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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pi_split_at (β : ι → Type*) [Π j, topological_space (β j)] :
(Π j, β j) ≃ₜ β i × Π j : {j // j ≠ i}, β j | { to_equiv := equiv.pi_split_at i β,
continuous_to_fun := (continuous_apply i).prod_mk (continuous_pi $ λ j, continuous_apply j),
continuous_inv_fun := continuous_pi $ λ j, by { dsimp only [equiv.pi_split_at],
split_ifs, subst h, exacts [continuous_fst, (continuous_apply _).comp continuous_snd] } } | def | homeomorph.pi_split_at | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_apply",
"continuous_fst",
"continuous_pi",
"continuous_snd",
"equiv.pi_split_at",
"topological_space"
] | A product of topological spaces can be split as the binary product of one of the spaces and
the product of all the remaining spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_split_at : (ι → β) ≃ₜ β × ({j // j ≠ i} → β) | pi_split_at i _ | def | homeomorph.fun_split_at | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | A product of copies of a topological space can be split as the binary product of one copy and
the product of all the remaining copies. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv.to_homeomorph_of_inducing [topological_space α] [topological_space β] (f : α ≃ β)
(hf : inducing f) :
α ≃ₜ β | { continuous_to_fun := hf.continuous,
continuous_inv_fun := hf.continuous_iff.2 $ by simpa using continuous_id,
.. f } | def | equiv.to_homeomorph_of_inducing | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_id",
"inducing",
"topological_space"
] | An inducing equiv between topological spaces is a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_symm_of_equiv_compact_to_t2 [compact_space α] [t2_space β]
{f : α ≃ β} (hf : continuous f) : continuous f.symm | begin
rw continuous_iff_is_closed,
intros C hC,
have hC' : is_closed (f '' C) := (hC.is_compact.image hf).is_closed,
rwa equiv.image_eq_preimage at hC',
end | lemma | continuous.continuous_symm_of_equiv_compact_to_t2 | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"compact_space",
"continuous",
"continuous_iff_is_closed",
"equiv.image_eq_preimage",
"is_closed",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeo_of_equiv_compact_to_t2 [compact_space α] [t2_space β]
{f : α ≃ β} (hf : continuous f) : α ≃ₜ β | { continuous_to_fun := hf,
continuous_inv_fun := hf.continuous_symm_of_equiv_compact_to_t2,
..f } | def | continuous.homeo_of_equiv_compact_to_t2 | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"compact_space",
"continuous",
"t2_space"
] | Continuous equivalences from a compact space to a T2 space are homeomorphisms.
This is not true when T2 is weakened to T1
(see `continuous.homeo_of_equiv_compact_to_t2.t1_counterexample`). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
specializes (x y : X) : Prop | 𝓝 x ≤ 𝓝 y | def | specializes | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | `x` specializes to `y` (notation: `x ⤳ y`) if either of the following equivalent properties
hold:
* `𝓝 x ≤ 𝓝 y`; this property is used as the definition;
* `pure x ≤ 𝓝 y`; in other words, any neighbourhood of `y` contains `x`;
* `y ∈ closure {x}`;
* `closure {y} ⊆ closure {x}`;
* for any closed set `s` we have `x ∈... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
specializes_tfae (x y : X) :
tfae [x ⤳ y,
pure x ≤ 𝓝 y,
∀ s : set X, is_open s → y ∈ s → x ∈ s,
∀ s : set X, is_closed s → x ∈ s → y ∈ s,
y ∈ closure ({x} : set X),
closure ({y} : set X) ⊆ closure {x},
cluster_pt y (pure x)] | begin
tfae_have : 1 → 2, from (pure_le_nhds _).trans,
tfae_have : 2 → 3, from λ h s hso hy, h (hso.mem_nhds hy),
tfae_have : 3 → 4, from λ h s hsc hx, of_not_not $ λ hy, h sᶜ hsc.is_open_compl hy hx,
tfae_have : 4 → 5, from λ h, h _ is_closed_closure (subset_closure $ mem_singleton _),
tfae_have : 6 ↔ 5, from... | lemma | specializes_tfae | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"closure",
"cluster_pt",
"is_closed",
"is_closed_closure",
"is_open",
"mem_closure_iff_cluster_pt",
"nhds_basis_opens",
"of_not_not",
"pure_le_nhds",
"subset_closure"
] | A collection of equivalent definitions of `x ⤳ y`. The public API is given by `iff` lemmas
below. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
specializes_iff_nhds : x ⤳ y ↔ 𝓝 x ≤ 𝓝 y | iff.rfl | lemma | specializes_iff_nhds | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_pure : x ⤳ y ↔ pure x ≤ 𝓝 y | (specializes_tfae x y).out 0 1 | lemma | specializes_iff_pure | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"specializes_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_forall_open : x ⤳ y ↔ ∀ s : set X, is_open s → y ∈ s → x ∈ s | (specializes_tfae x y).out 0 2 | lemma | specializes_iff_forall_open | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_open",
"specializes_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes.mem_open (h : x ⤳ y) (hs : is_open s) (hy : y ∈ s) : x ∈ s | specializes_iff_forall_open.1 h s hs hy | lemma | specializes.mem_open | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.not_specializes (hs : is_open s) (hx : x ∉ s) (hy : y ∈ s) : ¬ x ⤳ y | λ h, hx $ h.mem_open hs hy | lemma | is_open.not_specializes | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_forall_closed : x ⤳ y ↔ ∀ s : set X, is_closed s → x ∈ s → y ∈ s | (specializes_tfae x y).out 0 3 | lemma | specializes_iff_forall_closed | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_closed",
"specializes_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes.mem_closed (h : x ⤳ y) (hs : is_closed s) (hx : x ∈ s) : y ∈ s | specializes_iff_forall_closed.1 h s hs hx | lemma | specializes.mem_closed | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.not_specializes (hs : is_closed s) (hx : x ∈ s) (hy : y ∉ s) : ¬ x ⤳ y | λ h, hy $ h.mem_closed hs hx | lemma | is_closed.not_specializes | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_mem_closure : x ⤳ y ↔ y ∈ closure ({x} : set X) | (specializes_tfae x y).out 0 4 | lemma | specializes_iff_mem_closure | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"closure",
"specializes_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_iff_closure_subset :
x ⤳ y ↔ closure ({y} : set X) ⊆ closure {x} | (specializes_tfae x y).out 0 5 | lemma | specializes_iff_closure_subset | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"closure",
"specializes_tfae"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.specializes_iff {ι} {p : ι → Prop} {s : ι → set X}
(h : (𝓝 y).has_basis p s) :
x ⤳ y ↔ ∀ i, p i → x ∈ s i | specializes_iff_pure.trans h.ge_iff | lemma | filter.has_basis.specializes_iff | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_rfl : x ⤳ x | le_rfl | lemma | specializes_rfl | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_refl (x : X) : x ⤳ x | specializes_rfl | lemma | specializes_refl | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"specializes_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes.trans : x ⤳ y → y ⤳ z → x ⤳ z | le_trans | lemma | specializes.trans | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_of_eq (e : x = y) : x ⤳ y | e ▸ specializes_refl x | lemma | specializes_of_eq | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"specializes_refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_of_nhds_within (h₁ : 𝓝[s] x ≤ 𝓝[s] y) (h₂ : x ∈ s) : x ⤳ y | specializes_iff_pure.2 $
calc pure x ≤ 𝓝[s] x : le_inf (pure_le_nhds _) (le_principal_iff.2 h₂)
... ≤ 𝓝[s] y : h₁
... ≤ 𝓝 y : inf_le_left | lemma | specializes_of_nhds_within | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inf_le_left",
"le_inf",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes.map_of_continuous_at (h : x ⤳ y) (hy : continuous_at f y) : f x ⤳ f y | specializes_iff_pure.2 $ λ s hs, mem_pure.2 $ mem_preimage.1 $ mem_of_mem_nhds $ hy.mono_left h hs | lemma | specializes.map_of_continuous_at | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous_at",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes.map (h : x ⤳ y) (hf : continuous f) : f x ⤳ f y | h.map_of_continuous_at hf.continuous_at | lemma | specializes.map | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.specializes_iff (hf : inducing f) : f x ⤳ f y ↔ x ⤳ y | by simp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton,
mem_preimage] | lemma | inducing.specializes_iff | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inducing",
"specializes_iff_mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_specializes_iff {p : X → Prop} (x y : subtype p) : x ⤳ y ↔ (x : X) ⤳ y | inducing_coe.specializes_iff.symm | lemma | subtype_specializes_iff | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
(x₁, y₁) ⤳ (x₂, y₂) ↔ x₁ ⤳ x₂ ∧ y₁ ⤳ y₂ | by simp only [specializes, nhds_prod_eq, prod_le_prod] | lemma | specializes_prod | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"nhds_prod_eq",
"specializes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ⤳ x₂) (hy : y₁ ⤳ y₂) :
(x₁, y₁) ⤳ (x₂, y₂) | specializes_prod.2 ⟨hx, hy⟩ | lemma | specializes.prod | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes_pi {f g : Π i, π i} : f ⤳ g ↔ ∀ i, f i ⤳ g i | by simp only [specializes, nhds_pi, pi_le_pi] | lemma | specializes_pi | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"nhds_pi",
"specializes"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_specializes_iff_exists_open : ¬ x ⤳ y ↔ ∃ (S : set X), is_open S ∧ y ∈ S ∧ x ∉ S | by { rw [specializes_iff_forall_open], push_neg, refl } | lemma | not_specializes_iff_exists_open | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_open",
"specializes_iff_forall_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_specializes_iff_exists_closed : ¬ x ⤳ y ↔ ∃ (S : set X), is_closed S ∧ x ∈ S ∧ y ∉ S | by { rw [specializes_iff_forall_closed], push_neg, refl } | lemma | not_specializes_iff_exists_closed | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_closed",
"specializes_iff_forall_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specialization_preorder : preorder X | { le := λ x y, y ⤳ x,
lt := λ x y, y ⤳ x ∧ ¬(x ⤳ y),
.. preorder.lift (order_dual.to_dual ∘ 𝓝) } | def | specialization_preorder | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"order_dual.to_dual",
"preorder.lift"
] | Specialization forms a preorder on the topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.specialization_monotone (hf : continuous f) :
@monotone _ _ (specialization_preorder X) (specialization_preorder Y) f | λ x y h, h.map hf | lemma | continuous.specialization_monotone | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous",
"monotone",
"specialization_preorder"
] | A continuous function is monotone with respect to the specialization preorders on the domain and
the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inseparable (x y : X) : Prop | 𝓝 x = 𝓝 y | def | inseparable | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | Two points `x` and `y` in a topological space are `inseparable` if any of the following
equivalent properties hold:
- `𝓝 x = 𝓝 y`; we use this property as the definition;
- for any open set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_open`;
- for any closed set `s`, `x ∈ s ↔ y ∈ s`, see `inseparable_iff_closed`;
- `x... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inseparable_def : x ~ y ↔ 𝓝 x = 𝓝 y | iff.rfl | lemma | inseparable_def | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_iff_specializes_and : x ~ y ↔ x ⤳ y ∧ y ⤳ x | le_antisymm_iff | lemma | inseparable_iff_specializes_and | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable.specializes (h : x ~ y) : x ⤳ y | h.le | lemma | inseparable.specializes | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable.specializes' (h : x ~ y) : y ⤳ x | h.ge | lemma | inseparable.specializes' | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
specializes.antisymm (h₁ : x ⤳ y) (h₂ : y ⤳ x) : x ~ y | le_antisymm h₁ h₂ | lemma | specializes.antisymm | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_iff_forall_open : x ~ y ↔ ∀ s : set X, is_open s → (x ∈ s ↔ y ∈ s) | by simp only [inseparable_iff_specializes_and, specializes_iff_forall_open, ← forall_and_distrib,
← iff_def, iff.comm] | lemma | inseparable_iff_forall_open | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"forall_and_distrib",
"iff_def",
"inseparable_iff_specializes_and",
"is_open",
"specializes_iff_forall_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_inseparable_iff_exists_open : ¬(x ~ y) ↔ ∃ s : set X, is_open s ∧ xor (x ∈ s) (y ∈ s) | by simp [inseparable_iff_forall_open, ← xor_iff_not_iff] | lemma | not_inseparable_iff_exists_open | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inseparable_iff_forall_open",
"is_open",
"xor_iff_not_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_iff_forall_closed : x ~ y ↔ ∀ s : set X, is_closed s → (x ∈ s ↔ y ∈ s) | by simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and_distrib,
← iff_def] | lemma | inseparable_iff_forall_closed | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"forall_and_distrib",
"iff_def",
"inseparable_iff_specializes_and",
"is_closed",
"specializes_iff_forall_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_iff_mem_closure :
x ~ y ↔ x ∈ closure ({y} : set X) ∧ y ∈ closure ({x} : set X) | inseparable_iff_specializes_and.trans $ by simp only [specializes_iff_mem_closure, and_comm] | lemma | inseparable_iff_mem_closure | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"closure",
"specializes_iff_mem_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_iff_closure_eq : x ~ y ↔ closure ({x} : set X) = closure {y} | by simp only [inseparable_iff_specializes_and, specializes_iff_closure_subset,
← subset_antisymm_iff, eq_comm] | lemma | inseparable_iff_closure_eq | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"closure",
"inseparable_iff_specializes_and",
"specializes_iff_closure_subset",
"subset_antisymm_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_of_nhds_within_eq (hx : x ∈ s) (hy : y ∈ s) (h : 𝓝[s] x = 𝓝[s] y) : x ~ y | (specializes_of_nhds_within h.le hx).antisymm (specializes_of_nhds_within h.ge hy) | lemma | inseparable_of_nhds_within_eq | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"specializes_of_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.inseparable_iff (hf : inducing f) : f x ~ f y ↔ x ~ y | by simp only [inseparable_iff_specializes_and, hf.specializes_iff] | lemma | inducing.inseparable_iff | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inducing",
"inseparable_iff_specializes_and"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_inseparable_iff {p : X → Prop} (x y : subtype p) : x ~ y ↔ (x : X) ~ y | inducing_coe.inseparable_iff.symm | lemma | subtype_inseparable_iff | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_prod {x₁ x₂ : X} {y₁ y₂ : Y} :
(x₁, y₁) ~ (x₂, y₂) ↔ x₁ ~ x₂ ∧ y₁ ~ y₂ | by simp only [inseparable, nhds_prod_eq, prod_inj] | lemma | inseparable_prod | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inseparable",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable.prod {x₁ x₂ : X} {y₁ y₂ : Y} (hx : x₁ ~ x₂) (hy : y₁ ~ y₂) :
(x₁, y₁) ~ (x₂, y₂) | inseparable_prod.2 ⟨hx, hy⟩ | lemma | inseparable.prod | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_pi {f g : Π i, π i} : f ~ g ↔ ∀ i, f i ~ g i | by simp only [inseparable, nhds_pi, funext_iff, pi_inj] | lemma | inseparable_pi | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inseparable",
"nhds_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (x : X) : x ~ x | eq.refl (𝓝 x) | lemma | inseparable.refl | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rfl : x ~ x | refl x | lemma | inseparable.rfl | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_eq (e : x = y) : inseparable x y | e ▸ refl x | lemma | inseparable.of_eq | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inseparable",
"of_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : x ~ y) : y ~ x | h.symm | lemma | inseparable.symm | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (h₁ : x ~ y) (h₂ : y ~ z) : x ~ z | h₁.trans h₂ | lemma | inseparable.trans | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq (h : x ~ y) : 𝓝 x = 𝓝 y | h | lemma | inseparable.nhds_eq | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_open_iff (h : x ~ y) (hs : is_open s) : x ∈ s ↔ y ∈ s | inseparable_iff_forall_open.1 h s hs | lemma | inseparable.mem_open_iff | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closed_iff (h : x ~ y) (hs : is_closed s) : x ∈ s ↔ y ∈ s | inseparable_iff_forall_closed.1 h s hs | lemma | inseparable.mem_closed_iff | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_of_continuous_at (h : x ~ y) (hx : continuous_at f x) (hy : continuous_at f y) :
f x ~ f y | (h.specializes.map_of_continuous_at hy).antisymm (h.specializes'.map_of_continuous_at hx) | lemma | inseparable.map_of_continuous_at | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (h : x ~ y) (hf : continuous f) : f x ~ f y | h.map_of_continuous_at hf.continuous_at hf.continuous_at | lemma | inseparable.map | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.not_inseparable (hs : is_closed s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ~ y | λ h, hy $ (h.mem_closed_iff hs).1 hx | lemma | is_closed.not_inseparable | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.not_inseparable (hs : is_open s) (hx : x ∈ s) (hy : y ∉ s) : ¬x ~ y | λ h, hy $ (h.mem_open_iff hs).1 hx | lemma | is_open.not_inseparable | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inseparable_setoid : setoid X | { r := (~),
.. setoid.comap 𝓝 ⊥ } | def | inseparable_setoid | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"setoid.comap"
] | A `setoid` version of `inseparable`, used to define the `separation_quotient`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separation_quotient | quotient (inseparable_setoid X) | def | separation_quotient | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inseparable_setoid"
] | The quotient of a topological space by its `inseparable_setoid`. This quotient is guaranteed to
be a T₀ space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk : X → separation_quotient X | quotient.mk' | def | separation_quotient.mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"quotient.mk'",
"separation_quotient"
] | The natural map from a topological space to its separation quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_mk : quotient_map (mk : X → separation_quotient X) | quotient_map_quot_mk | lemma | separation_quotient.quotient_map_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"quotient_map",
"quotient_map_quot_mk",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_mk : continuous (mk : X → separation_quotient X) | continuous_quot_mk | lemma | separation_quotient.continuous_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous",
"continuous_quot_mk",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_mk : mk x = mk y ↔ x ~ y | quotient.eq' | lemma | separation_quotient.mk_eq_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"quotient.eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective_mk : surjective (mk : X → separation_quotient X) | surjective_quot_mk _ | lemma | separation_quotient.surjective_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"separation_quotient",
"surjective_quot_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_mk : range (mk : X → separation_quotient X) = univ | surjective_mk.range_eq | lemma | separation_quotient.range_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_image_mk_open (hs : is_open s) : mk ⁻¹' (mk '' s) = s | begin
refine subset.antisymm _ (subset_preimage_image _ _),
rintro x ⟨y, hys, hxy⟩,
exact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys
end | lemma | separation_quotient.preimage_image_mk_open | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_mk : is_open_map (mk : X → separation_quotient X) | λ s hs, quotient_map_mk.is_open_preimage.1 $ by rwa preimage_image_mk_open hs | lemma | separation_quotient.is_open_map_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_open_map",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_image_mk_closed (hs : is_closed s) : mk ⁻¹' (mk '' s) = s | begin
refine subset.antisymm _ (subset_preimage_image _ _),
rintro x ⟨y, hys, hxy⟩,
exact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys
end | lemma | separation_quotient.preimage_image_mk_closed | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_mk : inducing (mk : X → separation_quotient X) | ⟨le_antisymm (continuous_iff_le_induced.1 continuous_mk)
(λ s hs, ⟨mk '' s, is_open_map_mk s hs, preimage_image_mk_open hs⟩)⟩ | lemma | separation_quotient.inducing_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"inducing",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_mk : is_closed_map (mk : X → separation_quotient X) | inducing_mk.is_closed_map $ by { rw [range_mk], exact is_closed_univ } | lemma | separation_quotient.is_closed_map_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"is_closed_map",
"is_closed_univ",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mk_nhds_mk : comap mk (𝓝 (mk x)) = 𝓝 x | (inducing_mk.nhds_eq_comap _).symm | lemma | separation_quotient.comap_mk_nhds_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mk_nhds_set_image : comap mk (𝓝ˢ (mk '' s)) = 𝓝ˢ s | (inducing_mk.nhds_set_eq_comap _).symm | lemma | separation_quotient.comap_mk_nhds_set_image | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mk_nhds : map mk (𝓝 x) = 𝓝 (mk x) | by rw [← comap_mk_nhds_mk, map_comap_of_surjective surjective_mk] | lemma | separation_quotient.map_mk_nhds | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mk_nhds_set : map mk (𝓝ˢ s) = 𝓝ˢ (mk '' s) | by rw [← comap_mk_nhds_set_image, map_comap_of_surjective surjective_mk] | lemma | separation_quotient.map_mk_nhds_set | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mk_nhds_set : comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t) | by conv_lhs { rw [← image_preimage_eq t surjective_mk, comap_mk_nhds_set_image] } | lemma | separation_quotient.comap_mk_nhds_set | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mk_closure : mk ⁻¹' (closure t) = closure (mk ⁻¹' t) | is_open_map_mk.preimage_closure_eq_closure_preimage continuous_mk t | lemma | separation_quotient.preimage_mk_closure | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mk_interior : mk ⁻¹' (interior t) = interior (mk ⁻¹' t) | is_open_map_mk.preimage_interior_eq_interior_preimage continuous_mk t | lemma | separation_quotient.preimage_mk_interior | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_mk_frontier : mk ⁻¹' (frontier t) = frontier (mk ⁻¹' t) | is_open_map_mk.preimage_frontier_eq_frontier_preimage continuous_mk t | lemma | separation_quotient.preimage_mk_frontier | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mk_closure : mk '' closure s = closure (mk '' s) | (image_closure_subset_closure_image continuous_mk).antisymm $
is_closed_map_mk.closure_image_subset _ | lemma | separation_quotient.image_mk_closure | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"closure",
"image_closure_subset_closure_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_prod_map_mk_nhds (x : X) (y : Y) : map (prod.map mk mk) (𝓝 (x, y)) = 𝓝 (mk x, mk y) | by rw [nhds_prod_eq, ← prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq] | lemma | separation_quotient.map_prod_map_mk_nhds | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mk_nhds_within_preimage (s : set (separation_quotient X)) (x : X) :
map mk (𝓝[mk ⁻¹' s] x) = 𝓝[s] (mk x) | by rw [nhds_within, ← comap_principal, filter.push_pull, nhds_within, map_mk_nhds] | lemma | separation_quotient.map_mk_nhds_within_preimage | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"filter.push_pull",
"nhds_within",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (f : X → α) (hf : ∀ x y, x ~ y → f x = f y) : separation_quotient X → α | λ x, quotient.lift_on' x f hf | def | separation_quotient.lift | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"lift",
"quotient.lift_on'",
"separation_quotient"
] | Lift a map `f : X → α` such that `inseparable x y → f x = f y` to a map
`separation_quotient X → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk {f : X → α} (hf : ∀ x y, x ~ y → f x = f y) (x : X) :
lift f hf (mk x) = f x | rfl | lemma | separation_quotient.lift_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"lift",
"lift_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_comp_mk {f : X → α} (hf : ∀ x y, x ~ y → f x = f y) : lift f hf ∘ mk = f | rfl | lemma | separation_quotient.lift_comp_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lift_nhds_mk {f : X → α} {hf : ∀ x y, x ~ y → f x = f y} {x : X}
{l : filter α} : tendsto (lift f hf) (𝓝 $ mk x) l ↔ tendsto f (𝓝 x) l | by simp only [← map_mk_nhds, tendsto_map'_iff, lift_comp_mk] | lemma | separation_quotient.tendsto_lift_nhds_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"filter",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lift_nhds_within_mk {f : X → α} {hf : ∀ x y, x ~ y → f x = f y} {x : X}
{s : set (separation_quotient X)} {l : filter α} :
tendsto (lift f hf) (𝓝[s] (mk x)) l ↔ tendsto f (𝓝[mk ⁻¹' s] x) l | by simp only [← map_mk_nhds_within_preimage, tendsto_map'_iff, lift_comp_mk] | lemma | separation_quotient.tendsto_lift_nhds_within_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"filter",
"lift",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_lift {f : X → Y} {hf : ∀ x y, x ~ y → f x = f y} {x : X} :
continuous_at (lift f hf) (mk x) ↔ continuous_at f x | tendsto_lift_nhds_mk | lemma | separation_quotient.continuous_at_lift | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous_at",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_lift {f : X → Y} {hf : ∀ x y, x ~ y → f x = f y}
{s : set (separation_quotient X)} {x : X} :
continuous_within_at (lift f hf) s (mk x) ↔ continuous_within_at f (mk ⁻¹' s) x | tendsto_lift_nhds_within_mk | lemma | separation_quotient.continuous_within_at_lift | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous_within_at",
"lift",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_lift {f : X → Y} {hf : ∀ x y, x ~ y → f x = f y}
{s : set (separation_quotient X)} :
continuous_on (lift f hf) s ↔ continuous_on f (mk ⁻¹' s) | by simp only [continuous_on, surjective_mk.forall, continuous_within_at_lift, mem_preimage] | lemma | separation_quotient.continuous_on_lift | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous_on",
"lift",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_lift {f : X → Y} {hf : ∀ x y, x ~ y → f x = f y} :
continuous (lift f hf) ↔ continuous f | by simp only [continuous_iff_continuous_on_univ, continuous_on_lift, preimage_univ] | lemma | separation_quotient.continuous_lift | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"lift"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift₂ (f : X → Y → α) (hf : ∀ a b c d, a ~ c → b ~ d → f a b = f c d) :
separation_quotient X → separation_quotient Y → α | λ x y, quotient.lift_on₂' x y f hf | def | separation_quotient.lift₂ | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"quotient.lift_on₂'",
"separation_quotient"
] | Lift a map `f : X → Y → α` such that `inseparable a b → inseparable c d → f a c = f b d` to a
map `separation_quotient X → separation_quotient Y → α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift₂_mk {f : X → Y → α} (hf : ∀ a b c d, a ~ c → b ~ d → f a b = f c d) (x : X)
(y : Y) : lift₂ f hf (mk x) (mk y) = f x y | rfl | lemma | separation_quotient.lift₂_mk | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, a ~ c → b ~ d → f a b = f c d}
{x : X} {y : Y} {l : filter α} :
tendsto (uncurry $ lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ tendsto (uncurry f) (𝓝 (x, y)) l | by { rw [← map_prod_map_mk_nhds, tendsto_map'_iff], refl } | lemma | separation_quotient.tendsto_lift₂_nhds | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lift₂_nhds_within {f : X → Y → α}
{hf : ∀ a b c d, a ~ c → b ~ d → f a b = f c d} {x : X} {y : Y}
{s : set (separation_quotient X × separation_quotient Y)} {l : filter α} :
tendsto (uncurry $ lift₂ f hf) (𝓝[s] (mk x, mk y)) l ↔
tendsto (uncurry f) (𝓝[prod.map mk mk ⁻¹' s] (x, y)) l | by { rw [nhds_within, ← map_prod_map_mk_nhds, ← filter.push_pull, comap_principal], refl } | lemma | separation_quotient.tendsto_lift₂_nhds_within | topology | src/topology/inseparable.lean | [
"topology.continuous_on",
"data.setoid.basic",
"tactic.tfae"
] | [
"filter",
"filter.push_pull",
"nhds_within",
"separation_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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