statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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dense_inducing [topological_space α] [topological_space β] (i : α → β)
extends inducing i : Prop | (dense : dense_range i) | structure | dense_inducing | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense",
"dense_range",
"inducing",
"topological_space"
] | `i : α → β` is "dense inducing" if it has dense range and the topology on `α`
is the one induced by `i` from the topology on `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_eq_comap (di : dense_inducing i) :
∀ a : α, 𝓝 a = comap i (𝓝 $ i a) | di.to_inducing.nhds_eq_comap | lemma | dense_inducing.nhds_eq_comap | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (di : dense_inducing i) : continuous i | di.to_inducing.continuous | lemma | dense_inducing.continuous | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous",
"dense_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_range : closure (range i) = univ | di.dense.closure_range | lemma | dense_inducing.closure_range | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preconnected_space [preconnected_space α] (di : dense_inducing i) :
preconnected_space β | di.dense.preconnected_space di.continuous | lemma | dense_inducing.preconnected_space | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_inducing",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_image_mem_nhds {s : set α} {a : α} (di : dense_inducing i) (hs : s ∈ 𝓝 a) :
closure (i '' s) ∈ 𝓝 (i a) | begin
rw [di.nhds_eq_comap a, ((nhds_basis_opens _).comap _).mem_iff] at hs,
rcases hs with ⟨U, ⟨haU, hUo⟩, sub : i ⁻¹' U ⊆ s⟩,
refine mem_of_superset (hUo.mem_nhds haU) _,
calc U ⊆ closure (i '' (i ⁻¹' U)) : di.dense.subset_closure_image_preimage_of_is_open hUo
... ⊆ closure (i '' s) : closure_mon... | lemma | dense_inducing.closure_image_mem_nhds | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"closure",
"closure_mono",
"dense_inducing",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_image (di : dense_inducing i) {s : set α} : dense (i '' s) ↔ dense s | begin
refine ⟨λ H x, _, di.dense.dense_image di.continuous⟩,
rw [di.to_inducing.closure_eq_preimage_closure_image, H.closure_eq, preimage_univ],
trivial
end | lemma | dense_inducing.dense_image | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense",
"dense_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_compact_eq_empty [t2_space β] (di : dense_inducing i) (hd : dense (range i)ᶜ)
{s : set α} (hs : is_compact s) : interior s = ∅ | begin
refine eq_empty_iff_forall_not_mem.2 (λ x hx, _),
rw [mem_interior_iff_mem_nhds] at hx,
have := di.closure_image_mem_nhds hx,
rw (hs.image di.continuous).is_closed.closure_eq at this,
rcases hd.inter_nhds_nonempty this with ⟨y, hyi, hys⟩,
exact hyi (image_subset_range _ _ hys)
end | lemma | dense_inducing.interior_compact_eq_empty | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense",
"dense_inducing",
"interior",
"is_closed.closure_eq",
"is_compact",
"mem_interior_iff_mem_nhds",
"t2_space"
] | If `i : α → β` is a dense embedding with dense complement of the range, then any compact set in
`α` has empty interior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod [topological_space γ] [topological_space δ]
{e₁ : α → β} {e₂ : γ → δ} (de₁ : dense_inducing e₁) (de₂ : dense_inducing e₂) :
dense_inducing (λ(p : α × γ), (e₁ p.1, e₂ p.2)) | { induced := (de₁.to_inducing.prod_mk de₂.to_inducing).induced,
dense := de₁.dense.prod_map de₂.dense } | lemma | dense_inducing.prod | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense",
"dense_inducing",
"topological_space"
] | The product of two dense inducings is a dense inducing | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separable_space [separable_space α] : separable_space β | di.dense.separable_space di.continuous | lemma | dense_inducing.separable_space | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [] | If the domain of a `dense_inducing` map is a separable space, then so is the codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_comap_nhds_nhds {d : δ} {a : α} (di : dense_inducing i)
(H : tendsto h (𝓝 d) (𝓝 (i a))) (comm : h ∘ g = i ∘ f) : tendsto f (comap g (𝓝 d)) (𝓝 a) | begin
have lim1 : map g (comap g (𝓝 d)) ≤ 𝓝 d := map_comap_le,
replace lim1 : map h (map g (comap g (𝓝 d))) ≤ map h (𝓝 d) := map_mono lim1,
rw [filter.map_map, comm, ← filter.map_map, map_le_iff_le_comap] at lim1,
have lim2 : comap i (map h (𝓝 d)) ≤ comap i (𝓝 (i a)) := comap_mono H,
rw ← di.nhds_eq_... | lemma | dense_inducing.tendsto_comap_nhds_nhds | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"comm",
"dense_inducing",
"filter.map_map"
] | ```
γ -f→ α
g↓ ↓e
δ -h→ β
``` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_within_ne_bot (di : dense_inducing i) (b : β) :
ne_bot (𝓝[range i] b) | di.dense.nhds_within_ne_bot b | lemma | dense_inducing.nhds_within_ne_bot | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_nhds_ne_bot (di : dense_inducing i) (b : β) : ne_bot (comap i (𝓝 b)) | comap_ne_bot $ λ s hs,
let ⟨_, ⟨ha, a, rfl⟩⟩ := mem_closure_iff_nhds.1 (di.dense b) s hs in ⟨a, ha⟩ | lemma | dense_inducing.comap_nhds_ne_bot | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend (di : dense_inducing i) (f : α → γ) (b : β) : γ | @@lim _ ⟨f (di.dense.some b)⟩ (comap i (𝓝 b)) f | def | dense_inducing.extend | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_inducing",
"extend",
"lim"
] | If `i : α → β` is a dense inducing, then any function `f : α → γ` "extends"
to a function `g = extend di f : β → γ`. If `γ` is Hausdorff and `f` has a
continuous extension, then `g` is the unique such extension. In general,
`g` might not be continuous or even extend `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_eq_of_tendsto [t2_space γ] {b : β} {c : γ} {f : α → γ}
(hf : tendsto f (comap i (𝓝 b)) (𝓝 c)) :
di.extend f b = c | by haveI := di.comap_nhds_ne_bot; exact hf.lim_eq | lemma | dense_inducing.extend_eq_of_tendsto | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_eq_at [t2_space γ] {f : α → γ} {a : α} (hf : continuous_at f a) :
di.extend f (i a) = f a | extend_eq_of_tendsto _ $ di.nhds_eq_comap a ▸ hf | lemma | dense_inducing.extend_eq_at | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous_at",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_eq_at' [t2_space γ] {f : α → γ} {a : α} (c : γ) (hf : tendsto f (𝓝 a) (𝓝 c)) :
di.extend f (i a) = f a | di.extend_eq_at (continuous_at_of_tendsto_nhds hf) | lemma | dense_inducing.extend_eq_at' | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous_at_of_tendsto_nhds",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_eq [t2_space γ] {f : α → γ} (hf : continuous f) (a : α) :
di.extend f (i a) = f a | di.extend_eq_at hf.continuous_at | lemma | dense_inducing.extend_eq | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_eq' [t2_space γ] {f : α → γ}
(di : dense_inducing i) (hf : ∀ b, ∃ c, tendsto f (comap i (𝓝 b)) (𝓝 c)) (a : α) :
di.extend f (i a) = f a | begin
rcases hf (i a) with ⟨b, hb⟩,
refine di.extend_eq_at' b _,
rwa ← di.to_inducing.nhds_eq_comap at hb,
end | lemma | dense_inducing.extend_eq' | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_inducing",
"t2_space"
] | Variation of `extend_eq` where we ask that `f` has a limit along `comap i (𝓝 b)` for each
`b : β`. This is a strictly stronger assumption than continuity of `f`, but in a lot of cases
you'd have to prove it anyway to use `continuous_extend`, so this avoids doing the work twice. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_unique_at [t2_space γ] {b : β} {f : α → γ} {g : β → γ} (di : dense_inducing i)
(hf : ∀ᶠ x in comap i (𝓝 b), g (i x) = f x) (hg : continuous_at g b) :
di.extend f b = g b | begin
refine di.extend_eq_of_tendsto (λ s hs, mem_map.2 _),
suffices : ∀ᶠ (x : α) in comap i (𝓝 b), g (i x) ∈ s,
from hf.mp (this.mono $ λ x hgx hfx, hfx ▸ hgx),
clear hf f,
refine eventually_comap.2 ((hg.eventually hs).mono _),
rintros _ hxs x rfl,
exact hxs
end | lemma | dense_inducing.extend_unique_at | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous_at",
"dense_inducing",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_unique [t2_space γ] {f : α → γ} {g : β → γ} (di : dense_inducing i)
(hf : ∀ x, g (i x) = f x) (hg : continuous g) :
di.extend f = g | funext $ λ b, extend_unique_at di (eventually_of_forall hf) hg.continuous_at | lemma | dense_inducing.extend_unique | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous",
"dense_inducing",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_extend [t3_space γ] {b : β} {f : α → γ} (di : dense_inducing i)
(hf : ∀ᶠ x in 𝓝 b, ∃c, tendsto f (comap i $ 𝓝 x) (𝓝 c)) :
continuous_at (di.extend f) b | begin
set φ := di.extend f,
haveI := di.comap_nhds_ne_bot,
suffices : ∀ V' ∈ 𝓝 (φ b), is_closed V' → φ ⁻¹' V' ∈ 𝓝 b,
by simpa [continuous_at, (closed_nhds_basis _).tendsto_right_iff],
intros V' V'_in V'_closed,
set V₁ := {x | tendsto f (comap i $ 𝓝 x) (𝓝 $ φ x)},
have V₁_in : V₁ ∈ 𝓝 b,
{ filter_u... | lemma | dense_inducing.continuous_at_extend | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"closed_nhds_basis",
"continuous_at",
"dense_inducing",
"is_closed",
"is_open",
"is_open.mem_nhds",
"mem_of_mem_nhds",
"nhds_basis_opens'",
"t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_extend [t3_space γ] {f : α → γ} (di : dense_inducing i)
(hf : ∀b, ∃c, tendsto f (comap i (𝓝 b)) (𝓝 c)) : continuous (di.extend f) | continuous_iff_continuous_at.mpr $ assume b, di.continuous_at_extend $ univ_mem' hf | lemma | dense_inducing.continuous_extend | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous",
"dense_inducing",
"t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk'
(i : α → β)
(c : continuous i)
(dense : ∀x, x ∈ closure (range i))
(H : ∀ (a:α) s ∈ 𝓝 a,
∃t ∈ 𝓝 (i a), ∀ b, i b ∈ t → b ∈ s) :
dense_inducing i | { induced := (induced_iff_nhds_eq i).2 $
λ a, le_antisymm (tendsto_iff_comap.1 $ c.tendsto _) (by simpa [filter.le_def] using H a),
dense := dense } | lemma | dense_inducing.mk' | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"closure",
"continuous",
"dense",
"dense_inducing",
"filter.le_def",
"induced_iff_nhds_eq",
"mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_embedding [topological_space α] [topological_space β] (e : α → β)
extends dense_inducing e : Prop | (inj : function.injective e) | structure | dense_embedding | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_inducing",
"topological_space"
] | A dense embedding is an embedding with dense image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_embedding.mk'
[topological_space α] [topological_space β] (e : α → β)
(c : continuous e)
(dense : dense_range e)
(inj : function.injective e)
(H : ∀ (a:α) s ∈ 𝓝 a,
∃t ∈ 𝓝 (e a), ∀ b, e b ∈ t → b ∈ s) :
dense_embedding e | { inj := inj,
..dense_inducing.mk' e c dense H} | theorem | dense_embedding.mk' | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous",
"dense",
"dense_embedding",
"dense_inducing.mk'",
"dense_range",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_iff {x y} : e x = e y ↔ x = y | de.inj.eq_iff | lemma | dense_embedding.inj_iff | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_embedding : embedding e | { induced := de.induced,
inj := de.inj } | lemma | dense_embedding.to_embedding | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separable_space [separable_space α] : separable_space β | de.to_dense_inducing.separable_space | lemma | dense_embedding.separable_space | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [] | If the domain of a `dense_embedding` is a separable space, then so is its codomain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod {e₁ : α → β} {e₂ : γ → δ} (de₁ : dense_embedding e₁)
(de₂ : dense_embedding e₂) :
dense_embedding (λ(p : α × γ), (e₁ p.1, e₂ p.2)) | { inj := assume ⟨x₁, x₂⟩ ⟨y₁, y₂⟩,
by simp; exact assume h₁ h₂, ⟨de₁.inj h₁, de₂.inj h₂⟩,
..dense_inducing.prod de₁.to_dense_inducing de₂.to_dense_inducing } | lemma | dense_embedding.prod | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_embedding",
"dense_inducing.prod"
] | The product of two dense embeddings is a dense embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype_emb {α : Type*} (p : α → Prop) (e : α → β) (x : {x // p x}) :
{x // x ∈ closure (e '' {x | p x})} | ⟨e x, subset_closure $ mem_image_of_mem e x.prop⟩ | def | dense_embedding.subtype_emb | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"closure",
"subset_closure"
] | The dense embedding of a subtype inside its closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype (p : α → Prop) : dense_embedding (subtype_emb p e) | { dense := dense_iff_closure_eq.2 $
begin
ext ⟨x, hx⟩,
rw image_eq_range at hx,
simpa [closure_subtype, ← range_comp, (∘)],
end,
inj := (de.inj.comp subtype.coe_injective).cod_restrict _,
induced := (induced_iff_nhds_eq _).2 (assume ⟨x, hx⟩,
by simp [subtype_emb, nhds_subtype_eq_comap,... | lemma | dense_embedding.subtype | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"closure_subtype",
"dense",
"dense_embedding",
"induced_iff_nhds_eq",
"nhds_subtype_eq_comap",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_image {s : set α} : dense (e '' s) ↔ dense s | de.to_dense_inducing.dense_image | lemma | dense_embedding.dense_image | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_embedding_id {α : Type*} [topological_space α] : dense_embedding (id : α → α) | { dense := dense_range_id,
.. embedding_id } | lemma | dense_embedding_id | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense",
"dense_embedding",
"dense_range_id",
"embedding_id",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.dense_embedding_coe [topological_space α] {s : set α} (hs : dense s) :
dense_embedding (coe : s → α) | { dense := hs.dense_range_coe,
.. embedding_subtype_coe } | lemma | dense.dense_embedding_coe | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense",
"dense_embedding",
"embedding_subtype_coe",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_property [topological_space β] {e : α → β} {p : β → Prop}
(he : dense_range e) (hp : is_closed {x | p x}) (h : ∀a, p (e a)) :
∀b, p b | have univ ⊆ {b | p b},
from calc univ = closure (range e) : he.closure_range.symm
... ⊆ closure {b | p b} : closure_mono $ range_subset_iff.mpr h
... = _ : hp.closure_eq,
assume b, this trivial | lemma | is_closed_property | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"closure",
"closure_mono",
"dense_range",
"is_closed",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_property2 [topological_space β] {e : α → β} {p : β → β → Prop}
(he : dense_range e) (hp : is_closed {q:β×β | p q.1 q.2}) (h : ∀a₁ a₂, p (e a₁) (e a₂)) :
∀b₁ b₂, p b₁ b₂ | have ∀q:β×β, p q.1 q.2,
from is_closed_property (he.prod_map he) hp $ λ _, h _ _,
assume b₁ b₂, this ⟨b₁, b₂⟩ | lemma | is_closed_property2 | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_range",
"is_closed",
"is_closed_property",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_property3 [topological_space β] {e : α → β} {p : β → β → β → Prop}
(he : dense_range e) (hp : is_closed {q:β×β×β | p q.1 q.2.1 q.2.2})
(h : ∀a₁ a₂ a₃, p (e a₁) (e a₂) (e a₃)) :
∀b₁ b₂ b₃, p b₁ b₂ b₃ | have ∀q:β×β×β, p q.1 q.2.1 q.2.2,
from is_closed_property (he.prod_map $ he.prod_map he) hp $ λ _, h _ _ _,
assume b₁ b₂ b₃, this ⟨b₁, b₂, b₃⟩ | lemma | is_closed_property3 | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_range",
"is_closed",
"is_closed_property",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.induction_on [topological_space β] {e : α → β} (he : dense_range e) {p : β → Prop}
(b₀ : β) (hp : is_closed {b | p b}) (ih : ∀a:α, p $ e a) : p b₀ | is_closed_property he hp ih b₀ | lemma | dense_range.induction_on | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_range",
"ih",
"is_closed",
"is_closed_property",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.induction_on₂ [topological_space β] {e : α → β} {p : β → β → Prop}
(he : dense_range e) (hp : is_closed {q:β×β | p q.1 q.2}) (h : ∀a₁ a₂, p (e a₁) (e a₂))
(b₁ b₂ : β) : p b₁ b₂ | is_closed_property2 he hp h _ _ | lemma | dense_range.induction_on₂ | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_range",
"is_closed",
"is_closed_property2",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.induction_on₃ [topological_space β] {e : α → β} {p : β → β → β → Prop}
(he : dense_range e) (hp : is_closed {q:β×β×β | p q.1 q.2.1 q.2.2})
(h : ∀a₁ a₂ a₃, p (e a₁) (e a₂) (e a₃))
(b₁ b₂ b₃ : β) : p b₁ b₂ b₃ | is_closed_property3 he hp h _ _ _ | lemma | dense_range.induction_on₃ | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"dense_range",
"is_closed",
"is_closed_property3",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.equalizer (hfd : dense_range f)
{g h : β → γ} (hg : continuous g) (hh : continuous h) (H : g ∘ f = h ∘ f) :
g = h | funext $ λ y, hfd.induction_on y (is_closed_eq hg hh) $ congr_fun H | lemma | dense_range.equalizer | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"continuous",
"dense_range",
"is_closed_eq"
] | Two continuous functions to a t2-space that agree on the dense range of a function are equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.has_basis_of_dense_inducing
[topological_space α] [topological_space β] [t3_space β]
{ι : Type*} {s : ι → set α} {p : ι → Prop} {x : α} (h : (𝓝 x).has_basis p s)
{f : α → β} (hf : dense_inducing f) :
(𝓝 (f x)).has_basis p $ λ i, closure $ f '' (s i) | begin
rw filter.has_basis_iff at h ⊢,
intros T,
refine ⟨λ hT, _, λ hT, _⟩,
{ obtain ⟨T', hT₁, hT₂, hT₃⟩ := exists_mem_nhds_is_closed_subset hT,
have hT₄ : f⁻¹' T' ∈ 𝓝 x,
{ rw hf.to_inducing.nhds_eq_comap x,
exact ⟨T', hT₁, subset.rfl⟩, },
obtain ⟨i, hi, hi'⟩ := (h _).mp hT₄,
exact ⟨i, hi,... | lemma | filter.has_basis.has_basis_of_dense_inducing | topology | src/topology/dense_embedding.lean | [
"topology.separation",
"topology.bases"
] | [
"closure",
"closure_minimal",
"closure_mono",
"dense_inducing",
"exists_mem_nhds_is_closed_subset",
"filter.has_basis_iff",
"t3_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_quotient (X : Type*) [topological_space X] extends setoid X | (is_open_set_of_rel : ∀ x, is_open (set_of (to_setoid.rel x))) | structure | discrete_quotient | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"is_open",
"topological_space"
] | The type of discrete quotients of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_clopen {A : set X} (h : is_clopen A) : discrete_quotient X | { to_setoid := ⟨λ x y, x ∈ A ↔ y ∈ A, λ _, iff.rfl, λ _ _, iff.symm, λ _ _ _, iff.trans⟩,
is_open_set_of_rel := λ x,
by by_cases hx : x ∈ A; simp [setoid.rel, hx, h.1, h.2, ← compl_set_of] } | def | discrete_quotient.of_clopen | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_quotient",
"is_clopen",
"setoid.rel"
] | Construct a discrete quotient from a clopen set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl : ∀ x, S.rel x x | S.refl' | lemma | discrete_quotient.refl | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm {x y : X} : S.rel x y → S.rel y x | S.symm' | lemma | discrete_quotient.symm | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans {x y z} : S.rel x y → S.rel y z → S.rel x z | S.trans' | lemma | discrete_quotient.trans | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj : X → S | quotient.mk' | def | discrete_quotient.proj | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"quotient.mk'"
] | The projection from `X` to the given discrete quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber_eq (x : X) : S.proj ⁻¹' {S.proj x} = set_of (S.rel x) | set.ext $ λ y, eq_comm.trans quotient.eq' | lemma | discrete_quotient.fiber_eq | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"quotient.eq'",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_surjective : function.surjective S.proj | quotient.surjective_quotient_mk' | lemma | discrete_quotient.proj_surjective | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"quotient.surjective_quotient_mk'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_quotient_map : quotient_map S.proj | quotient_map_quot_mk | lemma | discrete_quotient.proj_quotient_map | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"quotient_map",
"quotient_map_quot_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_continuous : continuous S.proj | S.proj_quotient_map.continuous | lemma | discrete_quotient.proj_continuous | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_is_locally_constant : is_locally_constant S.proj | (is_locally_constant.iff_continuous S.proj).2 S.proj_continuous | lemma | discrete_quotient.proj_is_locally_constant | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"is_locally_constant",
"is_locally_constant.iff_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_preimage (A : set S) : is_clopen (S.proj ⁻¹' A) | (is_clopen_discrete A).preimage S.proj_continuous | lemma | discrete_quotient.is_clopen_preimage | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"is_clopen",
"is_clopen_discrete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_preimage (A : set S) : is_open (S.proj ⁻¹' A) | (S.is_clopen_preimage A).1 | lemma | discrete_quotient.is_open_preimage | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_preimage (A : set S) : is_closed (S.proj ⁻¹' A) | (S.is_clopen_preimage A).2 | lemma | discrete_quotient.is_closed_preimage | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_set_of_rel (x : X) : is_clopen (set_of (S.rel x)) | by { rw [← fiber_eq], apply is_clopen_preimage } | theorem | discrete_quotient.is_clopen_set_of_rel | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inhabited_quotient [inhabited X] : inhabited S | ⟨S.proj default⟩ | instance | discrete_quotient.inhabited_quotient | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (S : discrete_quotient Y) : discrete_quotient X | { to_setoid := setoid.comap f S.1,
is_open_set_of_rel := λ y, (S.2 _).preimage f.continuous } | def | discrete_quotient.comap | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_quotient",
"setoid.comap"
] | Comap a discrete quotient along a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_id : S.comap (continuous_map.id X) = S | by { ext, refl } | lemma | discrete_quotient.comap_id | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"continuous_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comp (S : discrete_quotient Z) : S.comap (g.comp f) = (S.comap g).comap f | rfl | lemma | discrete_quotient.comap_comp | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mono {A B : discrete_quotient Y} (h : A ≤ B) : A.comap f ≤ B.comap f | by tauto | lemma | discrete_quotient.comap_mono | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le (h : A ≤ B) : A → B | quotient.map' (λ x, x) h | def | discrete_quotient.of_le | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"quotient.map'"
] | The map induced by a refinement of a discrete quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_le_refl : of_le (le_refl A) = id | by { ext ⟨⟩, refl } | lemma | discrete_quotient.of_le_refl | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_refl_apply (a : A) : of_le (le_refl A) a = a | by simp | lemma | discrete_quotient.of_le_refl_apply | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_of_le (h₁ : A ≤ B) (h₂ : B ≤ C) (x : A) :
of_le h₂ (of_le h₁ x) = of_le (h₁.trans h₂) x | by { rcases x with ⟨⟩, refl } | lemma | discrete_quotient.of_le_of_le | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_comp_of_le (h₁ : A ≤ B) (h₂ : B ≤ C) :
of_le h₂ ∘ of_le h₁ = of_le (le_trans h₁ h₂) | funext $ of_le_of_le _ _ | lemma | discrete_quotient.of_le_comp_of_le | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_continuous (h : A ≤ B) : continuous (of_le h) | continuous_of_discrete_topology | lemma | discrete_quotient.of_le_continuous | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"continuous",
"continuous_of_discrete_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_proj (h : A ≤ B) (x : X) : of_le h (A.proj x) = B.proj x | quotient.sound' (B.refl _) | lemma | discrete_quotient.of_le_proj | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"quotient.sound'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_comp_proj (h : A ≤ B) : of_le h ∘ A.proj = B.proj | funext $ of_le_proj _ | lemma | discrete_quotient.of_le_comp_proj | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_bot_eq [locally_connected_space X] {x y : X} :
proj ⊥ x = proj ⊥ y ↔ connected_component x = connected_component y | quotient.eq' | theorem | discrete_quotient.proj_bot_eq | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"connected_component",
"locally_connected_space",
"quotient.eq'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_bot_inj [discrete_topology X] {x y : X} :
proj ⊥ x = proj ⊥ y ↔ x = y | by simp | theorem | discrete_quotient.proj_bot_inj | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_bot_injective [discrete_topology X] :
injective (⊥ : discrete_quotient X).proj | λ _ _, proj_bot_inj.1 | theorem | discrete_quotient.proj_bot_injective | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_quotient",
"discrete_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_bot_bijective [discrete_topology X] :
bijective (⊥ : discrete_quotient X).proj | ⟨proj_bot_injective, proj_surjective _⟩ | theorem | discrete_quotient.proj_bot_bijective | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_quotient",
"discrete_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap : Prop | A ≤ B.comap f | def | discrete_quotient.le_comap | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | Given `f : C(X, Y)`, `le_comap cont A B` is defined as `A ≤ B.comap f`. Mathematically this
means that `f` descends to a morphism `A → B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_comap_id : le_comap (continuous_map.id X) A A | λ _ _, id | theorem | discrete_quotient.le_comap_id | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"continuous_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap_id_iff : le_comap (continuous_map.id X) A A' ↔ A ≤ A' | iff.rfl | theorem | discrete_quotient.le_comap_id_iff | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"continuous_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap.comp :
le_comap g B C → le_comap f A B → le_comap (g.comp f) A C | by tauto | theorem | discrete_quotient.le_comap.comp | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_comap.mono (h : le_comap f A B) (hA : A' ≤ A) (hB : B ≤ B') : le_comap f A' B' | hA.trans $ le_trans h $ comap_mono _ hB | theorem | discrete_quotient.le_comap.mono | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : C(X, Y)) (cond : le_comap f A B) : A → B | quotient.map' f cond | def | discrete_quotient.map | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"quotient.map'"
] | Map a discrete quotient along a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_continuous (cond : le_comap f A B) : continuous (map f cond) | continuous_of_discrete_topology | theorem | discrete_quotient.map_continuous | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"continuous",
"continuous_of_discrete_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_proj (cond : le_comap f A B) : map f cond ∘ A.proj = B.proj ∘ f | rfl | theorem | discrete_quotient.map_comp_proj | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_proj (cond : le_comap f A B) (x : X) : map f cond (A.proj x) = B.proj (f x) | rfl | theorem | discrete_quotient.map_proj | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map _ (le_comap_id A) = id | by ext ⟨⟩; refl | theorem | discrete_quotient.map_id | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (h1 : le_comap g B C) (h2 : le_comap f A B) :
map (g.comp f) (h1.comp h2) = map g h1 ∘ map f h2 | by { ext ⟨⟩, refl } | theorem | discrete_quotient.map_comp | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_map (cond : le_comap f A B) (h : B ≤ B') (a : A) :
of_le h (map f cond a) = map f (cond.mono le_rfl h) a | by { rcases a with ⟨⟩, refl } | theorem | discrete_quotient.of_le_map | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_le_comp_map (cond : le_comap f A B) (h : B ≤ B') :
of_le h ∘ map f cond = map f (cond.mono le_rfl h) | funext $ of_le_map cond h | theorem | discrete_quotient.of_le_comp_map | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_of_le (cond : le_comap f A B) (h : A' ≤ A) (c : A') :
map f cond (of_le h c) = map f (cond.mono h le_rfl) c | by { rcases c with ⟨⟩, refl } | theorem | discrete_quotient.map_of_le | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_of_le (cond : le_comap f A B) (h : A' ≤ A) :
map f cond ∘ of_le h = map f (cond.mono h le_rfl) | funext $ map_of_le cond h | theorem | discrete_quotient.map_comp_of_le | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_forall_proj_eq [t2_space X] [compact_space X] [disc : totally_disconnected_space X]
{x y : X} (h : ∀ Q : discrete_quotient X, Q.proj x = Q.proj y) : x = y | begin
rw [← mem_singleton_iff, ← connected_component_eq_singleton, connected_component_eq_Inter_clopen,
mem_Inter],
rintro ⟨U, hU1, hU2⟩,
exact (quotient.exact' (h (of_clopen hU1))).mpr hU2
end | lemma | discrete_quotient.eq_of_forall_proj_eq | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"compact_space",
"connected_component_eq_Inter_clopen",
"connected_component_eq_singleton",
"discrete_quotient",
"quotient.exact'",
"t2_space",
"totally_disconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fiber_subset_of_le {A B : discrete_quotient X} (h : A ≤ B) (a : A) :
A.proj ⁻¹' {a} ⊆ B.proj ⁻¹' {of_le h a} | begin
rcases A.proj_surjective a with ⟨a, rfl⟩,
rw [fiber_eq, of_le_proj, fiber_eq],
exact λ _ h', h h'
end | lemma | discrete_quotient.fiber_subset_of_le | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_quotient"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_of_compat [compact_space X] (Qs : Π (Q : discrete_quotient X), Q)
(compat : ∀ (A B : discrete_quotient X) (h : A ≤ B), of_le h (Qs _) = Qs _) :
∃ x : X, ∀ Q : discrete_quotient X, Q.proj x = Qs _ | begin
obtain ⟨x,hx⟩ : (⋂ Q, proj Q ⁻¹' {Qs Q}).nonempty :=
is_compact.nonempty_Inter_of_directed_nonempty_compact_closed
(λ (Q : discrete_quotient X), Q.proj ⁻¹' {Qs _}) (directed_of_inf $ λ A B h, _)
(λ Q, (singleton_nonempty _).preimage Q.proj_surjective)
(λ i, (is_closed_preimage _ _).is_com... | lemma | discrete_quotient.exists_of_compat | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"compact_space",
"directed_of_inf",
"discrete_quotient",
"is_compact",
"is_compact.nonempty_Inter_of_directed_nonempty_compact_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_quotient : discrete_quotient X | { to_setoid := setoid.comap f ⊥,
is_open_set_of_rel := λ x, f.is_locally_constant _ } | def | locally_constant.discrete_quotient | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"discrete_quotient",
"setoid.comap"
] | Any locally constant function induces a discrete quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift : locally_constant f.discrete_quotient α | ⟨λ a, quotient.lift_on' a f (λ a b, id), λ A, is_open_discrete _⟩ | def | locally_constant.lift | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [
"is_open_discrete",
"lift",
"locally_constant",
"quotient.lift_on'"
] | The (locally constant) function from the discrete quotient associated to a locally constant
function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_comp_proj : f.lift ∘ f.discrete_quotient.proj = f | by { ext, refl } | lemma | locally_constant.lift_comp_proj | topology | src/topology/discrete_quotient.lean | [
"topology.separation",
"topology.subset_properties",
"topology.locally_constant.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_from (A : set X) (f : X → Y) : X → Y | λ x, @@lim _ ⟨f x⟩ (𝓝[A] x) f | def | extend_from | topology | src/topology/extend_from.lean | [
"topology.separation"
] | [
"lim"
] | Extend a function from a set `A`. The resulting function `g` is such that
at any `x₀`, if `f` converges to some `y` as `x` tends to `x₀` within `A`,
then `g x₀` is defined to be one of these `y`. Else, `g x₀` could be anything. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_extend_from {A : set X} {f : X → Y} {x : X}
(h : ∃ y, tendsto f (𝓝[A] x) (𝓝 y)) : tendsto f (𝓝[A] x) (𝓝 $ extend_from A f x) | tendsto_nhds_lim h | lemma | tendsto_extend_from | topology | src/topology/extend_from.lean | [
"topology.separation"
] | [
"extend_from",
"tendsto_nhds_lim"
] | If `f` converges to some `y` as `x` tends to `x₀` within `A`,
then `f` tends to `extend_from A f x` as `x` tends to `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_from_eq [t2_space Y] {A : set X} {f : X → Y} {x : X} {y : Y} (hx : x ∈ closure A)
(hf : tendsto f (𝓝[A] x) (𝓝 y)) : extend_from A f x = y | begin
haveI := mem_closure_iff_nhds_within_ne_bot.mp hx,
exact tendsto_nhds_unique (tendsto_nhds_lim ⟨y, hf⟩) hf,
end | lemma | extend_from_eq | topology | src/topology/extend_from.lean | [
"topology.separation"
] | [
"closure",
"extend_from",
"t2_space",
"tendsto_nhds_lim",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_from_extends [t2_space Y] {f : X → Y} {A : set X} (hf : continuous_on f A) :
∀ x ∈ A, extend_from A f x = f x | λ x x_in, extend_from_eq (subset_closure x_in) (hf x x_in) | lemma | extend_from_extends | topology | src/topology/extend_from.lean | [
"topology.separation"
] | [
"continuous_on",
"extend_from",
"extend_from_eq",
"subset_closure",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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