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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noetherian_space.finite_irreducible_components [noetherian_space α] :
(irreducible_components α).finite | begin
classical,
obtain ⟨S, hS₁, hS₂⟩ := noetherian_space.exists_finset_irreducible (⊤ : closeds α),
suffices : irreducible_components α ⊆ coe '' (S : set $ closeds α),
{ exact set.finite.subset ((set.finite.intro infer_instance).image _) this },
intros K hK,
obtain ⟨z, hz, hz'⟩ : ∃ (z : set α) (H : z ∈ fin... | lemma | topological_space.noetherian_space.finite_irreducible_components | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"and_imp",
"exists_prop",
"finite",
"finset.image",
"finset.mem_image",
"forall_exists_index",
"irreducible_components",
"set.finite.subset",
"set.subset_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ωSup {α : Type u} [preorder α] (c : chain α) (x : α) : Prop | (∀ i, c i ≤ x) ∧ (∀ y, (∀ i, c i ≤ y) → x ≤ y) | def | Scott.is_ωSup | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [] | `x` is an `ω`-Sup of a chain `c` if it is the least upper bound of the range of `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_ωSup_iff_is_lub {α : Type u} [preorder α] {c : chain α} {x : α} :
is_ωSup c x ↔ is_lub (range c) x | by simp [is_ωSup, is_lub, is_least, upper_bounds, lower_bounds] | lemma | Scott.is_ωSup_iff_is_lub | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"is_least",
"is_lub",
"lower_bounds",
"upper_bounds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open (s : set α) : Prop | continuous' (λ x, x ∈ s) | def | Scott.is_open | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"is_open"
] | The characteristic function of open sets is monotone and preserves
the limits of chains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_univ : is_open α univ | ⟨λ x y h hx, mem_univ _, @complete_lattice.top_continuous α Prop _ _⟩ | theorem | Scott.is_open_univ | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"complete_lattice.top_continuous",
"is_open",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.inter (s t : set α) : is_open α s → is_open α t → is_open α (s ∩ t) | complete_lattice.inf_continuous' | theorem | Scott.is_open.inter | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"complete_lattice.inf_continuous'",
"is_open",
"is_open.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_sUnion (s : set (set α)) (hs : ∀t∈s, is_open α t) : is_open α (⋃₀ s) | begin
simp only [is_open] at hs ⊢,
convert complete_lattice.Sup_continuous' (set_of ⁻¹' s) _,
{ ext1 x,
simp only [Sup_apply, set_of_bijective.surjective.exists, exists_prop, mem_preimage,
set_coe.exists, supr_Prop_eq, mem_set_of_eq, subtype.coe_mk, mem_sUnion] },
{ intros p hp,
exact hs (set_of p... | theorem | Scott.is_open_sUnion | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"Sup_apply",
"complete_lattice.Sup_continuous'",
"exists_prop",
"is_open",
"is_open_sUnion",
"set_coe.exists",
"subtype.coe_mk",
"supr_Prop_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scott (α : Type u) | α | def | Scott | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [] | A Scott topological space is defined on preorders
such that their open sets, seen as a function `α → Prop`,
preserves the joins of ω-chains | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Scott.topological_space (α : Type u) [omega_complete_partial_order α] :
topological_space (Scott α) | { is_open := Scott.is_open α,
is_open_univ := Scott.is_open_univ α,
is_open_inter := Scott.is_open.inter α,
is_open_sUnion := Scott.is_open_sUnion α } | instance | Scott.topological_space | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"Scott",
"Scott.is_open",
"Scott.is_open.inter",
"Scott.is_open_sUnion",
"Scott.is_open_univ",
"is_open",
"is_open_sUnion",
"is_open_univ",
"omega_complete_partial_order",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_below | { x | ¬ x ≤ y } | def | not_below | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [] | `not_below` is an open set in `Scott α` used
to prove the monotonicity of continuous functions | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_below_is_open : is_open (not_below y) | begin
have h : monotone (not_below y),
{ intros x y' h,
simp only [not_below, set_of, le_Prop_eq],
intros h₀ h₁, apply h₀ (le_trans h h₁) },
existsi h, rintros c,
apply eq_of_forall_ge_iff, intro z,
rw ωSup_le_iff,
simp only [ωSup_le_iff, not_below, mem_set_of_eq, le_Prop_eq, order_hom.coe_fun_mk,
... | lemma | not_below_is_open | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"eq_of_forall_ge_iff",
"exists_imp_distrib",
"is_open",
"le_Prop_eq",
"monotone",
"not_below",
"not_forall",
"order_hom.coe_fun_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_ωSup_ωSup {α} [omega_complete_partial_order α] (c : chain α) :
is_ωSup c (ωSup c) | begin
split,
{ apply le_ωSup, },
{ apply ωSup_le, },
end | lemma | is_ωSup_ωSup | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"omega_complete_partial_order"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Scott_continuous_of_continuous {α β}
[omega_complete_partial_order α]
[omega_complete_partial_order β]
(f : Scott α → Scott β) (hf : continuous f) :
omega_complete_partial_order.continuous' f | begin
simp only [continuous_def, (⁻¹')] at hf,
have h : monotone f,
{ intros x y h,
cases (hf {x | ¬ x ≤ f y} (not_below_is_open _)) with hf hf', clear hf',
specialize hf h, simp only [preimage, mem_set_of_eq, le_Prop_eq] at hf,
by_contradiction H, apply hf H le_rfl },
existsi h, intro c,
apply eq... | lemma | Scott_continuous_of_continuous | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"Scott",
"by_contradiction",
"continuous",
"continuous_def",
"eq_iff_iff",
"eq_of_forall_ge_iff",
"exists_prop",
"le_Prop_eq",
"le_rfl",
"monotone",
"not_below",
"not_below_is_open",
"not_forall",
"not_iff_not",
"omega_complete_partial_order",
"omega_complete_partial_order.continuous'"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_Scott_continuous {α β}
[omega_complete_partial_order α]
[omega_complete_partial_order β]
(f : Scott α → Scott β) (hf : omega_complete_partial_order.continuous' f) :
continuous f | begin
rw continuous_def,
intros s hs,
change continuous' (s ∘ f),
cases hs with hs hs',
cases hf with hf hf',
apply continuous.of_bundled,
apply continuous_comp _ _ hf' hs',
end | lemma | continuous_of_Scott_continuous | topology | src/topology/omega_complete_partial_order.lean | [
"topology.basic",
"order.omega_complete_partial_order"
] | [
"Scott",
"continuous",
"continuous_def",
"omega_complete_partial_order",
"omega_complete_partial_order.continuous'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_open (g : set (set α)) : set α → Prop
| basic : ∀s∈g, generate_open s
| univ : generate_open univ
| inter : ∀s t, generate_open s → generate_open t → generate_open (s ∩ t)
| sUnion : ∀k, (∀s∈k, generate_open s) → generate_open (⋃₀ k) | inductive | topological_space.generate_open | topology | src/topology/order.lean | [
"topology.tactic"
] | [] | The open sets of the least topology containing a collection of basic sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from (g : set (set α)) : topological_space α | { is_open := generate_open g,
is_open_univ := generate_open.univ,
is_open_inter := generate_open.inter,
is_open_sUnion := generate_open.sUnion } | def | topological_space.generate_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"is_open_sUnion",
"is_open_univ",
"topological_space"
] | The smallest topological space containing the collection `g` of basic sets | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_generate_from_of_mem {g : set (set α)} {s : set α} (hs : s ∈ g) :
is_open[generate_from g] s | generate_open.basic s hs | lemma | topological_space.is_open_generate_from_of_mem | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_generate_from {g : set (set α)} {a : α} :
@nhds α (generate_from g) a = (⨅s∈{s | a ∈ s ∧ s ∈ g}, 𝓟 s) | begin
rw nhds_def,
refine le_antisymm (binfi_mono $ λ s ⟨as, sg⟩, ⟨as, generate_open.basic _ sg⟩) _,
refine le_infi₂ (λ s hs, _), cases hs with ha hs,
induction hs,
case basic : s hs { exact infi₂_le _ ⟨ha, hs⟩ },
case univ : { exact le_top.trans_eq principal_univ.symm },
case inter : s t hs' ht' hs ht { ... | lemma | topological_space.nhds_generate_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"binfi_mono",
"infi₂_le",
"le_inf",
"le_infi₂",
"nhds",
"nhds_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_generate_from {β : Type*} {m : α → β} {f : filter α} {g : set (set β)} {b : β}
(h : ∀s∈g, b ∈ s → m ⁻¹' s ∈ f) : tendsto m f (@nhds β (generate_from g) b) | by rw [nhds_generate_from]; exact
(tendsto_infi.2 $ assume s, tendsto_infi.2 $ assume ⟨hbs, hsg⟩, tendsto_principal.2 $ h s hsg hbs) | lemma | topological_space.tendsto_nhds_generate_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_nhds (n : α → filter α) : topological_space α | { is_open := λs, ∀a∈s, s ∈ n a,
is_open_univ := assume x h, univ_mem,
is_open_inter := assume s t hs ht x ⟨hxs, hxt⟩, inter_mem (hs x hxs) (ht x hxt),
is_open_sUnion := assume s hs a ⟨x, hx, hxa⟩,
mem_of_superset (hs x hx _ hxa) (set.subset_sUnion_of_mem hx) } | def | topological_space.mk_of_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"is_open",
"is_open_sUnion",
"is_open_univ",
"set.subset_sUnion_of_mem",
"topological_space"
] | Construct a topology on α given the filter of neighborhoods of each point of α. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_mk_of_nhds (n : α → filter α) (a : α)
(h₀ : pure ≤ n) (h₁ : ∀ a s, s ∈ n a → ∃ t ∈ n a, t ⊆ s ∧ ∀a' ∈ t, s ∈ n a') :
@nhds α (topological_space.mk_of_nhds n) a = n a | begin
letI := topological_space.mk_of_nhds n,
refine le_antisymm (assume s hs, _) (assume s hs, _),
{ have h₀ : {b | s ∈ n b} ⊆ s := assume b hb, mem_pure.1 $ h₀ b hb,
have h₁ : {b | s ∈ n b} ∈ 𝓝 a,
{ refine is_open.mem_nhds (assume b (hb : s ∈ n b), _) hs,
rcases h₁ _ _ hb with ⟨t, ht, hts, h⟩,
... | lemma | topological_space.nhds_mk_of_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"is_open.mem_nhds",
"mem_nhds_iff",
"nhds",
"topological_space.mk_of_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_mk_of_nhds_single [decidable_eq α] {a₀ : α} {l : filter α} (h : pure a₀ ≤ l) (b : α) :
@nhds α (topological_space.mk_of_nhds $ update pure a₀ l) b =
(update pure a₀ l : α → filter α) b | begin
refine nhds_mk_of_nhds _ _ (le_update_iff.mpr ⟨h, λ _ _, le_rfl⟩) (λ a s hs, _),
rcases eq_or_ne a a₀ with rfl|ha,
{ refine ⟨s, hs, subset.rfl, λ b hb, _⟩,
rcases eq_or_ne b a with rfl|hb,
{ exact hs },
{ rwa [update_noteq hb] } },
{ have hs' := hs,
rw [update_noteq ha] at hs ⊢,
exact ... | lemma | topological_space.nhds_mk_of_nhds_single | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"eq_or_ne",
"filter",
"nhds",
"topological_space.mk_of_nhds",
"update",
"update_noteq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_mk_of_nhds_filter_basis (B : α → filter_basis α) (a : α) (h₀ : ∀ x (n ∈ B x), x ∈ n)
(h₁ : ∀ x (n ∈ B x), ∃ n₁ ∈ B x, n₁ ⊆ n ∧ ∀ x' ∈ n₁, ∃ n₂ ∈ B x', n₂ ⊆ n) :
@nhds α (topological_space.mk_of_nhds (λ x, (B x).filter)) a = (B a).filter | begin
rw topological_space.nhds_mk_of_nhds;
intros x n hn;
obtain ⟨m, hm₁, hm₂⟩ := (B x).mem_filter_iff.mp hn,
{ exact hm₂ (h₀ _ _ hm₁), },
{ obtain ⟨n₁, hn₁, hn₂, hn₃⟩ := h₁ x m hm₁,
refine ⟨n₁, (B x).mem_filter_of_mem hn₁, hn₂.trans hm₂, λ x' hx', (B x').mem_filter_iff.mp _⟩,
obtain ⟨n₂, hn₄, hn₅⟩ :... | lemma | topological_space.nhds_mk_of_nhds_filter_basis | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"filter_basis",
"nhds",
"topological_space.mk_of_nhds",
"topological_space.nhds_mk_of_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_def {α} {t s : topological_space α} : t ≤ s ↔ is_open[s] ≤ is_open[t] | iff.rfl | lemma | topological_space.le_def | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_generate_from_iff_subset_is_open {g : set (set α)} {t : topological_space α} :
t ≤ topological_space.generate_from g ↔ g ⊆ {s | is_open[t] s} | ⟨λ ht s hs, ht _ $ generate_open.basic s hs, λ hg s hs, hs.rec_on (assume v hv, hg hv)
t.is_open_univ (assume u v _ _, t.is_open_inter u v) (assume k _, t.is_open_sUnion k)⟩ | lemma | topological_space.le_generate_from_iff_subset_is_open | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_of_closure (s : set (set α)) (hs : {u | generate_open s u} = s) :
topological_space α | { is_open := λ u, u ∈ s,
is_open_univ := hs ▸ topological_space.generate_open.univ,
is_open_inter := hs ▸ topological_space.generate_open.inter,
is_open_sUnion := hs ▸ topological_space.generate_open.sUnion } | def | topological_space.mk_of_closure | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"is_open_sUnion",
"is_open_univ",
"topological_space"
] | If `s` equals the collection of open sets in the topology it generates, then `s` defines a
topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_of_closure_sets {s : set (set α)} {hs : {u | generate_open s u} = s} :
topological_space.mk_of_closure s hs = topological_space.generate_from s | topological_space_eq hs.symm | lemma | topological_space.mk_of_closure_sets | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space.generate_from",
"topological_space.mk_of_closure",
"topological_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_generate_from (α) :
galois_connection (λ t : topological_space α, order_dual.to_dual {s | is_open[t] s})
(generate_from ∘ order_dual.of_dual) | λ _ _, le_generate_from_iff_subset_is_open.symm | lemma | topological_space.gc_generate_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"galois_connection",
"is_open",
"order_dual.of_dual",
"order_dual.to_dual",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gci_generate_from (α : Type*) :
galois_coinsertion (λ t : topological_space α, order_dual.to_dual {s | is_open[t] s})
(generate_from ∘ order_dual.of_dual) | { gc := gc_generate_from α,
u_l_le := assume ts s hs, generate_open.basic s hs,
choice := λg hg, topological_space.mk_of_closure g
(subset.antisymm hg $ le_generate_from_iff_subset_is_open.1 $ le_rfl),
choice_eq := assume s hs, mk_of_closure_sets } | def | topological_space.gci_generate_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"galois_coinsertion",
"is_open",
"le_rfl",
"order_dual.of_dual",
"order_dual.to_dual",
"topological_space",
"topological_space.mk_of_closure"
] | The Galois coinsertion between `topological_space α` and `(set (set α))ᵒᵈ` whose lower part
sends a topology to its collection of open subsets, and whose upper part sends a collection of
subsets of α to the topology they generate. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
generate_from_anti {α} {g₁ g₂ : set (set α)} (h : g₁ ⊆ g₂) :
generate_from g₂ ≤ generate_from g₁ | (gc_generate_from _).monotone_u h | lemma | topological_space.generate_from_anti | topology | src/topology/order.lean | [
"topology.tactic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_set_of_is_open (t : topological_space α) :
generate_from {s | is_open[t] s} = t | (gci_generate_from α).u_l_eq t | lemma | topological_space.generate_from_set_of_is_open | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_inverse_generate_from :
left_inverse generate_from (λ t : topological_space α, {s | is_open[t] s}) | (gci_generate_from α).u_l_left_inverse | lemma | topological_space.left_inverse_generate_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
generate_from_surjective :
surjective (generate_from : set (set α) → topological_space α) | (gci_generate_from α).u_surjective | lemma | topological_space.generate_from_surjective | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_of_is_open_injective : injective (λ t : topological_space α, {s | is_open[t] s}) | (gci_generate_from α).l_injective | lemma | topological_space.set_of_is_open_injective | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.mono {α} {t₁ t₂ : topological_space α} {s : set α} (hs : is_open[t₂] s)
(h : t₁ ≤ t₂) : is_open[t₁] s | h s hs | lemma | is_open.mono | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.mono {α} {t₁ t₂ : topological_space α} {s : set α} (hs : is_closed[t₂] s)
(h : t₁ ≤ t₂) : is_closed[t₁] s | (@is_open_compl_iff α t₁ s).mp $ hs.is_open_compl.mono h | lemma | is_closed.mono | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_closed",
"is_open_compl_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_implies_is_open_iff {a b : topological_space α} :
(∀ s, is_open[a] s → is_open[b] s) ↔ b ≤ a | iff.rfl | lemma | is_open_implies_is_open_iff | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space.is_open_top_iff {α} (U : set α) :
is_open[⊤] U ↔ U = ∅ ∨ U = univ | ⟨λ h, begin
induction h with V h _ _ _ _ ih₁ ih₂ _ _ ih,
{ cases h }, { exact or.inr rfl },
{ obtain ⟨rfl|rfl, rfl|rfl⟩ := ⟨ih₁, ih₂⟩; simp },
{ rw [sUnion_eq_empty, or_iff_not_imp_left],
intro h, push_neg at h, obtain ⟨U, hU, hne⟩ := h,
have := (ih U hU).resolve_left hne, subst this,
refine sUnion_... | lemma | topological_space.is_open_top_iff | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"ih",
"is_open",
"is_open_empty",
"is_open_univ",
"or_iff_not_imp_left"
] | The only open sets in the indiscrete topology are the empty set and the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_topology (α : Type*) [t : topological_space α] : Prop | (eq_bot [] : t = ⊥) | class | discrete_topology | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space"
] | A topological space is discrete if every set is open, that is,
its topology equals the discrete topology `⊥`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_topology_bot (α : Type*) : @discrete_topology α ⊥ | @discrete_topology.mk α ⊥ rfl | lemma | discrete_topology_bot | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_discrete [topological_space α] [discrete_topology α] (s : set α) :
is_open s | (discrete_topology.eq_bot α).symm ▸ trivial | lemma | is_open_discrete | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_discrete [topological_space α] [discrete_topology α] (s : set α) :
is_closed s | is_open_compl_iff.1 $ is_open_discrete _ | lemma | is_closed_discrete | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"is_closed",
"is_open_discrete",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_discrete_topology [topological_space α] [discrete_topology α]
[topological_space β] {f : α → β} : continuous f | continuous_def.2 $ λ s hs, is_open_discrete _ | lemma | continuous_of_discrete_topology | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"continuous",
"discrete_topology",
"is_open_discrete",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_discrete (α : Type*) [topological_space α] [discrete_topology α] :
(@nhds α _) = pure | le_antisymm (λ _ s hs, (is_open_discrete s).mem_nhds hs) pure_le_nhds | lemma | nhds_discrete | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"is_open_discrete",
"nhds",
"pure_le_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_discrete [topological_space α] [discrete_topology α] {x : α} {s : set α} :
s ∈ 𝓝 x ↔ x ∈ s | by rw [nhds_discrete, mem_pure] | lemma | mem_nhds_discrete | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"nhds_discrete",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_nhds_le_nhds {t₁ t₂ : topological_space α} (h : ∀x, @nhds α t₁ x ≤ @nhds α t₂ x) :
t₁ ≤ t₂ | begin
intro s,
rw [@is_open_iff_mem_nhds _ t₁, @is_open_iff_mem_nhds α t₂],
exact λ hs a ha, h _ (hs _ ha)
end | lemma | le_of_nhds_le_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open_iff_mem_nhds",
"nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_nhds_eq_nhds {t₁ t₂ : topological_space α} (h : ∀x, @nhds α t₁ x = @nhds α t₂ x) :
t₁ = t₂ | le_antisymm
(le_of_nhds_le_nhds $ assume x, le_of_eq $ h x)
(le_of_nhds_le_nhds $ assume x, le_of_eq $ (h x).symm) | lemma | eq_of_nhds_eq_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"le_of_nhds_le_nhds",
"nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_of_singletons_open {t : topological_space α} (h : ∀ x, is_open[t] {x}) : t = ⊥ | bot_unique $ λ s hs, bUnion_of_singleton s ▸ is_open_bUnion (λ x _, h x) | lemma | eq_bot_of_singletons_open | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"bot_unique",
"is_open",
"is_open_bUnion",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_open_iff_discrete {X : Type*} [topological_space X] :
(∀ s : set X, is_open s) ↔ discrete_topology X | ⟨λ h, ⟨eq_bot_of_singletons_open $ λ _, h _⟩, @is_open_discrete _ _⟩ | lemma | forall_open_iff_discrete | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"is_open",
"is_open_discrete",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singletons_open_iff_discrete {X : Type*} [topological_space X] :
(∀ a : X, is_open ({a} : set X)) ↔ discrete_topology X | ⟨λ h, ⟨eq_bot_of_singletons_open h⟩, λ a _, @is_open_discrete _ _ a _⟩ | lemma | singletons_open_iff_discrete | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"is_open",
"is_open_discrete",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology_iff_singleton_mem_nhds [topological_space α] :
discrete_topology α ↔ ∀ x : α, {x} ∈ 𝓝 x | by simp only [← singletons_open_iff_discrete, is_open_iff_mem_nhds, mem_singleton_iff, forall_eq] | lemma | discrete_topology_iff_singleton_mem_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"forall_eq",
"is_open_iff_mem_nhds",
"singletons_open_iff_discrete",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology_iff_nhds [topological_space α] :
discrete_topology α ↔ ∀ x : α, 𝓝 x = pure x | by simp only [discrete_topology_iff_singleton_mem_nhds, ← nhds_ne_bot.le_pure_iff,
le_pure_iff] | lemma | discrete_topology_iff_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"discrete_topology_iff_singleton_mem_nhds",
"topological_space"
] | This lemma characterizes discrete topological spaces as those whose singletons are
neighbourhoods. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
discrete_topology_iff_nhds_ne [topological_space α] :
discrete_topology α ↔ ∀ x : α, 𝓝[≠] x = ⊥ | by simp only [discrete_topology_iff_singleton_mem_nhds, nhds_within, inf_principal_eq_bot,
compl_compl] | lemma | discrete_topology_iff_nhds_ne | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"compl_compl",
"discrete_topology",
"discrete_topology_iff_singleton_mem_nhds",
"nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space.induced {α : Type u} {β : Type v} (f : α → β) (t : topological_space β) :
topological_space α | { is_open := λs, ∃ s', is_open s' ∧ f ⁻¹' s' = s,
is_open_univ := ⟨univ, is_open_univ, preimage_univ⟩,
is_open_inter := by rintro s₁ s₂ ⟨s'₁, hs₁, rfl⟩ ⟨s'₂, hs₂, rfl⟩;
exact ⟨s'₁ ∩ s'₂, hs₁.inter hs₂, preimage_inter⟩,
is_open_sUnion := assume s h,
begin
simp only [classical.skolem] at h,
... | def | topological_space.induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"is_open_Union",
"is_open_sUnion",
"is_open_univ",
"topological_space"
] | Given `f : α → β` and a topology on `β`, the induced topology on `α` is the collection of
sets that are preimages of some open set in `β`. This is the coarsest topology that
makes `f` continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_induced_iff [t : topological_space β] {s : set α} {f : α → β} :
is_open[t.induced f] s ↔ (∃t, is_open t ∧ f ⁻¹' t = s) | iff.rfl | lemma | is_open_induced_iff | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_induced_iff [t : topological_space β] {s : set α} {f : α → β} :
is_closed[t.induced f] s ↔ (∃t, is_closed t ∧ f ⁻¹' t = s) | begin
simp only [← is_open_compl_iff, is_open_induced_iff],
exact compl_surjective.exists.trans (by simp only [preimage_compl, compl_inj_iff])
end | lemma | is_closed_induced_iff | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"compl_inj_iff",
"is_closed",
"is_open_compl_iff",
"is_open_induced_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space.coinduced {α : Type u} {β : Type v} (f : α → β) (t : topological_space α) :
topological_space β | { is_open := λ s, is_open[t] (f ⁻¹' s),
is_open_univ := t.is_open_univ,
is_open_inter := λ _ _ h₁ h₂, h₁.inter h₂,
is_open_sUnion := λ s h, by simpa only [preimage_sUnion] using is_open_bUnion h } | def | topological_space.coinduced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"is_open_bUnion",
"is_open_sUnion",
"is_open_univ",
"topological_space"
] | Given `f : α → β` and a topology on `α`, the coinduced topology on `β` is defined
such that `s:set β` is open if the preimage of `s` is open. This is the finest topology that
makes `f` continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_coinduced {t : topological_space α} {s : set β} {f : α → β} :
is_open[t.coinduced f] s ↔ is_open (f ⁻¹' s) | iff.rfl | lemma | is_open_coinduced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_nhds_coinduced [topological_space α] {π : α → β} {s : set β}
{a : α} (hs : s ∈ @nhds β (topological_space.coinduced π ‹_›) (π a)) : π ⁻¹' s ∈ 𝓝 a | begin
letI := topological_space.coinduced π ‹_›,
rcases mem_nhds_iff.mp hs with ⟨V, hVs, V_op, mem_V⟩,
exact mem_nhds_iff.mpr ⟨π ⁻¹' V, set.preimage_mono hVs, V_op, mem_V⟩
end | lemma | preimage_nhds_coinduced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"nhds",
"set.preimage_mono",
"topological_space",
"topological_space.coinduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.coinduced_le (h : @continuous α β t t' f) :
t.coinduced f ≤ t' | λ s hs, (continuous_def.1 h s hs : _) | lemma | continuous.coinduced_le | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_le_iff_le_induced {f : α → β} {tα : topological_space α}
{tβ : topological_space β} :
tα.coinduced f ≤ tβ ↔ tα ≤ tβ.induced f | ⟨λ h s ⟨t, ht, hst⟩, hst ▸ h _ ht, λ h s hs, h _ ⟨s, hs, rfl⟩⟩ | lemma | coinduced_le_iff_le_induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.le_induced (h : @continuous α β t t' f) :
t ≤ t'.induced f | coinduced_le_iff_le_induced.1 h.coinduced_le | lemma | continuous.le_induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gc_coinduced_induced (f : α → β) :
galois_connection (topological_space.coinduced f) (topological_space.induced f) | assume f g, coinduced_le_iff_le_induced | lemma | gc_coinduced_induced | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"coinduced_le_iff_le_induced",
"galois_connection",
"topological_space.coinduced",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_mono (h : t₁ ≤ t₂) : t₁.induced g ≤ t₂.induced g | (gc_coinduced_induced g).monotone_u h | lemma | induced_mono | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_coinduced_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_mono (h : t₁ ≤ t₂) : t₁.coinduced f ≤ t₂.coinduced f | (gc_coinduced_induced f).monotone_l h | lemma | coinduced_mono | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_coinduced_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_top : (⊤ : topological_space α).induced g = ⊤ | (gc_coinduced_induced g).u_top | lemma | induced_top | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_coinduced_induced",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_inf : (t₁ ⊓ t₂).induced g = t₁.induced g ⊓ t₂.induced g | (gc_coinduced_induced g).u_inf | lemma | induced_inf | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_coinduced_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_infi {ι : Sort w} {t : ι → topological_space α} :
(⨅i, t i).induced g = (⨅i, (t i).induced g) | (gc_coinduced_induced g).u_infi | lemma | induced_infi | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_coinduced_induced",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_bot : (⊥ : topological_space α).coinduced f = ⊥ | (gc_coinduced_induced f).l_bot | lemma | coinduced_bot | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_coinduced_induced",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_sup : (t₁ ⊔ t₂).coinduced f = t₁.coinduced f ⊔ t₂.coinduced f | (gc_coinduced_induced f).l_sup | lemma | coinduced_sup | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_coinduced_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_supr {ι : Sort w} {t : ι → topological_space α} :
(⨆i, t i).coinduced f = (⨆i, (t i).coinduced f) | (gc_coinduced_induced f).l_supr | lemma | coinduced_supr | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_coinduced_induced",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_id [t : topological_space α] : t.induced id = t | topological_space_eq $ funext $ assume s, propext $
⟨assume ⟨s', hs, h⟩, h ▸ hs, assume hs, ⟨s, hs, rfl⟩⟩ | lemma | induced_id | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space",
"topological_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_compose [tγ : topological_space γ]
{f : α → β} {g : β → γ} : (tγ.induced g).induced f = tγ.induced (g ∘ f) | topological_space_eq $ funext $ assume s, propext $
⟨assume ⟨s', ⟨s, hs, h₂⟩, h₁⟩, h₁ ▸ h₂ ▸ ⟨s, hs, rfl⟩,
assume ⟨s, hs, h⟩, ⟨preimage g s, ⟨s, hs, rfl⟩, h ▸ rfl⟩⟩ | lemma | induced_compose | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space",
"topological_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_const [t : topological_space α] {x : α} :
t.induced (λ y : β, x) = ⊤ | le_antisymm le_top (@continuous_const β α ⊤ t x).le_induced | lemma | induced_const | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"continuous_const",
"le_top",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_id [t : topological_space α] : t.coinduced id = t | topological_space_eq rfl | lemma | coinduced_id | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space",
"topological_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_compose [tα : topological_space α]
{f : α → β} {g : β → γ} : (tα.coinduced f).coinduced g = tα.coinduced (g ∘ f) | topological_space_eq rfl | lemma | coinduced_compose | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space",
"topological_space_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.induced_symm {α β : Type*} (e : α ≃ β) :
topological_space.induced e.symm = topological_space.coinduced e | begin
ext t U,
split,
{ rintros ⟨V, hV, rfl⟩,
rwa [is_open_coinduced, e.preimage_symm_preimage] },
{ exact λ hU, ⟨e ⁻¹' U, hU, e.symm_preimage_preimage _⟩ }
end | lemma | equiv.induced_symm | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open_coinduced",
"topological_space.coinduced",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.coinduced_symm {α β : Type*} (e : α ≃ β) :
topological_space.coinduced e.symm = topological_space.induced e | by rw [← e.symm.induced_symm, e.symm_symm] | lemma | equiv.coinduced_symm | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space.coinduced",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited_topological_space {α : Type u} : inhabited (topological_space α) | ⟨⊥⟩ | instance | inhabited_topological_space | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.unique_topological_space [subsingleton α] :
unique (topological_space α) | { default := ⊥,
uniq := λ t, eq_bot_of_singletons_open $ λ x, subsingleton.set_cases
(@is_open_empty _ t) (@is_open_univ _ t) ({x} : set α) } | instance | subsingleton.unique_topological_space | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"eq_bot_of_singletons_open",
"is_open_empty",
"is_open_univ",
"subsingleton.set_cases",
"topological_space",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.discrete_topology [t : topological_space α] [subsingleton α] :
discrete_topology α | ⟨unique.eq_default t⟩ | instance | subsingleton.discrete_topology | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"discrete_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sierpinski_space : topological_space Prop | generate_from {{true}} | instance | sierpinski_space | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_empty_function [topological_space α] [topological_space β] [is_empty β]
(f : α → β) : continuous f | by { letI := function.is_empty f, exact continuous_of_discrete_topology } | lemma | continuous_empty_function | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"continuous",
"continuous_of_discrete_topology",
"function.is_empty",
"is_empty",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_generate_from {t : topological_space α} { g : set (set α) } (h : ∀s∈g, is_open s) :
t ≤ generate_from g | le_generate_from_iff_subset_is_open.2 h | lemma | le_generate_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_generate_from_eq {α β} {b : set (set β)} {f : α → β} :
(generate_from b).induced f = topological_space.generate_from (preimage f '' b) | le_antisymm
(le_generate_from $ ball_image_iff.2 $ assume s hs, ⟨s, generate_open.basic _ hs, rfl⟩)
(coinduced_le_iff_le_induced.1 $ le_generate_from $ assume s hs,
generate_open.basic _ $ mem_image_of_mem _ hs) | lemma | induced_generate_from_eq | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"le_generate_from",
"topological_space.generate_from"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_induced_generate_from {α β} [t : topological_space α] {b : set (set β)}
{f : α → β} (h : ∀ (a : set β), a ∈ b → is_open (f ⁻¹' a)) : t ≤ induced f (generate_from b) | begin
rw induced_generate_from_eq,
apply le_generate_from,
simp only [mem_image, and_imp, forall_apply_eq_imp_iff₂, exists_imp_distrib],
exact h,
end | lemma | le_induced_generate_from | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"and_imp",
"exists_imp_distrib",
"forall_apply_eq_imp_iff₂",
"induced_generate_from_eq",
"is_open",
"le_generate_from",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_adjoint (a : α) (f : filter α) : topological_space α | { is_open := λs, a ∈ s → s ∈ f,
is_open_univ := assume s, univ_mem,
is_open_inter := assume s t hs ht ⟨has, hat⟩, inter_mem (hs has) (ht hat),
is_open_sUnion := assume k hk ⟨u, hu, hau⟩, mem_of_superset (hk u hu hau)
(subset_sUnion_of_mem hu) } | def | nhds_adjoint | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"is_open",
"is_open_sUnion",
"is_open_univ",
"topological_space"
] | This construction is left adjoint to the operation sending a topology on `α`
to its neighborhood filter at a fixed point `a : α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc_nhds (a : α) :
galois_connection (nhds_adjoint a) (λt, @nhds α t a) | assume f t, by { rw le_nhds_iff, exact ⟨λ H s hs has, H _ has hs, λ H s has hs, H _ hs has⟩ } | lemma | gc_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"galois_connection",
"le_nhds_iff",
"nhds",
"nhds_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_mono {t₁ t₂ : topological_space α} {a : α} (h : t₁ ≤ t₂) :
@nhds α t₁ a ≤ @nhds α t₂ a | (gc_nhds a).monotone_u h | lemma | nhds_mono | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_nhds",
"nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_iff_nhds {α : Type*} (t t' : topological_space α) :
t ≤ t' ↔ ∀ x, @nhds α t x ≤ @nhds α t' x | ⟨λ h x, nhds_mono h, le_of_nhds_le_nhds⟩ | lemma | le_iff_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"nhds",
"nhds_mono",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_adjoint_nhds {α : Type*} (a : α) (f : filter α) :
@nhds α (nhds_adjoint a f) a = pure a ⊔ f | begin
ext U,
rw mem_nhds_iff,
split,
{ rintros ⟨t, htU, ht, hat⟩,
exact ⟨htU hat, mem_of_superset (ht hat) htU⟩},
{ rintros ⟨haU, hU⟩,
exact ⟨U, subset.rfl, λ h, hU, haU⟩ }
end | lemma | nhds_adjoint_nhds | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"mem_nhds_iff",
"nhds",
"nhds_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_adjoint_nhds_of_ne {α : Type*} (a : α) (f : filter α) {b : α} (h : b ≠ a) :
@nhds α (nhds_adjoint a f) b = pure b | begin
apply le_antisymm,
{ intros U hU,
rw mem_nhds_iff,
use {b},
simp only [and_true, singleton_subset_iff, mem_singleton],
refine ⟨hU, λ ha, (h.symm ha).elim⟩ },
{ exact @pure_le_nhds α (nhds_adjoint a f) b },
end | lemma | nhds_adjoint_nhds_of_ne | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"mem_nhds_iff",
"nhds",
"nhds_adjoint",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_singleton_nhds_adjoint {α : Type*} {a b : α} (f : filter α) (hb : b ≠ a) :
is_open[nhds_adjoint a f] {b} | begin
rw is_open_singleton_iff_nhds_eq_pure,
exact nhds_adjoint_nhds_of_ne a f hb
end | lemma | is_open_singleton_nhds_adjoint | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"is_open",
"is_open_singleton_iff_nhds_eq_pure",
"nhds_adjoint",
"nhds_adjoint_nhds_of_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nhds_adjoint_iff' {α : Type*} (a : α) (f : filter α) (t : topological_space α) :
t ≤ nhds_adjoint a f ↔ @nhds α t a ≤ pure a ⊔ f ∧ ∀ b ≠ a, @nhds α t b = pure b | begin
rw le_iff_nhds,
split,
{ intros h,
split,
{ specialize h a,
rwa nhds_adjoint_nhds at h },
{ intros b hb,
apply le_antisymm _ (pure_le_nhds b),
specialize h b,
rwa nhds_adjoint_nhds_of_ne a f hb at h } },
{ rintros ⟨h, h'⟩ b,
by_cases hb : b = a,
{ rwa [hb, nhds_... | lemma | le_nhds_adjoint_iff' | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"filter",
"le_iff_nhds",
"nhds",
"nhds_adjoint",
"nhds_adjoint_nhds",
"nhds_adjoint_nhds_of_ne",
"pure_le_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nhds_adjoint_iff {α : Type*} (a : α) (f : filter α) (t : topological_space α) :
t ≤ nhds_adjoint a f ↔ (@nhds α t a ≤ pure a ⊔ f ∧ ∀ b, b ≠ a → is_open[t] {b}) | begin
change _ ↔ _ ∧ ∀ (b : α), b ≠ a → is_open {b},
rw [le_nhds_adjoint_iff', and.congr_right_iff],
apply λ h, forall_congr (λ b, _),
rw @is_open_singleton_iff_nhds_eq_pure α t b
end | lemma | le_nhds_adjoint_iff | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"and.congr_right_iff",
"filter",
"is_open",
"is_open_singleton_iff_nhds_eq_pure",
"le_nhds_adjoint_iff'",
"nhds",
"nhds_adjoint",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_infi {ι : Sort*} {t : ι → topological_space α} {a : α} :
@nhds α (infi t) a = (⨅i, @nhds α (t i) a) | (gc_nhds a).u_infi | lemma | nhds_infi | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_nhds",
"infi",
"nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_Inf {s : set (topological_space α)} {a : α} :
@nhds α (Inf s) a = (⨅t∈s, @nhds α t a) | (gc_nhds a).u_Inf | lemma | nhds_Inf | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_nhds",
"nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_inf {t₁ t₂ : topological_space α} {a : α} :
@nhds α (t₁ ⊓ t₂) a = @nhds α t₁ a ⊓ @nhds α t₂ a | (gc_nhds a).u_inf | lemma | nhds_inf | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_nhds",
"nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_top {a : α} : @nhds α ⊤ a = ⊤ | (gc_nhds a).u_top | lemma | nhds_top | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"gc_nhds",
"nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_sup {t₁ t₂ : topological_space α} {s : set α} :
is_open[t₁ ⊔ t₂] s ↔ is_open[t₁] s ∧ is_open[t₂] s | iff.rfl | lemma | is_open_sup | topology | src/topology/order.lean | [
"topology.tactic"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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