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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|---|---|---|---|---|---|---|---|---|---|---|
is_compact_of_finite_subcover
(h : Π {ι : Type u} (U : ι → (set α)), (∀ i, is_open (U i)) →
s ⊆ (⋃ i, U i) → (∃ (t : finset ι), s ⊆ (⋃ i ∈ t, U i))) :
is_compact s | is_compact_of_finite_subfamily_closed $
assume ι Z hZc hsZ,
let ⟨t, ht⟩ := h (λ i, (Z i)ᶜ) (assume i, is_open_compl_iff.mpr $ hZc i)
(by simpa only [subset_def, not_forall, eq_empty_iff_forall_not_mem, mem_Union,
exists_prop, mem_inter_iff, not_and, iff_self, mem_Inter, mem_compl_iff] using hsZ)
in
... | lemma | is_compact_of_finite_subcover | 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_prop",
"finset",
"is_compact",
"is_compact_of_finite_subfamily_closed",
"is_open",
"not_and",
"not_forall"
] | A set `s` is compact if for every open cover of `s`, there exists a finite subcover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_iff_finite_subcover :
is_compact s ↔ (Π {ι : Type u} (U : ι → (set α)), (∀ i, is_open (U i)) →
s ⊆ (⋃ i, U i) → (∃ (t : finset ι), s ⊆ (⋃ i ∈ t, U i))) | ⟨assume hs ι, hs.elim_finite_subcover, is_compact_of_finite_subcover⟩ | lemma | is_compact_iff_finite_subcover | 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_compact",
"is_open"
] | A set `s` is compact if and only if
for every open cover of `s`, there exists a finite subcover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_iff_finite_subfamily_closed :
is_compact s ↔ (Π {ι : Type u} (Z : ι → (set α)), (∀ i, is_closed (Z i)) →
s ∩ (⋂ i, Z i) = ∅ → (∃ (t : finset ι), s ∩ (⋂ i ∈ t, Z i) = ∅)) | ⟨assume hs ι, hs.elim_finite_subfamily_closed, is_compact_of_finite_subfamily_closed⟩ | theorem | is_compact_iff_finite_subfamily_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"
] | [
"finset",
"is_closed",
"is_compact"
] | A set `s` is compact if and only if
for every family of closed sets whose intersection avoids `s`,
there exists a finite subfamily whose intersection avoids `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.eventually_forall_of_forall_eventually {x₀ : α} {K : set β} (hK : is_compact K)
{P : α → β → Prop} (hP : ∀ y ∈ K, ∀ᶠ (z : α × β) in 𝓝 (x₀, y), P z.1 z.2):
∀ᶠ x in 𝓝 x₀, ∀ y ∈ K, P x y | begin
refine hK.induction_on _ _ _ _,
{ exact eventually_of_forall (λ x y, false.elim) },
{ intros s t hst ht, refine ht.mono (λ x h y hys, h y $ hst hys) },
{ intros s t hs ht, filter_upwards [hs, ht], rintro x h1 h2 y (hys|hyt),
exacts [h1 y hys, h2 y hyt] },
{ intros y hyK,
specialize hP y hyK,
... | lemma | is_compact.eventually_forall_of_forall_eventually | 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",
"mem_nhds_within_of_mem_nhds",
"nhds_prod_eq"
] | To show that `∀ y ∈ K, P x y` holds for `x` close enough to `x₀` when `K` is compact,
it is sufficient to show that for all `y₀ ∈ K` there `P x y` holds for `(x, y)` close enough
to `(x₀, y₀)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_empty : is_compact (∅ : set α) | assume f hnf hsf, not.elim hnf.ne $
empty_mem_iff_bot.1 $ le_principal_iff.1 hsf | lemma | is_compact_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_compact",
"not.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_singleton {a : α} : is_compact ({a} : set α) | λ f hf hfa, ⟨a, rfl, cluster_pt.of_le_nhds'
(hfa.trans $ by simpa only [principal_singleton] using pure_le_nhds a) hf⟩ | lemma | is_compact_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"
] | [
"cluster_pt.of_le_nhds'",
"is_compact",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.is_compact {s : set α} (hs : s.subsingleton) : is_compact s | subsingleton.induction_on hs is_compact_empty $ λ x, is_compact_singleton | lemma | set.subsingleton.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",
"is_compact_empty",
"is_compact_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.is_compact_bUnion {s : set ι} {f : ι → set α} (hs : s.finite)
(hf : ∀ i ∈ s, is_compact (f i)) :
is_compact (⋃ i ∈ s, f i) | is_compact_of_finite_subcover $ assume ι U hUo hsU,
have ∀ i : subtype s, ∃ t : finset ι, f i ⊆ (⋃ j ∈ t, U j), from
assume ⟨i, hi⟩, (hf i hi).elim_finite_subcover _ hUo
(calc f i ⊆ ⋃ i ∈ s, f i : subset_bUnion_of_mem hi
... ⊆ ⋃ j, U j : hsU),
let ⟨finite_subcovers, h⟩ := axiom_of_choice t... | lemma | set.finite.is_compact_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"
] | [
"finset",
"finset.bUnion",
"finset.mem_univ",
"finset.univ",
"fintype",
"is_compact",
"is_compact_of_finite_subcover"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.is_compact_bUnion (s : finset ι) {f : ι → set α} (hf : ∀ i ∈ s, is_compact (f i)) :
is_compact (⋃ i ∈ s, f i) | s.finite_to_set.is_compact_bUnion hf | lemma | finset.is_compact_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"
] | [
"finset",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_accumulate {K : ℕ → set α} (hK : ∀ n, is_compact (K n)) (n : ℕ) :
is_compact (accumulate K n) | (finite_le_nat n).is_compact_bUnion $ λ k _, hK k | lemma | is_compact_accumulate | 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 | |
is_compact_Union {f : ι → set α} [finite ι] (h : ∀ i, is_compact (f i)) :
is_compact (⋃ i, f i) | by rw ← bUnion_univ; exact finite_univ.is_compact_bUnion (λ i _, h i) | lemma | is_compact_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_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.is_compact (hs : s.finite) : is_compact s | bUnion_of_singleton s ▸ hs.is_compact_bUnion (λ _ _, is_compact_singleton) | lemma | set.finite.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",
"is_compact_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.finite_of_discrete [discrete_topology α] {s : set α} (hs : is_compact s) :
s.finite | begin
have : ∀ x : α, ({x} : set α) ∈ 𝓝 x, by simp [nhds_discrete],
rcases hs.elim_nhds_subcover (λ x, {x}) (λ x hx, this x) with ⟨t, hts, hst⟩,
simp only [← t.set_bUnion_coe, bUnion_of_singleton] at hst,
exact t.finite_to_set.subset hst
end | lemma | is_compact.finite_of_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_compact",
"nhds_discrete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_iff_finite [discrete_topology α] {s : set α} : is_compact s ↔ s.finite | ⟨λ h, h.finite_of_discrete, λ h, h.is_compact⟩ | lemma | is_compact_iff_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"
] | [
"discrete_topology",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.union (hs : is_compact s) (ht : is_compact t) : is_compact (s ∪ t) | by rw union_eq_Union; exact is_compact_Union (λ b, by cases b; assumption) | lemma | is_compact.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_compact",
"is_compact_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.insert (hs : is_compact s) (a) : is_compact (insert a s) | is_compact_singleton.union hs | lemma | is_compact.insert | 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 | |
exists_subset_nhds_of_is_compact' {ι : Type*} [nonempty ι]
{V : ι → set α} (hV : directed (⊇) V)
(hV_cpct : ∀ i, is_compact (V i)) (hV_closed : ∀ i, is_closed (V i))
{U : set α} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U | begin
obtain ⟨W, hsubW, W_op, hWU⟩ := exists_open_set_nhds hU,
rsuffices ⟨i, hi⟩ : ∃ i, V i ⊆ W,
{ exact ⟨i, hi.trans hWU⟩ },
by_contra' H,
replace H : ∀ i, (V i ∩ Wᶜ).nonempty := λ i, set.inter_compl_nonempty_iff.mpr (H i),
have : (⋂ i, V i ∩ Wᶜ).nonempty,
{ refine is_compact.nonempty_Inter_of_directed_n... | lemma | exists_subset_nhds_of_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"
] | [
"and_imp",
"directed",
"exists_open_set_nhds",
"is_closed",
"is_compact",
"is_compact.nonempty_Inter_of_directed_nonempty_compact_closed"
] | If `V : ι → set α` is a decreasing family of closed compact sets then any neighborhood of
`⋂ i, V i` contains some `V i`. We assume each `V i` is compact *and* closed because `α` is
not assumed to be Hausdorff. See `exists_subset_nhd_of_compact` for version assuming this. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_open_iff_eq_finite_Union_of_is_topological_basis (b : ι → set α)
(hb : is_topological_basis (set.range b))
(hb' : ∀ i, is_compact (b i)) (U : set α) :
is_compact U ∧ is_open U ↔ ∃ (s : set ι), s.finite ∧ U = ⋃ i ∈ s, b i | begin
classical,
split,
{ rintro ⟨h₁, h₂⟩,
obtain ⟨β, f, e, hf⟩ := hb.open_eq_Union h₂,
choose f' hf' using hf,
have : b ∘ f' = f := funext hf', subst this,
obtain ⟨t, ht⟩ := h₁.elim_finite_subcover (b ∘ f')
(λ i, hb.is_open (set.mem_range_self _)) (by rw e),
refine ⟨t.image f', set.fini... | lemma | is_compact_open_iff_eq_finite_Union_of_is_topological_basis | 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"
] | [
"coe_coe",
"finset.mem_image_of_mem",
"is_compact",
"is_open",
"is_open_bUnion",
"set.Union_subset_iff",
"set.Union_subtype",
"set.Union₂_subset",
"set.mem_range_self",
"set.range",
"set.subset.trans",
"set.subset_Union"
] | If `α` has a basis consisting of compact opens, then an open set in `α` is compact open iff
it is a finite union of some elements in the basis | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cocompact (α : Type*) [topological_space α] : filter α | ⨅ (s : set α) (hs : is_compact s), 𝓟 (sᶜ) | def | filter.cocompact | 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"
] | [
"filter",
"is_compact",
"topological_space"
] | `filter.cocompact` is the filter generated by complements to compact sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_cocompact : (cocompact α).has_basis is_compact compl | has_basis_binfi_principal'
(λ s hs t ht, ⟨s ∪ t, hs.union ht, compl_subset_compl.2 (subset_union_left s t),
compl_subset_compl.2 (subset_union_right s t)⟩)
⟨∅, is_compact_empty⟩ | lemma | filter.has_basis_cocompact | 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 | |
mem_cocompact : s ∈ cocompact α ↔ ∃ t, is_compact t ∧ tᶜ ⊆ s | has_basis_cocompact.mem_iff.trans $ exists_congr $ λ t, exists_prop | lemma | filter.mem_cocompact | 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_prop",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_cocompact' : s ∈ cocompact α ↔ ∃ t, is_compact t ∧ sᶜ ⊆ t | mem_cocompact.trans $ exists_congr $ λ t, and_congr_right $ λ ht, compl_subset_comm | lemma | filter.mem_cocompact' | 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 | |
_root_.is_compact.compl_mem_cocompact (hs : is_compact s) : sᶜ ∈ filter.cocompact α | has_basis_cocompact.mem_of_mem hs | lemma | is_compact.compl_mem_cocompact | 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"
] | [
"filter.cocompact",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocompact_le_cofinite : cocompact α ≤ cofinite | λ s hs, compl_compl s ▸ hs.is_compact.compl_mem_cocompact | lemma | filter.cocompact_le_cofinite | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocompact_eq_cofinite (α : Type*) [topological_space α] [discrete_topology α] :
cocompact α = cofinite | has_basis_cocompact.eq_of_same_basis $
by { convert has_basis_cofinite, ext s, exact is_compact_iff_finite } | lemma | filter.cocompact_eq_cofinite | 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_compact_iff_finite",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.nat.cocompact_eq : cocompact ℕ = at_top | (cocompact_eq_cofinite ℕ).trans nat.cofinite_eq_at_top | lemma | nat.cocompact_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"
] | [
"nat.cofinite_eq_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.is_compact_insert_range_of_cocompact {f : α → β} {b}
(hf : tendsto f (cocompact α) (𝓝 b)) (hfc : continuous f) :
is_compact (insert b (range f)) | begin
introsI l hne hle,
by_cases hb : cluster_pt b l, { exact ⟨b, or.inl rfl, hb⟩ },
simp only [cluster_pt_iff, not_forall, ← not_disjoint_iff_nonempty_inter, not_not] at hb,
rcases hb with ⟨s, hsb, t, htl, hd⟩,
rcases mem_cocompact.1 (hf hsb) with ⟨K, hKc, hKs⟩,
have : f '' K ∈ l,
{ filter_upwards [htl,... | lemma | filter.tendsto.is_compact_insert_range_of_cocompact | 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"
] | [
"cluster_pt",
"cluster_pt_iff",
"continuous",
"is_compact",
"not_forall",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.is_compact_insert_range_of_cofinite {f : ι → α} {a}
(hf : tendsto f cofinite (𝓝 a)) :
is_compact (insert a (range f)) | begin
letI : topological_space ι := ⊥, haveI := discrete_topology_bot ι,
rw ← cocompact_eq_cofinite at hf,
exact hf.is_compact_insert_range_of_cocompact continuous_of_discrete_topology
end | lemma | filter.tendsto.is_compact_insert_range_of_cofinite | 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_of_discrete_topology",
"discrete_topology_bot",
"is_compact",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto.is_compact_insert_range {f : ℕ → α} {a} (hf : tendsto f at_top (𝓝 a)) :
is_compact (insert a (range f)) | filter.tendsto.is_compact_insert_range_of_cofinite $ nat.cofinite_eq_at_top.symm ▸ hf | lemma | filter.tendsto.is_compact_insert_range | 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"
] | [
"filter.tendsto.is_compact_insert_range_of_cofinite",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coclosed_compact (α : Type*) [topological_space α] : filter α | ⨅ (s : set α) (h₁ : is_closed s) (h₂ : is_compact s), 𝓟 (sᶜ) | def | filter.coclosed_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"
] | [
"filter",
"is_closed",
"is_compact",
"topological_space"
] | `filter.coclosed_compact` is the filter generated by complements to closed compact sets.
In a Hausdorff space, this is the same as `filter.cocompact`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_basis_coclosed_compact :
(filter.coclosed_compact α).has_basis (λ s, is_closed s ∧ is_compact s) compl | begin
simp only [filter.coclosed_compact, infi_and'],
refine has_basis_binfi_principal' _ ⟨∅, is_closed_empty, is_compact_empty⟩,
rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩,
exact ⟨s ∪ t, ⟨⟨hs₁.union ht₁, hs₂.union ht₂⟩, compl_subset_compl.2 (subset_union_left _ _),
compl_subset_compl.2 (subset_union_right _ _)⟩⟩
end | lemma | filter.has_basis_coclosed_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"
] | [
"filter.coclosed_compact",
"infi_and'",
"is_closed",
"is_closed_empty",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coclosed_compact : s ∈ coclosed_compact α ↔ ∃ t, is_closed t ∧ is_compact t ∧ tᶜ ⊆ s | by simp [has_basis_coclosed_compact.mem_iff, and_assoc] | lemma | filter.mem_coclosed_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_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coclosed_compact' : s ∈ coclosed_compact α ↔ ∃ t, is_closed t ∧ is_compact t ∧ sᶜ ⊆ t | by simp only [mem_coclosed_compact, compl_subset_comm] | lemma | filter.mem_coclosed_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_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cocompact_le_coclosed_compact : cocompact α ≤ coclosed_compact α | infi_mono $ λ s, le_infi $ λ _, le_rfl | lemma | filter.cocompact_le_coclosed_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"
] | [
"infi_mono",
"le_infi",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.is_compact.compl_mem_coclosed_compact_of_is_closed (hs : is_compact s)
(hs' : is_closed s) :
sᶜ ∈ filter.coclosed_compact α | has_basis_coclosed_compact.mem_of_mem ⟨hs', hs⟩ | lemma | is_compact.compl_mem_coclosed_compact_of_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"
] | [
"filter.coclosed_compact",
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
in_compact : bornology α | { cobounded := filter.cocompact α,
le_cofinite := filter.cocompact_le_cofinite } | def | bornology.in_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"
] | [
"bornology",
"filter.cocompact",
"filter.cocompact_le_cofinite"
] | Sets that are contained in a compact set form a bornology. Its `cobounded` filter is
`filter.cocompact`. See also `bornology.relatively_compact` the bornology of sets with compact
closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
in_compact.is_bounded_iff : @is_bounded _ (in_compact α) s ↔ ∃ t, is_compact t ∧ s ⊆ t | begin
change sᶜ ∈ filter.cocompact α ↔ _,
rw filter.mem_cocompact,
simp
end | lemma | bornology.in_compact.is_bounded_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"
] | [
"filter.cocompact",
"filter.mem_cocompact",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_contain_boxes (s : set α) (t : set β) : Prop | ∀ (n : set (α × β)) (hn : is_open n) (hp : s ×ˢ t ⊆ n),
∃ (u : set α) (v : set β), is_open u ∧ is_open v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n | def | nhds_contain_boxes | 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"
] | `nhds_contain_boxes s t` means that any open neighborhood of `s × t` in `α × β` includes
a product of an open neighborhood of `s` by an open neighborhood of `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_contain_boxes.symm {s : set α} {t : set β} :
nhds_contain_boxes s t → nhds_contain_boxes t s | assume H n hn hp,
let ⟨u, v, uo, vo, su, tv, p⟩ :=
H (prod.swap ⁻¹' n)
(hn.preimage continuous_swap)
(by rwa [←image_subset_iff, image_swap_prod]) in
⟨v, u, vo, uo, tv, su,
by rwa [←image_subset_iff, image_swap_prod] at p⟩ | lemma | nhds_contain_boxes.symm | 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_swap",
"nhds_contain_boxes",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_contain_boxes.comm {s : set α} {t : set β} :
nhds_contain_boxes s t ↔ nhds_contain_boxes t s | iff.intro nhds_contain_boxes.symm nhds_contain_boxes.symm | lemma | nhds_contain_boxes.comm | 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"
] | [
"nhds_contain_boxes",
"nhds_contain_boxes.symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_contain_boxes_of_singleton {x : α} {y : β} :
nhds_contain_boxes ({x} : set α) ({y} : set β) | assume n hn hp,
let ⟨u, v, uo, vo, xu, yv, hp'⟩ :=
is_open_prod_iff.mp hn x y (hp $ by simp) in
⟨u, v, uo, vo, by simpa, by simpa, hp'⟩ | lemma | nhds_contain_boxes_of_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"
] | [
"nhds_contain_boxes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_contain_boxes_of_compact {s : set α} (hs : is_compact s) (t : set β)
(H : ∀ x ∈ s, nhds_contain_boxes ({x} : set α) t) : nhds_contain_boxes s t | assume n hn hp,
have ∀ x : s, ∃ uv : set α × set β,
is_open uv.1 ∧ is_open uv.2 ∧ {↑x} ⊆ uv.1 ∧ t ⊆ uv.2 ∧ uv.1 ×ˢ uv.2 ⊆ n,
from assume ⟨x, hx⟩,
have ({x} : set α) ×ˢ t ⊆ n, from
subset.trans (prod_mono (by simpa) subset.rfl) hp,
let ⟨ux,vx,H1⟩ := H x hx n hn this in ⟨⟨ux,vx⟩,H1⟩,
let ⟨uvs, h⟩ := ... | lemma | nhds_contain_boxes_of_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",
"is_open",
"is_open_bInter",
"is_open_bUnion",
"nhds_contain_boxes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generalized_tube_lemma {s : set α} (hs : is_compact s) {t : set β} (ht : is_compact t)
{n : set (α × β)} (hn : is_open n) (hp : s ×ˢ t ⊆ n) :
∃ (u : set α) (v : set β), is_open u ∧ is_open v ∧ s ⊆ u ∧ t ⊆ v ∧ u ×ˢ v ⊆ n | have _, from
nhds_contain_boxes_of_compact hs t $ assume x _, nhds_contain_boxes.symm $
nhds_contain_boxes_of_compact ht {x} $ assume y _, nhds_contain_boxes_of_singleton,
this n hn hp | lemma | generalized_tube_lemma | 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",
"is_open",
"nhds_contain_boxes.symm",
"nhds_contain_boxes_of_compact",
"nhds_contain_boxes_of_singleton"
] | If `s` and `t` are compact sets and `n` is an open neighborhood of `s × t`, then there exist
open neighborhoods `u ⊇ s` and `v ⊇ t` such that `u × v ⊆ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_space (α : Type*) [topological_space α] : Prop | (is_compact_univ : is_compact (univ : set α)) | class | 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",
"is_compact_univ",
"topological_space"
] | Type class for compact spaces. Separation is sometimes included in the definition, especially
in the French literature, but we do not include it here. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton.compact_space [subsingleton α] : compact_space α | ⟨subsingleton_univ.is_compact⟩ | instance | subsingleton.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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_univ_iff : is_compact (univ : set α) ↔ compact_space α | ⟨λ h, ⟨h⟩, λ h, h.1⟩ | lemma | is_compact_univ_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"
] | [
"compact_space",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_univ [h : compact_space α] : is_compact (univ : set α) | h.is_compact_univ | lemma | is_compact_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"
] | [
"compact_space",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cluster_point_of_compact [compact_space α] (f : filter α) [ne_bot f] :
∃ x, cluster_pt x f | by simpa using is_compact_univ (show f ≤ 𝓟 univ, by simp) | lemma | cluster_point_of_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"
] | [
"cluster_pt",
"compact_space",
"filter",
"is_compact_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_space.elim_nhds_subcover [compact_space α]
(U : α → set α) (hU : ∀ x, U x ∈ 𝓝 x) :
∃ t : finset α, (⋃ x ∈ t, U x) = ⊤ | begin
obtain ⟨t, -, s⟩ := is_compact.elim_nhds_subcover is_compact_univ U (λ x m, hU x),
exact ⟨t, by { rw eq_top_iff, exact s }⟩,
end | lemma | compact_space.elim_nhds_subcover | 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",
"eq_top_iff",
"finset",
"is_compact.elim_nhds_subcover",
"is_compact_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_space_of_finite_subfamily_closed
(h : Π {ι : Type u} (Z : ι → (set α)), (∀ i, is_closed (Z i)) →
(⋂ i, Z i) = ∅ → ∃ (t : finset ι), (⋂ i ∈ t, Z i) = ∅) :
compact_space α | { is_compact_univ :=
begin
apply is_compact_of_finite_subfamily_closed,
intros ι Z, specialize h Z,
simpa using h
end } | theorem | compact_space_of_finite_subfamily_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"
] | [
"compact_space",
"finset",
"is_closed",
"is_compact_of_finite_subfamily_closed",
"is_compact_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.is_compact [compact_space α] {s : set α} (h : is_closed s) :
is_compact s | is_compact_of_is_closed_subset is_compact_univ h (subset_univ _) | lemma | is_closed.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"
] | [
"compact_space",
"is_closed",
"is_compact",
"is_compact_of_is_closed_subset",
"is_compact_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noncompact_space (α : Type*) [topological_space α] : Prop | (noncompact_univ [] : ¬is_compact (univ : set α)) | class | noncompact_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"
] | `α` is a noncompact topological space if it not a compact space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.ne_univ [noncompact_space α] {s : set α} (hs : is_compact s) : s ≠ univ | λ h, noncompact_univ α (h ▸ hs) | lemma | is_compact.ne_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_compact",
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.cocompact_eq_bot [compact_space α] : filter.cocompact α = ⊥ | filter.has_basis_cocompact.eq_bot_iff.mpr ⟨set.univ, is_compact_univ, set.compl_univ⟩ | lemma | filter.cocompact_eq_bot | 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",
"filter.cocompact",
"is_compact_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noncompact_space_of_ne_bot (h : ne_bot (filter.cocompact α)) : noncompact_space α | ⟨λ h', (filter.nonempty_of_mem h'.compl_mem_cocompact).ne_empty compl_univ⟩ | lemma | noncompact_space_of_ne_bot | 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"
] | [
"filter.cocompact",
"filter.nonempty_of_mem",
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.cocompact_ne_bot_iff : ne_bot (filter.cocompact α) ↔ noncompact_space α | ⟨noncompact_space_of_ne_bot, @filter.cocompact.filter.ne_bot _ _⟩ | lemma | filter.cocompact_ne_bot_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"
] | [
"filter.cocompact",
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_compact_space_iff : ¬compact_space α ↔ noncompact_space α | ⟨λ h₁, ⟨λ h₂, h₁ ⟨h₂⟩⟩, λ ⟨h₁⟩ ⟨h₂⟩, h₁ h₂⟩ | lemma | not_compact_space_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"
] | [
"compact_space",
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_of_compact_of_discrete [compact_space α] [discrete_topology α] : finite α | finite.of_finite_univ $ is_compact_univ.finite_of_discrete | lemma | finite_of_compact_of_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"
] | [
"compact_space",
"discrete_topology",
"finite"
] | A compact discrete space is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_nhds_ne_ne_bot (α : Type*) [topological_space α] [compact_space α] [infinite α] :
∃ z : α, (𝓝[≠] z).ne_bot | begin
by_contra' H,
simp_rw not_ne_bot at H,
haveI := discrete_topology_iff_nhds_ne.mpr H,
exact infinite.not_finite (finite_of_compact_of_discrete : finite α),
end | lemma | exists_nhds_ne_ne_bot | 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",
"finite",
"finite_of_compact_of_discrete",
"infinite",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_cover_nhds_interior [compact_space α] {U : α → set α} (hU : ∀ x, U x ∈ 𝓝 x) :
∃ t : finset α, (⋃ x ∈ t, interior (U x)) = univ | let ⟨t, ht⟩ := is_compact_univ.elim_finite_subcover (λ x, interior (U x)) (λ x, is_open_interior)
(λ x _, mem_Union.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x)⟩)
in ⟨t, univ_subset_iff.1 ht⟩ | lemma | finite_cover_nhds_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"
] | [
"compact_space",
"finset",
"interior",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_cover_nhds [compact_space α] {U : α → set α} (hU : ∀ x, U x ∈ 𝓝 x) :
∃ t : finset α, (⋃ x ∈ t, U x) = univ | let ⟨t, ht⟩ := finite_cover_nhds_interior hU in ⟨t, univ_subset_iff.1 $ ht.symm.subset.trans $
Union₂_mono $ λ x hx, interior_subset⟩ | lemma | finite_cover_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"
] | [
"compact_space",
"finite_cover_nhds_interior",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_finite.finite_nonempty_of_compact {ι : Type*} [compact_space α] {f : ι → set α}
(hf : locally_finite f) :
{i | (f i).nonempty}.finite | by simpa only [inter_univ] using hf.finite_nonempty_inter_compact is_compact_univ | lemma | locally_finite.finite_nonempty_of_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",
"finite",
"is_compact_univ",
"locally_finite"
] | If `α` is a compact space, then a locally finite family of sets of `α` can have only finitely
many nonempty elements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_finite.finite_of_compact {ι : Type*} [compact_space α] {f : ι → set α}
(hf : locally_finite f) (hne : ∀ i, (f i).nonempty) :
(univ : set ι).finite | by simpa only [hne] using hf.finite_nonempty_of_compact | lemma | locally_finite.finite_of_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",
"finite",
"locally_finite"
] | If `α` is a compact space, then a locally finite family of nonempty sets of `α` can have only
finitely many elements, `set.finite` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_finite.fintype_of_compact {ι : Type*} [compact_space α] {f : ι → set α}
(hf : locally_finite f) (hne : ∀ i, (f i).nonempty) :
fintype ι | fintype_of_finite_univ (hf.finite_of_compact hne) | def | locally_finite.fintype_of_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",
"fintype",
"locally_finite"
] | If `α` is a compact space, then a locally finite family of nonempty sets of `α` can have only
finitely many elements, `fintype` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.comap_cocompact_le {f : α → β} (hf : continuous f) :
(filter.cocompact β).comap f ≤ filter.cocompact α | begin
rw (filter.has_basis_cocompact.comap f).le_basis_iff filter.has_basis_cocompact,
intros t ht,
refine ⟨f '' t, ht.image hf, _⟩,
simpa using t.subset_preimage_image f
end | lemma | filter.comap_cocompact_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"
] | [
"continuous",
"filter.cocompact",
"filter.has_basis_cocompact"
] | The comap of the cocompact filter on `β` by a continuous function `f : α → β` is less than or
equal to the cocompact filter on `α`.
This is a reformulation of the fact that images of compact sets are compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_range [compact_space α] {f : α → β} (hf : continuous f) :
is_compact (range f) | by rw ← image_univ; exact is_compact_univ.image hf | lemma | is_compact_range | 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",
"continuous",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_diagonal [compact_space α] : is_compact (diagonal α) | @range_diag α ▸ is_compact_range (continuous_id.prod_mk continuous_id) | lemma | is_compact_diagonal | 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",
"continuous_id",
"is_compact",
"is_compact_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_proj_of_is_compact
{X : Type*} [topological_space X] [compact_space X]
{Y : Type*} [topological_space Y] :
is_closed_map (prod.snd : X × Y → Y) | begin
set πX := (prod.fst : X × Y → X),
set πY := (prod.snd : X × Y → Y),
assume C (hC : is_closed C),
rw is_closed_iff_cluster_pt at hC ⊢,
assume y (y_closure : cluster_pt y $ 𝓟 (πY '' C)),
haveI : ne_bot (map πX (comap πY (𝓝 y) ⊓ 𝓟 C)),
{ suffices : ne_bot (map πY (comap πY (𝓝 y) ⊓ 𝓟 C)),
by ... | theorem | is_closed_proj_of_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"
] | [
"cluster_point_of_compact",
"cluster_pt",
"compact_space",
"filter.map_ne_bot_iff",
"filter.prod",
"filter.push_pull",
"filter.push_pull'",
"inf_comm",
"is_closed",
"is_closed_iff_cluster_pt",
"is_closed_map",
"nhds_prod_eq",
"topological_space"
] | If X is is_compact then pr₂ : X × Y → Y is a closed map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_subset_nhds_of_compact_space [compact_space α] {ι : Type*} [nonempty ι]
{V : ι → set α} (hV : directed (⊇) V) (hV_closed : ∀ i, is_closed (V i))
{U : set α} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U | exists_subset_nhds_of_is_compact' hV (λ i, (hV_closed i).is_compact) hV_closed hU | lemma | exists_subset_nhds_of_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",
"directed",
"exists_subset_nhds_of_is_compact'",
"is_closed",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.is_compact_iff {f : α → β} (hf : inducing f) {s : set α} :
is_compact (f '' s) ↔ is_compact s | begin
refine ⟨_, λ hs, hs.image hf.continuous⟩,
introsI hs F F_ne_bot F_le,
obtain ⟨_, ⟨x, x_in : x ∈ s, rfl⟩, hx : cluster_pt (f x) (map f F)⟩ :=
hs (calc map f F ≤ map f (𝓟 s) : map_mono F_le
... = 𝓟 (f '' s) : map_principal),
use [x, x_in],
suffices : (map f (𝓝 x ⊓ F)).ne_bot, by sim... | lemma | inducing.is_compact_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"
] | [
"cluster_pt",
"filter.map_ne_bot_iff",
"filter.push_pull'",
"inducing",
"is_compact"
] | If `f : α → β` is an `inducing` map, then the image `f '' s` of a set `s` is compact if and only
if the set `s` is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding.is_compact_iff_is_compact_image {f : α → β} (hf : embedding f) :
is_compact s ↔ is_compact (f '' s) | hf.to_inducing.is_compact_iff.symm | lemma | embedding.is_compact_iff_is_compact_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"
] | [
"embedding",
"is_compact"
] | If `f : α → β` is an `embedding` (or more generally, an `inducing` map, see
`inducing.is_compact_iff`), then the image `f '' s` of a set `s` is compact if and only if the set
`s` is closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding.is_compact_preimage {f : α → β} (hf : closed_embedding f) {K : set β}
(hK : is_compact K) : is_compact (f ⁻¹' K) | begin
replace hK := hK.inter_right hf.closed_range,
rwa [← hf.to_inducing.is_compact_iff, image_preimage_eq_inter_range]
end | lemma | closed_embedding.is_compact_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"
] | [
"closed_embedding",
"is_compact"
] | The preimage of a compact set under a closed embedding is a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding.tendsto_cocompact
{f : α → β} (hf : closed_embedding f) : tendsto f (filter.cocompact α) (filter.cocompact β) | filter.has_basis_cocompact.tendsto_right_iff.mpr $ λ K hK,
(hf.is_compact_preimage hK).compl_mem_cocompact | lemma | closed_embedding.tendsto_cocompact | 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",
"filter.cocompact"
] | A closed embedding is proper, ie, inverse images of compact sets are contained in compacts.
Moreover, the preimage of a compact set is compact, see `closed_embedding.is_compact_preimage`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_iff_is_compact_in_subtype {p : α → Prop} {s : set {a // p a}} :
is_compact s ↔ is_compact ((coe : _ → α) '' s) | embedding_subtype_coe.is_compact_iff_is_compact_image | lemma | is_compact_iff_is_compact_in_subtype | 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 | |
is_compact_iff_is_compact_univ {s : set α} : is_compact s ↔ is_compact (univ : set s) | by rw [is_compact_iff_is_compact_in_subtype, image_univ, subtype.range_coe]; refl | lemma | is_compact_iff_is_compact_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_compact",
"is_compact_iff_is_compact_in_subtype",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_iff_compact_space {s : set α} : is_compact s ↔ compact_space s | is_compact_iff_is_compact_univ.trans ⟨λ h, ⟨h⟩, @compact_space.is_compact_univ _ _⟩ | lemma | is_compact_iff_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",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.finite {s : set α} (hs : is_compact s) (hs' : discrete_topology s) : s.finite | finite_coe_iff.mp (@finite_of_compact_of_discrete _ _ (is_compact_iff_compact_space.mp hs) hs') | lemma | is_compact.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"
] | [
"discrete_topology",
"finite_of_compact_of_discrete",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nhds_ne_inf_principal_ne_bot {s : set α} (hs : is_compact s) (hs' : s.infinite) :
∃ z ∈ s, (𝓝[≠] z ⊓ 𝓟 s).ne_bot | begin
by_contra' H,
simp_rw not_ne_bot at H,
exact hs' (hs.finite $ discrete_topology_subtype_iff.mpr H),
end | lemma | exists_nhds_ne_inf_principal_ne_bot | 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 | |
closed_embedding.noncompact_space [noncompact_space α] {f : α → β}
(hf : closed_embedding f) : noncompact_space β | noncompact_space_of_ne_bot hf.tendsto_cocompact.ne_bot | lemma | closed_embedding.noncompact_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",
"noncompact_space",
"noncompact_space_of_ne_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.compact_space [h : compact_space β] {f : α → β}
(hf : closed_embedding f) : compact_space α | by { unfreezingI { contrapose! h, rw not_compact_space_iff at h ⊢ }, exact hf.noncompact_space } | lemma | closed_embedding.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_space",
"not_compact_space_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.prod {s : set α} {t : set β} (hs : is_compact s) (ht : is_compact t) :
is_compact (s ×ˢ t) | begin
rw is_compact_iff_ultrafilter_le_nhds at hs ht ⊢,
intros f hfs,
rw le_principal_iff at hfs,
obtain ⟨a : α, sa : a ∈ s, ha : map prod.fst ↑f ≤ 𝓝 a⟩ :=
hs (f.map prod.fst) (le_principal_iff.2 $ mem_map.2 $ mem_of_superset hfs (λ x, and.left)),
obtain ⟨b : β, tb : b ∈ t, hb : map prod.snd ↑f ≤ 𝓝 b⟩ :... | lemma | is_compact.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_compact",
"is_compact_iff_ultrafilter_le_nhds",
"le_inf",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite.compact_space [finite α] : compact_space α | { is_compact_univ := finite_univ.is_compact } | instance | finite.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",
"finite",
"is_compact_univ"
] | Finite topological spaces are compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.coprod_cocompact :
(filter.cocompact α).coprod (filter.cocompact β) = filter.cocompact (α × β) | begin
ext S,
simp only [mem_coprod_iff, exists_prop, mem_comap, filter.mem_cocompact],
split,
{ rintro ⟨⟨A, ⟨t, ht, hAt⟩, hAS⟩, B, ⟨t', ht', hBt'⟩, hBS⟩,
refine ⟨t ×ˢ t', ht.prod ht', _⟩,
refine subset.trans _ (union_subset hAS hBS),
rw compl_subset_comm at ⊢ hAt hBt',
refine subset.trans _ (set... | lemma | filter.coprod_cocompact | 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_fst",
"continuous_snd",
"exists_prop",
"filter.cocompact",
"filter.mem_cocompact",
"set.prod_mono"
] | The coproduct of the cocompact filters on two topological spaces is the cocompact filter on
their product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod.noncompact_space_iff :
noncompact_space (α × β) ↔ noncompact_space α ∧ nonempty β ∨ nonempty α ∧ noncompact_space β | by simp [← filter.cocompact_ne_bot_iff, ← filter.coprod_cocompact, filter.coprod_ne_bot_iff] | lemma | prod.noncompact_space_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"
] | [
"filter.cocompact_ne_bot_iff",
"filter.coprod_cocompact",
"filter.coprod_ne_bot_iff",
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.noncompact_space_left [noncompact_space α] [nonempty β] : noncompact_space (α × β) | prod.noncompact_space_iff.2 (or.inl ⟨‹_›, ‹_›⟩) | instance | prod.noncompact_space_left | 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"
] | [
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.noncompact_space_right [nonempty α] [noncompact_space β] : noncompact_space (α × β) | prod.noncompact_space_iff.2 (or.inr ⟨‹_›, ‹_›⟩) | instance | prod.noncompact_space_right | 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"
] | [
"noncompact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_pi_infinite {s : Π i, set (π i)} :
(∀ i, is_compact (s i)) → is_compact {x : Π i, π i | ∀ i, x i ∈ s i} | begin
simp only [is_compact_iff_ultrafilter_le_nhds, nhds_pi, filter.pi, exists_prop, mem_set_of_eq,
le_infi_iff, le_principal_iff],
intros h f hfs,
have : ∀ i:ι, ∃ a, a ∈ s i ∧ tendsto (λx:Πi:ι, π i, x i) f (𝓝 a),
{ refine λ i, h i (f.map _) (mem_map.2 _),
exact mem_of_superset hfs (λ x hx, hx i) },
... | lemma | is_compact_pi_infinite | 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_prop",
"filter.pi",
"is_compact",
"is_compact_iff_ultrafilter_le_nhds",
"le_infi_iff",
"nhds_pi"
] | **Tychonoff's theorem**: product of compact sets is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_univ_pi {s : Π i, set (π i)} (h : ∀ i, is_compact (s i)) :
is_compact (pi univ s) | by { convert is_compact_pi_infinite h, simp only [← mem_univ_pi, set_of_mem_eq] } | lemma | is_compact_univ_pi | 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",
"is_compact_pi_infinite"
] | **Tychonoff's theorem** formulated using `set.pi`: product of compact sets is compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.compact_space [∀ i, compact_space (π i)] : compact_space (Πi, π i) | ⟨by { rw [← pi_univ univ], exact is_compact_univ_pi (λ i, is_compact_univ) }⟩ | instance | pi.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",
"is_compact_univ",
"is_compact_univ_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.compact_space [compact_space β] : compact_space (ι → β) | pi.compact_space | instance | function.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",
"pi.compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.Coprod_cocompact {δ : Type*} {κ : δ → Type*} [Π d, topological_space (κ d)] :
filter.Coprod (λ d, filter.cocompact (κ d)) = filter.cocompact (Π d, κ d) | begin
refine le_antisymm (supr_le $ λ i, filter.comap_cocompact_le (continuous_apply i)) _,
refine compl_surjective.forall.2 (λ s H, _),
simp only [compl_mem_Coprod, filter.mem_cocompact, compl_subset_compl, image_subset_iff] at H ⊢,
choose K hKc htK using H,
exact ⟨set.pi univ K, is_compact_univ_pi hKc, λ f ... | lemma | filter.Coprod_cocompact | 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_apply",
"filter.Coprod",
"filter.cocompact",
"filter.comap_cocompact_le",
"filter.mem_cocompact",
"is_compact_univ_pi",
"supr_le",
"topological_space"
] | **Tychonoff's theorem** formulated in terms of filters: `filter.cocompact` on an indexed product
type `Π d, κ d` the `filter.Coprod` of filters `filter.cocompact` on `κ d`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quot.compact_space {r : α → α → Prop} [compact_space α] :
compact_space (quot r) | ⟨by { rw ← range_quot_mk, exact is_compact_range continuous_quot_mk }⟩ | instance | quot.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",
"continuous_quot_mk",
"is_compact_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient.compact_space {s : setoid α} [compact_space α] :
compact_space (quotient s) | quot.compact_space | instance | quotient.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",
"quot.compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_compact_space (α : Type*) [topological_space α] : Prop | (local_compact_nhds : ∀ (x : α) (n ∈ 𝓝 x), ∃ s ∈ 𝓝 x, s ⊆ n ∧ is_compact s) | class | 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_compact",
"local_compact_nhds",
"topological_space"
] | There are various definitions of "locally compact space" in the literature, which agree for
Hausdorff spaces but not in general. This one is the precise condition on X needed for the
evaluation `map C(X, Y) × X → Y` to be continuous for all `Y` when `C(X, Y)` is given the
compact-open topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compact_basis_nhds [locally_compact_space α] (x : α) :
(𝓝 x).has_basis (λ s, s ∈ 𝓝 x ∧ is_compact s) (λ s, s) | has_basis_self.2 $ by simpa only [and_comm] using locally_compact_space.local_compact_nhds x | lemma | compact_basis_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"
] | [
"is_compact",
"locally_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_compact_nhds [locally_compact_space α] {x : α} {n : set α} (h : n ∈ 𝓝 x) :
∃ s ∈ 𝓝 x, s ⊆ n ∧ is_compact s | locally_compact_space.local_compact_nhds _ _ h | lemma | local_compact_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"
] | [
"is_compact",
"locally_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_compact_space_of_has_basis {ι : α → Type*} {p : Π x, ι x → Prop}
{s : Π x, ι x → set α} (h : ∀ x, (𝓝 x).has_basis (p x) (s x))
(hc : ∀ x i, p x i → is_compact (s x i)) :
locally_compact_space α | ⟨λ x t ht, let ⟨i, hp, ht⟩ := (h x).mem_iff.1 ht in ⟨s x i, (h x).mem_of_mem hp, ht, hc x i hp⟩⟩ | lemma | locally_compact_space_of_has_basis | 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",
"locally_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.locally_compact_space (α : Type*) (β : Type*) [topological_space α]
[topological_space β] [locally_compact_space α] [locally_compact_space β] :
locally_compact_space (α × β) | have _ := λ x : α × β, (compact_basis_nhds x.1).prod_nhds' (compact_basis_nhds x.2),
locally_compact_space_of_has_basis this $ λ x s ⟨⟨_, h₁⟩, _, h₂⟩, h₁.prod h₂ | instance | prod.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",
"locally_compact_space",
"locally_compact_space_of_has_basis",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.locally_compact_space_of_finite [finite ι] : locally_compact_space (Π i, π i) | ⟨λ t n hn, begin
rw [nhds_pi, filter.mem_pi] at hn,
obtain ⟨s, hs, n', hn', hsub⟩ := hn,
choose n'' hn'' hsub' hc using λ i, locally_compact_space.local_compact_nhds (t i) (n' i) (hn' i),
refine ⟨(set.univ : set ι).pi n'', _, subset_trans (λ _ h, _) hsub, is_compact_univ_pi hc⟩,
{ exact (set_pi_mem_nhds_iff (... | instance | pi.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"
] | [
"filter.mem_pi",
"finite",
"is_compact_univ_pi",
"locally_compact_space",
"nhds_pi",
"set.finite_univ",
"set_pi_mem_nhds_iff",
"subset_trans"
] | In general it suffices that all but finitely many of the spaces are compact,
but that's not straightforward to state and use. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.locally_compact_space [∀ i, compact_space (π i)] : locally_compact_space (Π i, π i) | ⟨λ t n hn, begin
rw [nhds_pi, filter.mem_pi] at hn,
obtain ⟨s, hs, n', hn', hsub⟩ := hn,
choose n'' hn'' hsub' hc using λ i, locally_compact_space.local_compact_nhds (t i) (n' i) (hn' i),
refine ⟨s.pi n'', _, subset_trans (λ _, _) hsub, _⟩,
{ exact (set_pi_mem_nhds_iff hs _).mpr (λ i _, hn'' i), },
{ exact ... | instance | pi.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",
"filter.mem_pi",
"forall₂_imp",
"is_compact_univ_pi",
"locally_compact_space",
"nhds_pi",
"set.univ_pi_ite",
"set_pi_mem_nhds_iff",
"subset_trans"
] | For spaces that are not Hausdorff. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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