}
(hf : quotient_map f) {s : set β} : is_clopen (f ⁻¹' s) ↔ is_clopen s | and_congr hf.is_open_preimage hf.is_closed_preimage | lemma | quotient_map.is_clopen_preimage | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_clopen",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_bool_indicator_iff_clopen (U : set X) :
continuous U.bool_indicator ↔ is_clopen U | begin
split,
{ intros hc,
rw ← U.preimage_bool_indicator_tt,
exact
⟨hc.is_open_preimage _ trivial, continuous_iff_is_closed.mp hc _ (is_closed_discrete _)⟩ },
{ refine λ hU, ⟨λ s hs, _⟩,
rcases U.preimage_bool_indicator s with (h|h|h|h) ; rw h,
exacts [is_open_univ, hU.1, hU.2.is_open_compl,... | lemma | continuous_bool_indicator_iff_clopen | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"continuous",
"is_clopen",
"is_closed_discrete",
"is_open_empty",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_indicator_iff_clopen (s U : set X) :
continuous_on U.bool_indicator s ↔ is_clopen ((coe : s → X) ⁻¹' U) | begin
rw [continuous_on_iff_continuous_restrict, ← continuous_bool_indicator_iff_clopen],
refl
end | lemma | continuous_on_indicator_iff_clopen | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"continuous_bool_indicator_iff_clopen",
"continuous_on",
"continuous_on_iff_continuous_restrict",
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preirreducible (s : set α) : Prop | ∀ (u v : set α), is_open u → is_open v →
(s ∩ u).nonempty → (s ∩ v).nonempty → (s ∩ (u ∩ v)).nonempty | def | is_preirreducible | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_open"
] | A preirreducible set `s` is one where there is no non-trivial pair of disjoint opens on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_irreducible (s : set α) : Prop | s.nonempty ∧ is_preirreducible s | def | is_irreducible | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_preirreducible"
] | An irreducible set `s` is one that is nonempty and
where there is no non-trivial pair of disjoint opens on `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_irreducible.nonempty {s : set α} (h : is_irreducible s) :
s.nonempty | h.1 | lemma | is_irreducible.nonempty | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible.is_preirreducible {s : set α} (h : is_irreducible s) :
is_preirreducible s | h.2 | lemma | is_irreducible.is_preirreducible | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_irreducible",
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preirreducible_empty : is_preirreducible (∅ : set α) | λ _ _ _ _ _ ⟨x, h1, h2⟩, h1.elim | theorem | is_preirreducible_empty | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.is_preirreducible {s : set α} (hs : s.subsingleton) :
is_preirreducible s | λ u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩, ⟨y, hys, hs hxs hys ▸ hxu, hyv⟩ | lemma | set.subsingleton.is_preirreducible | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_singleton {x} : is_irreducible ({x} : set α) | ⟨singleton_nonempty x, subsingleton_singleton.is_preirreducible⟩ | theorem | is_irreducible_singleton | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preirreducible_iff_closure {s : set α} :
is_preirreducible (closure s) ↔ is_preirreducible s | forall₄_congr $ λ u v hu hv,
by { iterate 3 { rw closure_inter_open_nonempty_iff }, exacts [hu.inter hv, hv, hu] } | theorem | is_preirreducible_iff_closure | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"closure",
"closure_inter_open_nonempty_iff",
"forall₄_congr",
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_iff_closure {s : set α} :
is_irreducible (closure s) ↔ is_irreducible s | and_congr closure_nonempty_iff is_preirreducible_iff_closure | theorem | is_irreducible_iff_closure | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"closure",
"closure_nonempty_iff",
"is_irreducible",
"is_preirreducible_iff_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_preirreducible (s : set α) (H : is_preirreducible s) :
∃ t : set α, is_preirreducible t ∧ s ⊆ t ∧ ∀ u, is_preirreducible u → t ⊆ u → u = t | let ⟨m, hm, hsm, hmm⟩ := zorn_subset_nonempty {t : set α | is_preirreducible t}
(λ c hc hcc hcn, let ⟨t, htc⟩ := hcn in
⟨⋃₀ c, λ u v hu hv ⟨y, hy, hyu⟩ ⟨z, hz, hzv⟩,
let ⟨p, hpc, hyp⟩ := mem_sUnion.1 hy,
⟨q, hqc, hzq⟩ := mem_sUnion.1 hz in
or.cases_on (hcc.total hpc hqc)
(assume hpq ... | theorem | exists_preirreducible | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_preirreducible",
"zorn_subset_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_components (α : Type*) [topological_space α] : set (set α) | maximals (≤) { s : set α | is_irreducible s } | def | irreducible_components | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_irreducible",
"maximals",
"topological_space"
] | The set of irreducible components of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_of_mem_irreducible_components (s ∈ irreducible_components α) :
is_closed s | begin
rw [← closure_eq_iff_is_closed, eq_comm],
exact subset_closure.antisymm (H.2 H.1.closure subset_closure),
end | lemma | is_closed_of_mem_irreducible_components | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"closure_eq_iff_is_closed",
"irreducible_components",
"is_closed",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_components_eq_maximals_closed (α : Type*) [topological_space α] :
irreducible_components α = maximals (≤) { s : set α | is_closed s ∧ is_irreducible s } | begin
ext s,
split,
{ intro H, exact ⟨⟨is_closed_of_mem_irreducible_components _ H, H.1⟩, λ x h e, H.2 h.2 e⟩ },
{ intro H, refine ⟨H.1.2, λ x h e, _⟩,
have : closure x ≤ s,
{ exact H.2 ⟨is_closed_closure, h.closure⟩ (e.trans subset_closure) },
exact le_trans subset_closure this }
end | lemma | irreducible_components_eq_maximals_closed | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"closure",
"irreducible_components",
"is_closed",
"is_irreducible",
"maximals",
"subset_closure",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_component (x : α) : set α | classical.some (exists_preirreducible {x} is_irreducible_singleton.is_preirreducible) | def | irreducible_component | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"exists_preirreducible"
] | A maximal irreducible set that contains a given point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_component_property (x : α) :
is_preirreducible (irreducible_component x) ∧ {x} ⊆ (irreducible_component x) ∧
∀ u, is_preirreducible u → (irreducible_component x) ⊆ u → u = (irreducible_component x) | classical.some_spec (exists_preirreducible {x} is_irreducible_singleton.is_preirreducible) | lemma | irreducible_component_property | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"exists_preirreducible",
"irreducible_component",
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_irreducible_component {x : α} : x ∈ irreducible_component x | singleton_subset_iff.1 (irreducible_component_property x).2.1 | theorem | mem_irreducible_component | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"irreducible_component",
"irreducible_component_property"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_irreducible_component {x : α} : is_irreducible (irreducible_component x) | ⟨⟨x, mem_irreducible_component⟩, (irreducible_component_property x).1⟩ | theorem | is_irreducible_irreducible_component | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"irreducible_component",
"irreducible_component_property",
"is_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_irreducible_component {x : α} :
∀ {s : set α}, is_preirreducible s → irreducible_component x ⊆ s → s = irreducible_component x | (irreducible_component_property x).2.2 | theorem | eq_irreducible_component | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"irreducible_component",
"irreducible_component_property",
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_component_mem_irreducible_components (x : α) :
irreducible_component x ∈ irreducible_components α | ⟨is_irreducible_irreducible_component, λ s h₁ h₂,(eq_irreducible_component h₁.2 h₂).le⟩ | lemma | irreducible_component_mem_irreducible_components | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"eq_irreducible_component",
"irreducible_component",
"irreducible_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_irreducible_component {x : α} :
is_closed (irreducible_component x) | is_closed_of_mem_irreducible_components _ (irreducible_component_mem_irreducible_components x) | theorem | is_closed_irreducible_component | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"irreducible_component",
"irreducible_component_mem_irreducible_components",
"is_closed",
"is_closed_of_mem_irreducible_components"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preirreducible_space (α : Type u) [topological_space α] : Prop | (is_preirreducible_univ [] : is_preirreducible (univ : set α)) | class | preirreducible_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_preirreducible",
"topological_space"
] | A preirreducible space is one where there is no non-trivial pair of disjoint opens. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_space (α : Type u) [topological_space α] extends preirreducible_space α : Prop | (to_nonempty [] : nonempty α) | class | irreducible_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"preirreducible_space",
"topological_space"
] | An irreducible space is one that is nonempty
and where there is no non-trivial pair of disjoint opens. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
irreducible_space.is_irreducible_univ (α : Type u) [topological_space α]
[irreducible_space α] : is_irreducible (⊤ : set α) | ⟨by simp, preirreducible_space.is_preirreducible_univ α⟩ | lemma | irreducible_space.is_irreducible_univ | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"irreducible_space",
"is_irreducible",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_space_def (α : Type u) [topological_space α] :
irreducible_space α ↔ is_irreducible (⊤ : set α) | ⟨@@irreducible_space.is_irreducible_univ α _,
λ h, by { haveI : preirreducible_space α := ⟨h.2⟩, exact ⟨⟨h.1.some⟩⟩ }⟩ | lemma | irreducible_space_def | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"irreducible_space",
"irreducible_space.is_irreducible_univ",
"is_irreducible",
"preirreducible_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_preirreducible_inter [preirreducible_space α] {s t : set α} :
is_open s → is_open t → s.nonempty → t.nonempty → (s ∩ t).nonempty | by simpa only [univ_inter, univ_subset_iff] using
@preirreducible_space.is_preirreducible_univ α _ _ s t | theorem | nonempty_preirreducible_inter | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_open",
"preirreducible_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.dense [preirreducible_space α] {s : set α} (ho : is_open s)
(hne : s.nonempty) : dense s | dense_iff_inter_open.2 $ λ t hto htne, nonempty_preirreducible_inter hto ho htne hne | theorem | is_open.dense | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"dense",
"is_open",
"nonempty_preirreducible_inter",
"preirreducible_space"
] | In a (pre)irreducible space, a nonempty open set is dense. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preirreducible.image {s : set α} (H : is_preirreducible s)
(f : α → β) (hf : continuous_on f s) : is_preirreducible (f '' s) | begin
rintros u v hu hv ⟨_, ⟨⟨x, hx, rfl⟩, hxu⟩⟩ ⟨_, ⟨⟨y, hy, rfl⟩, hyv⟩⟩,
rw ← mem_preimage at hxu hyv,
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⟩,
have := H u' v' hu' hv',
rw [inter_comm s u', ← u'_eq] at this,
rw [inter_comm s v'... | theorem | is_preirreducible.image | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"continuous_on",
"is_preirreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible.image {s : set α} (H : is_irreducible s)
(f : α → β) (hf : continuous_on f s) : is_irreducible (f '' s) | ⟨H.nonempty.image _, H.is_preirreducible.image f hf⟩ | theorem | is_irreducible.image | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"continuous_on",
"is_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.preirreducible_space {s : set α} (h : is_preirreducible s) :
preirreducible_space s | { is_preirreducible_univ :=
begin
intros u v hu hv hsu hsv,
rw is_open_induced_iff at hu hv,
rcases hu with ⟨u, hu, rfl⟩,
rcases hv with ⟨v, hv, rfl⟩,
rcases hsu with ⟨⟨x, hxs⟩, hxs', hxu⟩,
rcases hsv with ⟨⟨y, hys⟩, hys', hyv⟩,
rcases h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩ with ⟨z, hzs, ... | lemma | subtype.preirreducible_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"is_open_induced_iff",
"is_preirreducible",
"preirreducible_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.irreducible_space {s : set α} (h : is_irreducible s) :
irreducible_space s | { is_preirreducible_univ :=
(subtype.preirreducible_space h.is_preirreducible).is_preirreducible_univ,
to_nonempty := h.nonempty.to_subtype } | lemma | subtype.irreducible_space | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"irreducible_space",
"is_irreducible",
"subtype.preirreducible_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible_iff_sInter {s : set α} :
is_irreducible s ↔
∀ (U : finset (set α)) (hU : ∀ u ∈ U, is_open u) (H : ∀ u ∈ U, (s ∩ u).nonempty),
(s ∩ ⋂₀ ↑U).nonempty | begin
split; intro h,
{ intro U, apply finset.induction_on U,
{ intros, simpa using h.nonempty },
{ intros u U hu IH hU H,
rw [finset.coe_insert, sInter_insert],
apply h.2,
{ solve_by_elim [finset.mem_insert_self] },
{ apply is_open_sInter (finset.finite_to_set U),
intros, so... | lemma | is_irreducible_iff_sInter | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"finset",
"finset.coe_insert",
"finset.finite_to_set",
"finset.induction_on",
"finset.mem_insert",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.mem_singleton",
"is_irreducible",
"is_open",
"is_open_sInter"
] | A set `s` is irreducible if and only if
for every finite collection of open sets all of whose members intersect `s`,
`s` also intersects the intersection of the entire collection
(i.e., there is an element of `s` contained in every member of the collection). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preirreducible_iff_closed_union_closed {s : set α} :
is_preirreducible s ↔
∀ (z₁ z₂ : set α), is_closed z₁ → is_closed z₂ → s ⊆ z₁ ∪ z₂ → s ⊆ z₁ ∨ s ⊆ z₂ | begin
split,
all_goals
{ intros h t₁ t₂ ht₁ ht₂,
specialize h t₁ᶜ t₂ᶜ,
simp only [is_open_compl_iff, is_closed_compl_iff] at h,
specialize h ht₁ ht₂ },
{ contrapose!, simp only [not_subset],
rintro ⟨⟨x, hx, hx'⟩, ⟨y, hy, hy'⟩⟩,
rcases h ⟨x, hx, hx'⟩ ⟨y, hy, hy'⟩ with ⟨z, hz, hz'⟩,
rw ← c... | lemma | is_preirreducible_iff_closed_union_closed | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"imp_iff_not_or",
"is_closed",
"is_closed_compl_iff",
"is_open_compl_iff",
"is_preirreducible",
"set.nonempty"
] | A set is preirreducible if and only if
for every cover by two closed sets, it is contained in one of the two covering sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_irreducible_iff_sUnion_closed {s : set α} :
is_irreducible s ↔
∀ (Z : finset (set α)) (hZ : ∀ z ∈ Z, is_closed z) (H : s ⊆ ⋃₀ ↑Z),
∃ z ∈ Z, s ⊆ z | begin
rw [is_irreducible, is_preirreducible_iff_closed_union_closed],
split; intro h,
{ intro Z, apply finset.induction_on Z,
{ intros, rw [finset.coe_empty, sUnion_empty] at H,
rcases h.1 with ⟨x, hx⟩,
exfalso, tauto },
{ intros z Z hz IH hZ H,
cases h.2 z (⋃₀ ↑Z) _ _ _
with h' ... | lemma | is_irreducible_iff_sUnion_closed | topology | src/topology/subset_properties.lean | [
"order.filter.pi",
"topology.bases",
"data.finset.order",
"data.set.accumulate",
"data.set.bool_indicator",
"topology.bornology.basic",
"topology.locally_finite",
"order.minimal"
] | [
"by_contradiction",
"exists_prop",
"finset",
"finset.coe_empty",
"finset.finite_to_set",
"finset.induction_on",
"finset.mem_insert",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.mem_singleton",
"is_closed",
"is_closed_bUnion",
"is_irreducible",
"is_preirreducible_iff_close... | A set is irreducible if and only if
for every cover by a finite collection of closed sets,
it is contained in one of the members of the collection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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