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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apply_eq (v : partial_refinement u s) {i : ι} (hi : i ∉ v.carrier) : v i = u i | v.apply_eq' i hi | lemma | shrinking_lemma.partial_refinement.apply_eq | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open (v : partial_refinement u s) (i : ι) : is_open (v i) | v.is_open' i | lemma | shrinking_lemma.partial_refinement.is_open | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset (v : partial_refinement u s) (i : ι) : v i ⊆ u i | if h : i ∈ v.carrier then subset.trans subset_closure (v.closure_subset h)
else (v.apply_eq h).le | lemma | shrinking_lemma.partial_refinement.subset | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_of_chain {c : set (partial_refinement u s)} (hc : is_chain (≤) c) {v₁ v₂}
(h₁ : v₁ ∈ c) (h₂ : v₂ ∈ c) {i} (hi₁ : i ∈ v₁.carrier) (hi₂ : i ∈ v₂.carrier) :
v₁ i = v₂ i | begin
wlog hle : v₁ ≤ v₂,
{ cases hc.total h₁ h₂; [skip, symmetry]; apply_assumption; assumption' },
exact hle.2 _ hi₁,
end | lemma | shrinking_lemma.partial_refinement.apply_eq_of_chain | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"is_chain"
] | If two partial refinements `v₁`, `v₂` belong to a chain (hence, they are comparable)
and `i` belongs to the carriers of both partial refinements, then `v₁ i = v₂ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
chain_Sup_carrier (c : set (partial_refinement u s)) : set ι | ⋃ v ∈ c, carrier v | def | shrinking_lemma.partial_refinement.chain_Sup_carrier | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [] | The carrier of the least upper bound of a non-empty chain of partial refinements
is the union of their carriers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
find (c : set (partial_refinement u s)) (ne : c.nonempty) (i : ι) :
partial_refinement u s | if hi : ∃ v ∈ c, i ∈ carrier v then hi.some else ne.some | def | shrinking_lemma.partial_refinement.find | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [] | Choice of an element of a nonempty chain of partial refinements. If `i` belongs to one of
`carrier v`, `v ∈ c`, then `find c ne i` is one of these partial refinements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
find_mem {c : set (partial_refinement u s)} (i : ι) (ne : c.nonempty) :
find c ne i ∈ c | by { rw find, split_ifs, exacts [h.some_spec.fst, ne.some_spec] } | lemma | shrinking_lemma.partial_refinement.find_mem | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_find_carrier_iff {c : set (partial_refinement u s)} {i : ι} (ne : c.nonempty) :
i ∈ (find c ne i).carrier ↔ i ∈ chain_Sup_carrier c | begin
rw find,
split_ifs,
{ have : i ∈ h.some.carrier ∧ i ∈ chain_Sup_carrier c,
from ⟨h.some_spec.snd, mem_Union₂.2 h⟩,
simp only [this] },
{ have : i ∉ ne.some.carrier ∧ i ∉ chain_Sup_carrier c,
from ⟨λ hi, h ⟨_, ne.some_spec, hi⟩, mt mem_Union₂.1 h⟩,
simp only [this] }
end | lemma | shrinking_lemma.partial_refinement.mem_find_carrier_iff | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
find_apply_of_mem {c : set (partial_refinement u s)} (hc : is_chain (≤) c) (ne : c.nonempty)
{i v} (hv : v ∈ c) (hi : i ∈ carrier v) :
find c ne i i = v i | apply_eq_of_chain hc (find_mem _ _) hv
((mem_find_carrier_iff _).2 $ mem_Union₂.2 ⟨v, hv, hi⟩) hi | lemma | shrinking_lemma.partial_refinement.find_apply_of_mem | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"is_chain"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
chain_Sup (c : set (partial_refinement u s)) (hc : is_chain (≤) c)
(ne : c.nonempty) (hfin : ∀ x ∈ s, {i | x ∈ u i}.finite) (hU : s ⊆ ⋃ i, u i) :
partial_refinement u s | begin
refine ⟨λ i, find c ne i i, chain_Sup_carrier c,
λ i, (find _ _ _).is_open i,
λ x hxs, mem_Union.2 _,
λ i hi, (find c ne i).closure_subset ((mem_find_carrier_iff _).2 hi),
λ i hi, (find c ne i).apply_eq (mt (mem_find_carrier_iff _).1 hi)⟩,
rcases em (∃ i ∉ chain_Sup_carrier c, x ∈ u i) with ⟨i... | def | shrinking_lemma.partial_refinement.chain_Sup | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"em",
"finite",
"is_chain",
"is_open",
"not_exists",
"not_imp_not"
] | Least upper bound of a nonempty chain of partial refinements. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_chain_Sup {c : set (partial_refinement u s)} (hc : is_chain (≤) c)
(ne : c.nonempty) (hfin : ∀ x ∈ s, {i | x ∈ u i}.finite) (hU : s ⊆ ⋃ i, u i)
{v} (hv : v ∈ c) :
v ≤ chain_Sup c hc ne hfin hU | ⟨λ i hi, mem_bUnion hv hi, λ i hi, (find_apply_of_mem hc _ hv hi).symm⟩ | lemma | shrinking_lemma.partial_refinement.le_chain_Sup | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"finite",
"is_chain"
] | `chain_Sup hu c hc ne hfin hU` is an upper bound of the chain `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_gt (v : partial_refinement u s) (hs : is_closed s) (i : ι) (hi : i ∉ v.carrier) :
∃ v' : partial_refinement u s, v < v' | begin
have I : s ∩ (⋂ j ≠ i, (v j)ᶜ) ⊆ v i,
{ simp only [subset_def, mem_inter_iff, mem_Inter, and_imp],
intros x hxs H,
rcases mem_Union.1 (v.subset_Union hxs) with ⟨j, hj⟩,
exact (em (j = i)).elim (λ h, h ▸ hj) (λ h, (H j h hj).elim) },
have C : is_closed (s ∩ (⋂ j ≠ i, (v j)ᶜ)),
from is_closed.... | lemma | shrinking_lemma.partial_refinement.exists_gt | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"and_imp",
"em",
"is_closed",
"is_closed.inter",
"is_closed_bInter",
"ne_of_mem_of_not_mem",
"normal_exists_closure_subset",
"not_or_distrib",
"update_noteq",
"update_same"
] | If `s` is a closed set, `v` is a partial refinement, and `i` is an index such that
`i ∉ v.carrier`, then there exists a partial refinement that is strictly greater than `v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_subset_Union_closure_subset (hs : is_closed s) (uo : ∀ i, is_open (u i))
(uf : ∀ x ∈ s, {i | x ∈ u i}.finite) (us : s ⊆ ⋃ i, u i) :
∃ v : ι → set X, s ⊆ Union v ∧ (∀ i, is_open (v i)) ∧ ∀ i, closure (v i) ⊆ u i | begin
classical,
haveI : nonempty (partial_refinement u s) := ⟨⟨u, ∅, uo, us, λ _, false.elim, λ _ _, rfl⟩⟩,
have : ∀ c : set (partial_refinement u s), is_chain (≤) c → c.nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub,
from λ c hc ne, ⟨partial_refinement.chain_Sup c hc ne uf us,
λ v hv, partial_refinement.le_chain_Su... | lemma | exists_subset_Union_closure_subset | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"closure",
"finite",
"is_chain",
"is_closed",
"is_open",
"zorn_nonempty_partial_order"
] | Shrinking lemma. A point-finite open cover of a closed subset of a normal space can be "shrunk"
to a new open cover so that the closure of each new open set is contained in the corresponding
original open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_subset_Union_closed_subset (hs : is_closed s) (uo : ∀ i, is_open (u i))
(uf : ∀ x ∈ s, {i | x ∈ u i}.finite) (us : s ⊆ ⋃ i, u i) :
∃ v : ι → set X, s ⊆ Union v ∧ (∀ i, is_closed (v i)) ∧ ∀ i, v i ⊆ u i | let ⟨v, hsv, hvo, hv⟩ := exists_subset_Union_closure_subset hs uo uf us
in ⟨λ i, closure (v i), subset.trans hsv (Union_mono $ λ i, subset_closure),
λ i, is_closed_closure, hv⟩ | lemma | exists_subset_Union_closed_subset | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"closure",
"exists_subset_Union_closure_subset",
"finite",
"is_closed",
"is_closed_closure",
"is_open",
"subset_closure"
] | Shrinking lemma. A point-finite open cover of a closed subset of a normal space can be "shrunk"
to a new closed cover so that each new closed set is contained in the corresponding original open
set. See also `exists_subset_Union_closure_subset` for a stronger statement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_Union_eq_closure_subset (uo : ∀ i, is_open (u i)) (uf : ∀ x, {i | x ∈ u i}.finite)
(uU : (⋃ i, u i) = univ) :
∃ v : ι → set X, Union v = univ ∧ (∀ i, is_open (v i)) ∧ ∀ i, closure (v i) ⊆ u i | let ⟨v, vU, hv⟩ := exists_subset_Union_closure_subset is_closed_univ uo (λ x _, uf x) uU.ge
in ⟨v, univ_subset_iff.1 vU, hv⟩ | lemma | exists_Union_eq_closure_subset | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"closure",
"exists_subset_Union_closure_subset",
"finite",
"is_closed_univ",
"is_open"
] | Shrinking lemma. A point-finite open cover of a closed subset of a normal space can be "shrunk"
to a new open cover so that the closure of each new open set is contained in the corresponding
original open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_Union_eq_closed_subset (uo : ∀ i, is_open (u i)) (uf : ∀ x, {i | x ∈ u i}.finite)
(uU : (⋃ i, u i) = univ) :
∃ v : ι → set X, Union v = univ ∧ (∀ i, is_closed (v i)) ∧ ∀ i, v i ⊆ u i | let ⟨v, vU, hv⟩ := exists_subset_Union_closed_subset is_closed_univ uo (λ x _, uf x) uU.ge
in ⟨v, univ_subset_iff.1 vU, hv⟩ | lemma | exists_Union_eq_closed_subset | topology | src/topology/shrinking_lemma.lean | [
"topology.separation"
] | [
"exists_subset_Union_closed_subset",
"finite",
"is_closed",
"is_closed_univ",
"is_open"
] | Shrinking lemma. A point-finite open cover of a closed subset of a normal space can be "shrunk"
to a new closed cover so that each of the new closed sets is contained in the corresponding
original open set. See also `exists_Union_eq_closure_subset` for a stronger statement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_generic_point (x : α) (S : set α) : Prop | closure ({x} : set α) = S | def | is_generic_point | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure"
] | `x` is a generic point of `S` if `S` is the closure of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_generic_point_def {x : α} {S : set α} : is_generic_point x S ↔ closure ({x} : set α) = S | iff.rfl | lemma | is_generic_point_def | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure",
"is_generic_point"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_generic_point.def {x : α} {S : set α} (h : is_generic_point x S) :
closure ({x} : set α) = S | h | lemma | is_generic_point.def | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure",
"is_generic_point"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_generic_point_closure {x : α} : is_generic_point x (closure ({x} : set α)) | refl _ | lemma | is_generic_point_closure | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure",
"is_generic_point"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_generic_point_iff_specializes :
is_generic_point x S ↔ ∀ y, x ⤳ y ↔ y ∈ S | by simp only [specializes_iff_mem_closure, is_generic_point, set.ext_iff] | lemma | is_generic_point_iff_specializes | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_generic_point",
"set.ext_iff",
"specializes_iff_mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_mem (h : is_generic_point x S) : x ⤳ y ↔ y ∈ S | is_generic_point_iff_specializes.1 h y | lemma | is_generic_point.specializes_iff_mem | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_generic_point"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes (h : is_generic_point x S) (h' : y ∈ S) : x ⤳ y | h.specializes_iff_mem.2 h' | lemma | is_generic_point.specializes | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_generic_point",
"specializes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem (h : is_generic_point x S) : x ∈ S | h.specializes_iff_mem.1 specializes_rfl | lemma | is_generic_point.mem | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_generic_point",
"specializes_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed (h : is_generic_point x S) : is_closed S | h.def ▸ is_closed_closure | lemma | is_generic_point.is_closed | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_closed",
"is_closed_closure",
"is_generic_point"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible (h : is_generic_point x S) : is_irreducible S | h.def ▸ is_irreducible_singleton.closure | lemma | is_generic_point.is_irreducible | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_generic_point",
"is_irreducible"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq [t0_space α] (h : is_generic_point x S) (h' : is_generic_point y S) : x = y | ((h.specializes h'.mem).antisymm (h'.specializes h.mem)).eq | lemma | is_generic_point.eq | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_generic_point",
"t0_space"
] | In a T₀ space, each set has at most one generic point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_open_set_iff (h : is_generic_point x S) (hU : is_open U) :
x ∈ U ↔ (S ∩ U).nonempty | ⟨λ h', ⟨x, h.mem, h'⟩, λ ⟨y, hyS, hyU⟩, (h.specializes hyS).mem_open hU hyU⟩ | lemma | is_generic_point.mem_open_set_iff | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_generic_point",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_iff (h : is_generic_point x S) (hU : is_open U) : disjoint S U ↔ x ∉ U | by rw [h.mem_open_set_iff hU, ← not_disjoint_iff_nonempty_inter, not_not] | lemma | is_generic_point.disjoint_iff | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"disjoint",
"disjoint_iff",
"is_generic_point",
"is_open",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_set_iff (h : is_generic_point x S) (hZ : is_closed Z) :
x ∈ Z ↔ S ⊆ Z | by rw [← h.def, hZ.closure_subset_iff, singleton_subset_iff] | lemma | is_generic_point.mem_closed_set_iff | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_closed",
"is_generic_point"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image (h : is_generic_point x S) {f : α → β} (hf : continuous f) :
is_generic_point (f x) (closure (f '' S)) | begin
rw [is_generic_point_def, ← h.def, ← image_singleton],
exact subset.antisymm (closure_mono (image_subset _ subset_closure))
(closure_minimal (image_closure_subset_closure_image hf) is_closed_closure)
end | lemma | is_generic_point.image | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure",
"closure_minimal",
"closure_mono",
"continuous",
"image_closure_subset_closure_image",
"is_closed_closure",
"is_generic_point",
"is_generic_point_def",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_generic_point_iff_forall_closed (hS : is_closed S) (hxS : x ∈ S) :
is_generic_point x S ↔ ∀ Z : set α, is_closed Z → x ∈ Z → S ⊆ Z | have closure {x} ⊆ S, from closure_minimal (singleton_subset_iff.2 hxS) hS,
by simp_rw [is_generic_point, subset_antisymm_iff, this, true_and, closure, subset_sInter_iff,
mem_set_of_eq, and_imp, singleton_subset_iff] | lemma | is_generic_point_iff_forall_closed | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"and_imp",
"closure",
"closure_minimal",
"is_closed",
"is_generic_point",
"subset_antisymm_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_sober (α : Type*) [topological_space α] : Prop | (sober : ∀ {S : set α} (hS₁ : is_irreducible S) (hS₂ : is_closed S), ∃ x, is_generic_point x S) | class | quasi_sober | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_closed",
"is_generic_point",
"is_irreducible",
"topological_space"
] | A space is sober if every irreducible closed subset has a generic point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_irreducible.generic_point [quasi_sober α] {S : set α} (hS : is_irreducible S) : α | (quasi_sober.sober hS.closure is_closed_closure).some | def | is_irreducible.generic_point | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"is_closed_closure",
"is_irreducible",
"quasi_sober"
] | A generic point of the closure of an irreducible space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_irreducible.generic_point_spec [quasi_sober α] {S : set α} (hS : is_irreducible S) :
is_generic_point hS.generic_point (closure S) | (quasi_sober.sober hS.closure is_closed_closure).some_spec | lemma | is_irreducible.generic_point_spec | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure",
"is_closed_closure",
"is_generic_point",
"is_irreducible",
"quasi_sober"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_irreducible.generic_point_closure_eq [quasi_sober α] {S : set α}
(hS : is_irreducible S) : closure ({hS.generic_point} : set α) = closure S | hS.generic_point_spec | lemma | is_irreducible.generic_point_closure_eq | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure",
"is_irreducible",
"quasi_sober"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generic_point [quasi_sober α] [irreducible_space α] : α | (irreducible_space.is_irreducible_univ α).generic_point | def | generic_point | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"irreducible_space",
"irreducible_space.is_irreducible_univ",
"quasi_sober"
] | A generic point of a sober irreducible space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
generic_point_spec [quasi_sober α] [irreducible_space α] :
is_generic_point (generic_point α) ⊤ | by simpa using (irreducible_space.is_irreducible_univ α).generic_point_spec | lemma | generic_point_spec | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"generic_point",
"irreducible_space",
"irreducible_space.is_irreducible_univ",
"is_generic_point",
"quasi_sober"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generic_point_closure [quasi_sober α] [irreducible_space α] :
closure ({generic_point α} : set α) = ⊤ | generic_point_spec α | lemma | generic_point_closure | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure",
"generic_point",
"generic_point_spec",
"irreducible_space",
"quasi_sober"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generic_point_specializes [quasi_sober α] [irreducible_space α] (x : α) :
generic_point α ⤳ x | (is_irreducible.generic_point_spec _).specializes (by simp) | lemma | generic_point_specializes | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"generic_point",
"irreducible_space",
"is_irreducible.generic_point_spec",
"quasi_sober",
"specializes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_set_equiv_points [quasi_sober α] [t0_space α] :
{ s : set α | is_irreducible s ∧ is_closed s } ≃o α | { to_fun := λ s, s.prop.1.generic_point,
inv_fun := λ x, ⟨closure ({x} : set α), is_irreducible_singleton.closure, is_closed_closure⟩,
left_inv := λ s,
subtype.eq $ eq.trans (s.prop.1.generic_point_spec) $ closure_eq_iff_is_closed.mpr s.2.2,
right_inv := λ x, is_irreducible_singleton.closure.generic_point_spe... | def | irreducible_set_equiv_points | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure_closure",
"inv_fun",
"is_closed",
"is_generic_point_closure",
"is_irreducible",
"quasi_sober",
"specializes_iff_closure_subset",
"subtype.coe_le_coe",
"t0_space"
] | The closed irreducible subsets of a sober space bijects with the points of the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding.quasi_sober {f : α → β} (hf : closed_embedding f) [quasi_sober β] :
quasi_sober α | begin
constructor,
intros S hS hS',
have hS'' := hS.image f hf.continuous.continuous_on,
obtain ⟨x, hx⟩ := quasi_sober.sober hS'' (hf.is_closed_map _ hS'),
obtain ⟨y, hy, rfl⟩ := hx.mem,
use y,
change _ = _ at hx,
apply set.image_injective.mpr hf.inj,
rw [← hx, ← hf.closure_image_eq, set.image_singlet... | lemma | closed_embedding.quasi_sober | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closed_embedding",
"quasi_sober",
"set.image_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.quasi_sober {f : α → β} (hf : open_embedding f) [quasi_sober β] :
quasi_sober α | begin
constructor,
intros S hS hS',
have hS'' := hS.image f hf.continuous.continuous_on,
obtain ⟨x, hx⟩ := quasi_sober.sober hS''.closure is_closed_closure,
obtain ⟨T, hT, rfl⟩ := hf.to_inducing.is_closed_iff.mp hS',
rw set.image_preimage_eq_inter_range at hx hS'',
have hxT : x ∈ T,
{ rw ← hT.closure_eq... | lemma | open_embedding.quasi_sober | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"and.congr_left_iff",
"closure_mono",
"is_closed_closure",
"open_embedding",
"quasi_sober",
"set.image_preimage_eq_inter_range",
"set.image_singleton",
"set.inter_subset_left",
"set.mem_inter_iff",
"set.nonempty.mono",
"set.range",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quasi_sober_of_open_cover (S : set (set α)) (hS : ∀ s : S, is_open (s : set α))
[hS' : ∀ s : S, quasi_sober s] (hS'' : ⋃₀ S = ⊤) : quasi_sober α | begin
rw quasi_sober_iff,
intros t h h',
obtain ⟨x, hx⟩ := h.1,
obtain ⟨U, hU, hU'⟩ : x ∈ ⋃₀S := by { rw hS'', trivial },
haveI : quasi_sober U := hS' ⟨U, hU⟩,
have H : is_preirreducible (coe ⁻¹' t : set U) :=
h.2.preimage (hS ⟨U, hU⟩).open_embedding_subtype_coe,
replace H : is_irreducible (coe ⁻¹' t ... | lemma | quasi_sober_of_open_cover | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"closure_closure",
"closure_mono",
"continuous_subtype_coe",
"image_closure_subset_closure_image",
"is_irreducible",
"is_open",
"is_preirreducible",
"quasi_sober",
"set.image_singleton",
"set.image_subset",
"set.subset.trans",
"subset_closure",
"subset_closure_inter_of_is_preirreducible_of_i... | A space is quasi sober if it can be covered by open quasi sober subsets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t2_space.quasi_sober [t2_space α] : quasi_sober α | begin
constructor,
rintro S h -,
obtain ⟨x, rfl⟩ := is_irreducible_iff_singleton.mp h,
exact ⟨x, closure_singleton⟩
end | instance | t2_space.quasi_sober | topology | src/topology/sober.lean | [
"topology.separation"
] | [
"quasi_sober",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter_basis (α : Type u) : set (set (ultrafilter α)) | range $ λ s : set α, {u | s ∈ u} | def | ultrafilter_basis | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"ultrafilter"
] | Basis for the topology on `ultrafilter α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter_basis_is_basis :
topological_space.is_topological_basis (ultrafilter_basis α) | ⟨begin
rintros _ ⟨a, rfl⟩ _ ⟨b, rfl⟩ u ⟨ua, ub⟩,
refine ⟨_, ⟨a ∩ b, rfl⟩, inter_mem ua ub, assume v hv, ⟨_, _⟩⟩;
apply mem_of_superset hv; simp [inter_subset_right a b]
end,
eq_univ_of_univ_subset $ subset_sUnion_of_mem $
⟨univ, eq_univ_of_forall (λ u, univ_mem)⟩,
rfl⟩ | lemma | ultrafilter_basis_is_basis | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"topological_space.is_topological_basis",
"ultrafilter_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter_is_open_basic (s : set α) :
is_open {u : ultrafilter α | s ∈ u} | ultrafilter_basis_is_basis.is_open ⟨s, rfl⟩ | lemma | ultrafilter_is_open_basic | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"is_open",
"ultrafilter"
] | The basic open sets for the topology on ultrafilters are open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter_is_closed_basic (s : set α) :
is_closed {u : ultrafilter α | s ∈ u} | begin
rw ← is_open_compl_iff,
convert ultrafilter_is_open_basic sᶜ,
ext u,
exact ultrafilter.compl_mem_iff_not_mem.symm
end | lemma | ultrafilter_is_closed_basic | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"is_closed",
"is_open_compl_iff",
"ultrafilter",
"ultrafilter_is_open_basic"
] | The basic open sets for the topology on ultrafilters are also closed. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter_converges_iff {u : ultrafilter (ultrafilter α)} {x : ultrafilter α} :
↑u ≤ 𝓝 x ↔ x = mjoin u | begin
rw [eq_comm, ← ultrafilter.coe_le_coe],
change ↑u ≤ 𝓝 x ↔ ∀ s ∈ x, {v : ultrafilter α | s ∈ v} ∈ u,
simp only [topological_space.nhds_generate_from, le_infi_iff, ultrafilter_basis,
le_principal_iff, mem_set_of_eq],
split,
{ intros h a ha, exact h _ ⟨ha, a, rfl⟩ },
{ rintros h a ⟨xi, a, rfl⟩, exac... | lemma | ultrafilter_converges_iff | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"le_infi_iff",
"topological_space.nhds_generate_from",
"ultrafilter",
"ultrafilter.coe_le_coe",
"ultrafilter_basis"
] | Every ultrafilter `u` on `ultrafilter α` converges to a unique
point of `ultrafilter α`, namely `mjoin u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter_compact : compact_space (ultrafilter α) | ⟨is_compact_iff_ultrafilter_le_nhds.mpr $ assume f _,
⟨mjoin f, trivial, ultrafilter_converges_iff.mpr rfl⟩⟩ | instance | ultrafilter_compact | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"compact_space",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter.t2_space : t2_space (ultrafilter α) | t2_iff_ultrafilter.mpr $ assume x y f fx fy,
have hx : x = mjoin f, from ultrafilter_converges_iff.mp fx,
have hy : y = mjoin f, from ultrafilter_converges_iff.mp fy,
hx.trans hy.symm | instance | ultrafilter.t2_space | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"t2_space",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter_comap_pure_nhds (b : ultrafilter α) : comap pure (𝓝 b) ≤ b | begin
rw topological_space.nhds_generate_from,
simp only [comap_infi, comap_principal],
intros s hs,
rw ←le_principal_iff,
refine infi_le_of_le {u | s ∈ u} _,
refine infi_le_of_le ⟨hs, ⟨s, rfl⟩⟩ _,
exact principal_mono.2 (λ a, id)
end | lemma | ultrafilter_comap_pure_nhds | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"infi_le_of_le",
"topological_space.nhds_generate_from",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter_pure_injective : function.injective (pure : α → ultrafilter α) | begin
intros x y h,
have : {x} ∈ (pure x : ultrafilter α) := singleton_mem_pure,
rw h at this,
exact (mem_singleton_iff.mp (mem_pure.mp this)).symm
end | lemma | ultrafilter_pure_injective | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range_pure : dense_range (pure : α → ultrafilter α) | λ x, mem_closure_iff_ultrafilter.mpr
⟨x.map pure, range_mem_map, ultrafilter_converges_iff.mpr (bind_pure x).symm⟩ | lemma | dense_range_pure | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"dense_range",
"ultrafilter"
] | The range of `pure : α → ultrafilter α` is dense in `ultrafilter α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
induced_topology_pure :
topological_space.induced (pure : α → ultrafilter α) ultrafilter.topological_space = ⊥ | begin
apply eq_bot_of_singletons_open,
intros x,
use [{u : ultrafilter α | {x} ∈ u}, ultrafilter_is_open_basic _],
simp,
end | lemma | induced_topology_pure | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"eq_bot_of_singletons_open",
"topological_space.induced",
"ultrafilter",
"ultrafilter_is_open_basic"
] | The map `pure : α → ultra_filter α` induces on `α` the discrete topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_inducing_pure : @dense_inducing _ _ ⊥ _ (pure : α → ultrafilter α) | by letI : topological_space α := ⊥; exact ⟨⟨induced_topology_pure.symm⟩, dense_range_pure⟩ | lemma | dense_inducing_pure | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"dense_inducing",
"topological_space",
"ultrafilter"
] | `pure : α → ultrafilter α` defines a dense inducing of `α` in `ultrafilter α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_embedding_pure : @dense_embedding _ _ ⊥ _ (pure : α → ultrafilter α) | by letI : topological_space α := ⊥ ;
exact { inj := ultrafilter_pure_injective, ..dense_inducing_pure } | lemma | dense_embedding_pure | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"dense_embedding",
"dense_inducing_pure",
"topological_space",
"ultrafilter",
"ultrafilter_pure_injective"
] | `pure : α → ultrafilter α` defines a dense embedding of `α` in `ultrafilter α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter.extend (f : α → γ) : ultrafilter α → γ | by letI : topological_space α := ⊥; exact dense_inducing_pure.extend f | def | ultrafilter.extend | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"topological_space",
"ultrafilter"
] | The extension of a function `α → γ` to a function `ultrafilter α → γ`.
When `γ` is a compact Hausdorff space it will be continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ultrafilter_extend_extends (f : α → γ) : ultrafilter.extend f ∘ pure = f | begin
letI : topological_space α := ⊥,
haveI : discrete_topology α := ⟨rfl⟩,
exact funext (dense_inducing_pure.extend_eq continuous_of_discrete_topology)
end | lemma | ultrafilter_extend_extends | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"continuous_of_discrete_topology",
"discrete_topology",
"topological_space",
"ultrafilter.extend"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_ultrafilter_extend (f : α → γ) : continuous (ultrafilter.extend f) | have ∀ (b : ultrafilter α), ∃ c, tendsto f (comap pure (𝓝 b)) (𝓝 c) := assume b,
-- b.map f is an ultrafilter on γ, which is compact, so it converges to some c in γ.
let ⟨c, _, h⟩ := is_compact_univ.ultrafilter_le_nhds (b.map f)
(by rw [le_principal_iff]; exact univ_mem) in
⟨c, le_trans (map_mono (ultrafilt... | lemma | continuous_ultrafilter_extend | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"continuous",
"normal_of_compact_t2",
"normal_space",
"topological_space",
"ultrafilter",
"ultrafilter.extend",
"ultrafilter_comap_pure_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ultrafilter_extend_eq_iff {f : α → γ} {b : ultrafilter α} {c : γ} :
ultrafilter.extend f b = c ↔ ↑(b.map f) ≤ 𝓝 c | ⟨assume h, begin
-- Write b as an ultrafilter limit of pure ultrafilters, and use
-- the facts that ultrafilter.extend is a continuous extension of f.
let b' : ultrafilter (ultrafilter α) := b.map pure,
have t : ↑b' ≤ 𝓝 b,
from ultrafilter_converges_iff.mpr (bind_pure _).symm,
rw ←h,
have := (co... | lemma | ultrafilter_extend_eq_iff | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"continuous_ultrafilter_extend",
"le_rfl",
"topological_space",
"ultrafilter",
"ultrafilter.extend",
"ultrafilter_comap_pure_nhds",
"ultrafilter_extend_extends"
] | The value of `ultrafilter.extend f` on an ultrafilter `b` is the
unique limit of the ultrafilter `b.map f` in `γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stone_cech_setoid : setoid (ultrafilter α) | { r := λ x y, ∀ (γ : Type u) [topological_space γ], by exactI
∀ [t2_space γ] [compact_space γ] (f : α → γ) (hf : continuous f),
ultrafilter.extend f x = ultrafilter.extend f y,
iseqv :=
⟨assume x γ tγ h₁ h₂ f hf, rfl,
assume x y xy γ tγ h₁ h₂ f hf, by exactI (xy γ f hf).symm,
assume x y z xy yz ... | instance | stone_cech_setoid | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"compact_space",
"continuous",
"t2_space",
"topological_space",
"ultrafilter",
"ultrafilter.extend"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stone_cech : Type u | quotient (stone_cech_setoid α) | def | stone_cech | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"stone_cech_setoid"
] | The Stone-Čech compactification of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stone_cech_unit (x : α) : stone_cech α | ⟦pure x⟧ | def | stone_cech_unit | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"stone_cech"
] | The natural map from α to its Stone-Čech compactification. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range_stone_cech_unit : dense_range (stone_cech_unit : α → stone_cech α) | dense_range_pure.quotient | lemma | dense_range_stone_cech_unit | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"dense_range",
"stone_cech",
"stone_cech_unit"
] | The image of stone_cech_unit is dense. (But stone_cech_unit need
not be an embedding, for example if α is not Hausdorff.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stone_cech_extend : stone_cech α → γ | quotient.lift (ultrafilter.extend f) (λ x y xy, xy γ f hf) | def | stone_cech_extend | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"stone_cech",
"ultrafilter.extend"
] | The extension of a continuous function from α to a compact
Hausdorff space γ to the Stone-Čech compactification of α. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stone_cech_extend_extends : stone_cech_extend hf ∘ stone_cech_unit = f | ultrafilter_extend_extends f | lemma | stone_cech_extend_extends | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"stone_cech_extend",
"stone_cech_unit",
"ultrafilter_extend_extends"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_stone_cech_extend : continuous (stone_cech_extend hf) | continuous_quot_lift _ (continuous_ultrafilter_extend f) | lemma | continuous_stone_cech_extend | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"continuous",
"continuous_quot_lift",
"continuous_ultrafilter_extend",
"stone_cech_extend"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stone_cech_hom_ext {g₁ g₂ : stone_cech α → γ'}
(h₁ : continuous g₁) (h₂ : continuous g₂)
(h : g₁ ∘ stone_cech_unit = g₂ ∘ stone_cech_unit) : g₁ = g₂ | begin
apply continuous.ext_on dense_range_stone_cech_unit h₁ h₂,
rintros x ⟨x, rfl⟩,
apply (congr_fun h x)
end | lemma | stone_cech_hom_ext | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"continuous",
"continuous.ext_on",
"dense_range_stone_cech_unit",
"stone_cech",
"stone_cech_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convergent_eqv_pure {u : ultrafilter α} {x : α} (ux : ↑u ≤ 𝓝 x) : u ≈ pure x | assume γ tγ h₁ h₂ f hf, begin
resetI,
transitivity f x, swap, symmetry,
all_goals { refine ultrafilter_extend_eq_iff.mpr (le_trans (map_mono _) (hf.tendsto _)) },
{ apply pure_le_nhds }, { exact ux }
end | lemma | convergent_eqv_pure | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"pure_le_nhds",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_stone_cech_unit : continuous (stone_cech_unit : α → stone_cech α) | continuous_iff_ultrafilter.mpr $ λ x g gx,
have ↑(g.map pure) ≤ 𝓝 g,
by rw ultrafilter_converges_iff; exact (bind_pure _).symm,
have (g.map stone_cech_unit : filter (stone_cech α)) ≤ 𝓝 ⟦g⟧, from
continuous_at_iff_ultrafilter.mp (continuous_quotient_mk.tendsto g) _ this,
by rwa (show ⟦g⟧ = ⟦pure x⟧, from... | lemma | continuous_stone_cech_unit | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"continuous",
"convergent_eqv_pure",
"filter",
"stone_cech",
"stone_cech_unit",
"ultrafilter_converges_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stone_cech.t2_space : t2_space (stone_cech α) | begin
rw t2_iff_ultrafilter,
rintros ⟨x⟩ ⟨y⟩ g gx gy,
apply quotient.sound,
intros γ tγ h₁ h₂ f hf,
resetI,
let ff := stone_cech_extend hf,
change ff ⟦x⟧ = ff ⟦y⟧,
have lim := λ (z : ultrafilter α) (gz : (g : filter (stone_cech α)) ≤ 𝓝 ⟦z⟧),
((continuous_stone_cech_extend hf).tendsto _).mono_left g... | instance | stone_cech.t2_space | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"continuous_stone_cech_extend",
"filter",
"lim",
"stone_cech",
"stone_cech_extend",
"t2_iff_ultrafilter",
"t2_space",
"tendsto_nhds_unique",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stone_cech.compact_space : compact_space (stone_cech α) | quotient.compact_space | instance | stone_cech.compact_space | topology | src/topology/stone_cech.lean | [
"topology.bases",
"topology.dense_embedding"
] | [
"compact_space",
"quotient.compact_space",
"stone_cech"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact (s : set α) | ∀ ⦃f⦄ [ne_bot f], f ≤ 𝓟 s → ∃ a ∈ s, cluster_pt a f | def | 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_pt"
] | A set `s` is compact if for every nontrivial filter `f` that contains `s`,
there exists `a ∈ s` such that every set of `f` meets every neighborhood of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.compl_mem_sets (hs : is_compact s) {f : filter α} (hf : ∀ a ∈ s, sᶜ ∈ 𝓝 a ⊓ f) :
sᶜ ∈ f | begin
contrapose! hf,
simp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc, ← exists_prop] at hf ⊢,
exact @hs _ hf inf_le_right
end | lemma | is_compact.compl_mem_sets | 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",
"exists_prop",
"filter",
"inf_assoc",
"inf_le_right",
"is_compact"
] | The complement to a compact set belongs to a filter `f` if it belongs to each filter
`𝓝 a ⊓ f`, `a ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.compl_mem_sets_of_nhds_within (hs : is_compact s) {f : filter α}
(hf : ∀ a ∈ s, ∃ t ∈ 𝓝[s] a, tᶜ ∈ f) :
sᶜ ∈ f | begin
refine hs.compl_mem_sets (λ a ha, _),
rcases hf a ha with ⟨t, ht, hst⟩,
replace ht := mem_inf_principal.1 ht,
apply mem_inf_of_inter ht hst,
rintros x ⟨h₁, h₂⟩ hs,
exact h₂ (h₁ hs)
end | lemma | is_compact.compl_mem_sets_of_nhds_within | 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"
] | The complement to a compact set belongs to a filter `f` if each `a ∈ s` has a neighborhood `t`
within `s` such that `tᶜ` belongs to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.induction_on {s : set α} (hs : is_compact s) {p : set α → Prop} (he : p ∅)
(hmono : ∀ ⦃s t⦄, s ⊆ t → p t → p s) (hunion : ∀ ⦃s t⦄, p s → p t → p (s ∪ t))
(hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t) :
p s | let f : filter α :=
{ sets := {t | p tᶜ},
univ_sets := by simpa,
sets_of_superset := λ t₁ t₂ ht₁ ht, hmono (compl_subset_compl.2 ht) ht₁,
inter_sets := λ t₁ t₂ ht₁ ht₂, by simp [compl_inter, hunion ht₁ ht₂] } in
have sᶜ ∈ f, from hs.compl_mem_sets_of_nhds_within (by simpa using hnhds),
by simpa | lemma | is_compact.induction_on | 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"
] | If `p : set α → Prop` is stable under restriction and union, and each point `x`
of a compact set `s` has a neighborhood `t` within `s` such that `p t`, then `p s` holds. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.inter_right (hs : is_compact s) (ht : is_closed t) :
is_compact (s ∩ t) | begin
introsI f hnf hstf,
obtain ⟨a, hsa, ha⟩ : ∃ a ∈ s, cluster_pt a f :=
hs (le_trans hstf (le_principal_iff.2 (inter_subset_left _ _))),
have : a ∈ t :=
(ht.mem_of_nhds_within_ne_bot $ ha.mono $
le_trans hstf (le_principal_iff.2 (inter_subset_right _ _))),
exact ⟨a, ⟨hsa, this⟩, ha⟩
end | lemma | is_compact.inter_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"
] | [
"cluster_pt",
"is_closed",
"is_compact"
] | The intersection of a compact set and a closed set is a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.inter_left (ht : is_compact t) (hs : is_closed s) : is_compact (s ∩ t) | inter_comm t s ▸ ht.inter_right hs | lemma | is_compact.inter_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"
] | [
"is_closed",
"is_compact"
] | The intersection of a closed set and a compact set is a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.diff (hs : is_compact s) (ht : is_open t) : is_compact (s \ t) | hs.inter_right (is_closed_compl_iff.mpr ht) | lemma | is_compact.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_compact",
"is_open"
] | The set difference of a compact set and an open set is a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_of_is_closed_subset (hs : is_compact s) (ht : is_closed t) (h : t ⊆ s) :
is_compact t | inter_eq_self_of_subset_right h ▸ hs.inter_right ht | lemma | is_compact_of_is_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"
] | [
"is_closed",
"is_compact"
] | A closed subset of a compact set is a compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.image_of_continuous_on {f : α → β} (hs : is_compact s) (hf : continuous_on f s) :
is_compact (f '' s) | begin
intros l lne ls,
have : ne_bot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_ne_bot_of_image_mem lne (le_principal_iff.1 ls),
obtain ⟨a, has, ha⟩ : ∃ a ∈ s, cluster_pt a (l.comap f ⊓ 𝓟 s) := @@hs this inf_le_right,
use [f a, mem_image_of_mem f has],
have : tendsto f (𝓝 a ⊓ (comap f l ⊓ 𝓟 s)) (𝓝 (f a... | lemma | is_compact.image_of_continuous_on | 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",
"continuous_on",
"inf_le_right",
"is_compact",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.image {f : α → β} (hs : is_compact s) (hf : continuous f) :
is_compact (f '' s) | hs.image_of_continuous_on hf.continuous_on | lemma | 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"
] | [
"continuous",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.adherence_nhdset {f : filter α}
(hs : is_compact s) (hf₂ : f ≤ 𝓟 s) (ht₁ : is_open t) (ht₂ : ∀ a ∈ s, cluster_pt a f → a ∈ t) :
t ∈ f | classical.by_cases mem_of_eq_bot $
assume : f ⊓ 𝓟 tᶜ ≠ ⊥,
let ⟨a, ha, (hfa : cluster_pt a $ f ⊓ 𝓟 tᶜ)⟩ := @@hs ⟨this⟩ $ inf_le_of_left_le hf₂ in
have a ∈ t,
from ht₂ a ha (hfa.of_inf_left),
have tᶜ ∩ t ∈ 𝓝[tᶜ] a,
from inter_mem_nhds_within _ (is_open.mem_nhds ht₁ this),
have A : 𝓝[tᶜ] a = ⊥,
f... | lemma | is_compact.adherence_nhdset | 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",
"inf_le_of_left_le",
"inter_mem_nhds_within",
"is_compact",
"is_open",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_iff_ultrafilter_le_nhds :
is_compact s ↔ (∀ f : ultrafilter α, ↑f ≤ 𝓟 s → ∃ a ∈ s, ↑f ≤ 𝓝 a) | begin
refine (forall_ne_bot_le_iff _).trans _,
{ rintro f g hle ⟨a, has, haf⟩,
exact ⟨a, has, haf.mono hle⟩ },
{ simp only [ultrafilter.cluster_pt_iff] }
end | lemma | is_compact_iff_ultrafilter_le_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",
"ultrafilter",
"ultrafilter.cluster_pt_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.elim_directed_cover {ι : Type v} [hι : nonempty ι] (hs : is_compact s)
(U : ι → set α) (hUo : ∀ i, is_open (U i)) (hsU : s ⊆ ⋃ i, U i) (hdU : directed (⊆) U) :
∃ i, s ⊆ U i | hι.elim $ λ i₀, is_compact.induction_on hs ⟨i₀, empty_subset _⟩
(λ s₁ s₂ hs ⟨i, hi⟩, ⟨i, subset.trans hs hi⟩)
(λ s₁ s₂ ⟨i, hi⟩ ⟨j, hj⟩, let ⟨k, hki, hkj⟩ := hdU i j in
⟨k, union_subset (subset.trans hi hki) (subset.trans hj hkj)⟩)
(λ x hx, let ⟨i, hi⟩ := mem_Union.1 (hsU hx) in
⟨U i, mem_nhds_within_of_me... | lemma | is_compact.elim_directed_cover | 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"
] | [
"directed",
"is_compact",
"is_compact.induction_on",
"is_open",
"is_open.mem_nhds",
"mem_nhds_within_of_mem_nhds"
] | For every open directed cover of a compact set, there exists a single element of the
cover which itself includes the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.elim_finite_subcover {ι : Type v} (hs : is_compact s)
(U : ι → set α) (hUo : ∀ i, is_open (U i)) (hsU : s ⊆ ⋃ i, U i) :
∃ t : finset ι, s ⊆ ⋃ i ∈ t, U i | hs.elim_directed_cover _ (λ t, is_open_bUnion $ λ i _, hUo i) (Union_eq_Union_finset U ▸ hsU)
(directed_of_sup $ λ t₁ t₂ h, bUnion_subset_bUnion_left h) | lemma | is_compact.elim_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"
] | [
"directed_of_sup",
"finset",
"is_compact",
"is_open",
"is_open_bUnion"
] | For every open cover of a compact set, there exists a finite subcover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.elim_nhds_subcover' (hs : is_compact s) (U : Π x ∈ s, set α)
(hU : ∀ x ∈ s, U x ‹x ∈ s› ∈ 𝓝 x) :
∃ t : finset s, s ⊆ ⋃ x ∈ t, U (x : s) x.2 | (hs.elim_finite_subcover (λ x : s, interior (U x x.2)) (λ x, is_open_interior)
(λ x hx, mem_Union.2 ⟨⟨x, hx⟩, mem_interior_iff_mem_nhds.2 $ hU _ _⟩)).imp $ λ t ht,
subset.trans ht $ Union₂_mono $ λ _ _, interior_subset | lemma | is_compact.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"
] | [
"finset",
"interior",
"interior_subset",
"is_compact",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.elim_nhds_subcover (hs : is_compact s) (U : α → set α) (hU : ∀ x ∈ s, U x ∈ 𝓝 x) :
∃ t : finset α, (∀ x ∈ t, x ∈ s) ∧ s ⊆ ⋃ x ∈ t, U x | let ⟨t, ht⟩ := hs.elim_nhds_subcover' (λ x _, U x) hU
in ⟨t.image coe, λ x hx, let ⟨y, hyt, hyx⟩ := finset.mem_image.1 hx in hyx ▸ y.2,
by rwa finset.set_bUnion_finset_image⟩ | lemma | is_compact.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"
] | [
"finset",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.disjoint_nhds_set_left {l : filter α} (hs : is_compact s) :
disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, disjoint (𝓝 x) l | begin
refine ⟨λ h x hx, h.mono_left $ nhds_le_nhds_set hx, λ H, _⟩,
choose! U hxU hUl using λ x hx, (nhds_basis_opens x).disjoint_iff_left.1 (H x hx),
choose hxU hUo using hxU,
rcases hs.elim_nhds_subcover U (λ x hx, (hUo x hx).mem_nhds (hxU x hx)) with ⟨t, hts, hst⟩,
refine (has_basis_nhds_set _).disjoint_if... | lemma | is_compact.disjoint_nhds_set_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"
] | [
"disjoint",
"filter",
"has_basis_nhds_set",
"is_compact",
"nhds_basis_opens",
"nhds_le_nhds_set"
] | The neighborhood filter of a compact set is disjoint with a filter `l` if and only if the
neighborhood filter of each point of this set is disjoint with `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.disjoint_nhds_set_right {l : filter α} (hs : is_compact s) :
disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, disjoint l (𝓝 x) | by simpa only [disjoint.comm] using hs.disjoint_nhds_set_left | lemma | is_compact.disjoint_nhds_set_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"
] | [
"disjoint",
"disjoint.comm",
"filter",
"is_compact"
] | A filter `l` is disjoint with the neighborhood filter of a compact set if and only if it is
disjoint with the neighborhood filter of each point of this set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.elim_directed_family_closed {ι : Type v} [hι : nonempty ι] (hs : is_compact s)
(Z : ι → set α) (hZc : ∀ i, is_closed (Z i)) (hsZ : s ∩ (⋂ i, Z i) = ∅) (hdZ : directed (⊇) Z) :
∃ i : ι, s ∩ Z i = ∅ | let ⟨t, ht⟩ := hs.elim_directed_cover (compl ∘ Z) (λ i, (hZc i).is_open_compl)
(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)
(hdZ.mono_comp _ $ λ _ _, compl_subset_compl.mpr)
in
⟨t, by simpa... | lemma | is_compact.elim_directed_family_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"
] | [
"directed",
"exists_prop",
"is_closed",
"is_compact",
"not_and",
"not_forall"
] | For every directed family of closed sets whose intersection avoids a compact set,
there exists a single element of the family which itself avoids this compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.elim_finite_subfamily_closed {s : set α} {ι : Type v} (hs : is_compact s)
(Z : ι → set α) (hZc : ∀ i, is_closed (Z i)) (hsZ : s ∩ (⋂ i, Z i) = ∅) :
∃ t : finset ι, s ∩ (⋂ i ∈ t, Z i) = ∅ | let ⟨t, ht⟩ := hs.elim_finite_subcover (λ i, (Z i)ᶜ) (λ i, (hZc i).is_open_compl)
(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
⟨t, by simpa only [subset_def, not_forall, eq_empty_iff_foral... | lemma | is_compact.elim_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"
] | [
"exists_prop",
"finset",
"is_closed",
"is_compact",
"not_and",
"not_forall"
] | For every family of closed sets whose intersection avoids a compact set,
there exists a finite subfamily whose intersection avoids this compact set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_finite.finite_nonempty_inter_compact {ι : Type*} {f : ι → set α}
(hf : locally_finite f) {s : set α} (hs : is_compact s) :
{i | (f i ∩ s).nonempty}.finite | begin
choose U hxU hUf using hf,
rcases hs.elim_nhds_subcover U (λ x _, hxU x) with ⟨t, -, hsU⟩,
refine (t.finite_to_set.bUnion (λ x _, hUf x)).subset _,
rintro i ⟨x, hx⟩,
rcases mem_Union₂.1 (hsU hx.2) with ⟨c, hct, hcx⟩,
exact mem_bUnion hct ⟨x, hx.1, hcx⟩
end | lemma | locally_finite.finite_nonempty_inter_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"
] | [
"finite",
"is_compact",
"locally_finite"
] | If `s` is a compact set in a topological space `α` and `f : ι → set α` is a locally finite
family of sets, then `f i ∩ s` is nonempty only for a finitely many `i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.inter_Inter_nonempty {s : set α} {ι : Type v} (hs : is_compact s)
(Z : ι → set α) (hZc : ∀ i, is_closed (Z i)) (hsZ : ∀ t : finset ι, (s ∩ ⋂ i ∈ t, Z i).nonempty) :
(s ∩ ⋂ i, Z i).nonempty | begin
simp only [nonempty_iff_ne_empty] at hsZ ⊢,
apply mt (hs.elim_finite_subfamily_closed Z hZc), push_neg, exact hsZ
end | lemma | is_compact.inter_Inter_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"
] | [
"finset",
"is_closed",
"is_compact"
] | To show that a compact set intersects the intersection of a family of closed sets,
it is sufficient to show that it intersects every finite subfamily. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.nonempty_Inter_of_directed_nonempty_compact_closed
{ι : Type v} [hι : nonempty ι] (Z : ι → set α) (hZd : directed (⊇) Z)
(hZn : ∀ i, (Z i).nonempty) (hZc : ∀ i, is_compact (Z i)) (hZcl : ∀ i, is_closed (Z i)) :
(⋂ i, Z i).nonempty | begin
let i₀ := hι.some,
suffices : (Z i₀ ∩ ⋂ i, Z i).nonempty,
by rwa inter_eq_right_iff_subset.mpr (Inter_subset _ i₀) at this,
simp only [nonempty_iff_ne_empty] at hZn ⊢,
apply mt ((hZc i₀).elim_directed_family_closed Z hZcl),
push_neg,
simp only [← nonempty_iff_ne_empty] at hZn ⊢,
refine ⟨hZd, λ i... | lemma | is_compact.nonempty_Inter_of_directed_nonempty_compact_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"
] | [
"directed",
"is_closed",
"is_compact"
] | Cantor's intersection theorem:
the intersection of a directed family of nonempty compact closed sets is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.nonempty_Inter_of_sequence_nonempty_compact_closed
(Z : ℕ → set α) (hZd : ∀ i, Z (i+1) ⊆ Z i)
(hZn : ∀ i, (Z i).nonempty) (hZ0 : is_compact (Z 0)) (hZcl : ∀ i, is_closed (Z i)) :
(⋂ i, Z i).nonempty | have Zmono : antitone Z := antitone_nat_of_succ_le hZd,
have hZd : directed (⊇) Z, from directed_of_sup Zmono,
have ∀ i, Z i ⊆ Z 0, from assume i, Zmono $ zero_le i,
have hZc : ∀ i, is_compact (Z i),
from assume i, is_compact_of_is_closed_subset hZ0 (hZcl i) (this i),
is_compact.nonempty_Inter_of_directed_nonempty_co... | lemma | is_compact.nonempty_Inter_of_sequence_nonempty_compact_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"
] | [
"antitone",
"antitone_nat_of_succ_le",
"directed",
"directed_of_sup",
"is_closed",
"is_compact",
"is_compact.nonempty_Inter_of_directed_nonempty_compact_closed",
"is_compact_of_is_closed_subset"
] | Cantor's intersection theorem for sequences indexed by `ℕ`:
the intersection of a decreasing sequence of nonempty compact closed sets is nonempty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact.elim_finite_subcover_image {b : set ι} {c : ι → set α}
(hs : is_compact s) (hc₁ : ∀ i ∈ b, is_open (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i) :
∃ b' ⊆ b, set.finite b' ∧ s ⊆ ⋃ i ∈ b', c i | begin
rcases hs.elim_finite_subcover (λ i, c i : b → set α) _ _ with ⟨d, hd⟩;
[skip, simpa using hc₁, simpa using hc₂],
refine ⟨↑(d.image coe), _, finset.finite_to_set _, _⟩,
{ simp },
{ rwa [finset.coe_image, bUnion_image] }
end | lemma | is_compact.elim_finite_subcover_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"
] | [
"finset.coe_image",
"finset.finite_to_set",
"is_compact",
"is_open",
"set.finite"
] | For every open cover of a compact set, there exists a finite subcover. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_of_finite_subfamily_closed
(h : 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) = ∅)) :
is_compact s | assume f hfn hfs, classical.by_contradiction $ assume : ¬ (∃ x ∈ s, cluster_pt x f),
have hf : ∀ x ∈ s, 𝓝 x ⊓ f = ⊥,
by simpa only [cluster_pt, not_exists, not_not, ne_bot_iff],
have ¬ ∃ x ∈ s, ∀ t ∈ f.sets, x ∈ closure t,
from assume ⟨x, hxs, hx⟩,
have ∅ ∈ 𝓝 x ⊓ f, by rw [empty_mem_iff_bot, hf x hxs]... | theorem | is_compact_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"
] | [
"closure",
"closure_eq_cluster_pts",
"cluster_pt",
"finset",
"inter_mem_nhds_within",
"is_closed",
"is_closed_closure",
"is_compact",
"not_exists",
"not_forall",
"not_not",
"subset_closure"
] | A set `s` is compact 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 |
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