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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
trans_equiv_eq_trans (e' : β ≃ₜ γ) : e.trans_homeomorph e' = e.trans e'.to_local_homeomorph | to_local_equiv_injective $ local_equiv.trans_equiv_eq_trans _ _ | lemma | local_homeomorph.trans_equiv_eq_trans | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.trans_equiv_eq_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.homeomorph.trans_local_homeomorph (e : α ≃ₜ β) : local_homeomorph α γ | { to_local_equiv := e.to_equiv.trans_local_equiv e'.to_local_equiv,
open_source := e'.open_source.preimage e.continuous,
open_target := e'.open_target,
continuous_to_fun := e'.continuous_on.comp e.continuous.continuous_on (λ x h, h),
continuous_inv_fun := e.symm.continuous.comp_continuous_on e'.symm.continuous_... | def | homeomorph.trans_local_homeomorph | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | Precompose a local homeomorphism with an homeomorphism.
We modify the source and target to have better definitional behavior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.homeomorph.trans_local_homeomorph_eq_trans (e : α ≃ₜ β) :
e.trans_local_homeomorph e' = e.to_local_homeomorph.trans e' | to_local_equiv_injective $ equiv.trans_local_equiv_eq_trans _ _ | lemma | homeomorph.trans_local_homeomorph_eq_trans | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"equiv.trans_local_equiv_eq_trans"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_source (e e' : local_homeomorph α β) : Prop | e.source = e'.source ∧ (eq_on e e' e.source) | def | local_homeomorph.eq_on_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | `eq_on_source e e'` means that `e` and `e'` have the same source, and coincide there. They
should really be considered the same local equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_source_iff (e e' : local_homeomorph α β) :
eq_on_source e e' ↔ local_equiv.eq_on_source e.to_local_equiv e'.to_local_equiv | iff.rfl | lemma | local_homeomorph.eq_on_source_iff | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.eq_on_source",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_source.symm' {e e' : local_homeomorph α β} (h : e ≈ e') : e.symm ≈ e'.symm | local_equiv.eq_on_source.symm' h | lemma | local_homeomorph.eq_on_source.symm' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.eq_on_source.symm'",
"local_homeomorph"
] | If two local homeomorphisms are equivalent, so are their inverses | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_source.source_eq {e e' : local_homeomorph α β} (h : e ≈ e') : e.source = e'.source | h.1 | lemma | local_homeomorph.eq_on_source.source_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | Two equivalent local homeomorphisms have the same source | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_source.target_eq {e e' : local_homeomorph α β} (h : e ≈ e') : e.target = e'.target | h.symm'.1 | lemma | local_homeomorph.eq_on_source.target_eq | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | Two equivalent local homeomorphisms have the same target | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_source.eq_on {e e' : local_homeomorph α β} (h : e ≈ e') :
eq_on e e' e.source | h.2 | lemma | local_homeomorph.eq_on_source.eq_on | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | Two equivalent local homeomorphisms have coinciding `to_fun` on the source | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_source.symm_eq_on_target {e e' : local_homeomorph α β} (h : e ≈ e') :
eq_on e.symm e'.symm e.target | h.symm'.2 | lemma | local_homeomorph.eq_on_source.symm_eq_on_target | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | Two equivalent local homeomorphisms have coinciding `inv_fun` on the target | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_source.trans' {e e' : local_homeomorph α β} {f f' : local_homeomorph β γ}
(he : e ≈ e') (hf : f ≈ f') : e.trans f ≈ e'.trans f' | local_equiv.eq_on_source.trans' he hf | lemma | local_homeomorph.eq_on_source.trans' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.eq_on_source.trans'",
"local_homeomorph"
] | Composition of local homeomorphisms respects equivalence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_on_source.restr {e e' : local_homeomorph α β} (he : e ≈ e') (s : set α) :
e.restr s ≈ e'.restr s | local_equiv.eq_on_source.restr he _ | lemma | local_homeomorph.eq_on_source.restr | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.eq_on_source.restr",
"local_homeomorph"
] | Restriction of local homeomorphisms respects equivalence | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.eq_on.restr_eq_on_source {e e' : local_homeomorph α β}
(h : eq_on e e' (e.source ∩ e'.source)) :
e.restr e'.source ≈ e'.restr e.source | begin
split,
{ rw e'.restr_source' _ e.open_source,
rw e.restr_source' _ e'.open_source,
exact set.inter_comm _ _ },
{ rw e.restr_source' _ e'.open_source,
refine (eq_on.trans _ h).trans _;
simp only with mfld_simps },
end | lemma | local_homeomorph.set.eq_on.restr_eq_on_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph",
"set.inter_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_self_symm :
e.trans e.symm ≈ local_homeomorph.of_set e.source e.open_source | local_equiv.trans_self_symm _ | lemma | local_homeomorph.trans_self_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.trans_self_symm",
"local_homeomorph.of_set"
] | Composition of a local homeomorphism and its inverse is equivalent to the restriction of the
identity to the source | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_symm_self :
e.symm.trans e ≈ local_homeomorph.of_set e.target e.open_target | e.symm.trans_self_symm | lemma | local_homeomorph.trans_symm_self | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph.of_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_eq_on_source_univ {e e' : local_homeomorph α β} (h : e ≈ e')
(s : e.source = univ) (t : e.target = univ) : e = e' | eq_of_local_equiv_eq $ local_equiv.eq_of_eq_on_source_univ _ _ h s t | lemma | local_homeomorph.eq_of_eq_on_source_univ | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.eq_of_eq_on_source_univ",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (e : local_homeomorph α β) (e' : local_homeomorph γ δ) :
local_homeomorph (α × γ) (β × δ) | { open_source := e.open_source.prod e'.open_source,
open_target := e.open_target.prod e'.open_target,
continuous_to_fun := e.continuous_on.prod_map e'.continuous_on,
continuous_inv_fun := e.continuous_on_symm.prod_map e'.continuous_on_symm,
to_local_equiv := e.to_local_equiv.prod e'.to_local_equiv } | def | local_homeomorph.prod | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | The product of two local homeomorphisms, as a local homeomorphism on the product space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_symm (e : local_homeomorph α β) (e' : local_homeomorph γ δ) :
(e.prod e').symm = (e.symm.prod e'.symm) | rfl | lemma | local_homeomorph.prod_symm | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_prod_refl {α β : Type*} [topological_space α] [topological_space β] :
(local_homeomorph.refl α).prod (local_homeomorph.refl β) = local_homeomorph.refl (α × β) | by { ext1 ⟨x, y⟩, { refl }, { rintro ⟨x, y⟩, refl }, exact univ_prod_univ } | lemma | local_homeomorph.refl_prod_refl | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph.refl",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_trans
{η : Type*} {ε : Type*} [topological_space η] [topological_space ε]
(e : local_homeomorph α β) (f : local_homeomorph β γ)
(e' : local_homeomorph δ η) (f' : local_homeomorph η ε) :
(e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f') | local_homeomorph.eq_of_local_equiv_eq $
by dsimp only [trans_to_local_equiv, prod_to_local_equiv]; apply local_equiv.prod_trans | lemma | local_homeomorph.prod_trans | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_equiv.prod_trans",
"local_homeomorph",
"local_homeomorph.eq_of_local_equiv_eq",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_prod_of_nonempty {e₁ e₁' : local_homeomorph α β} {e₂ e₂' : local_homeomorph γ δ}
(h : (e₁.prod e₂).source.nonempty) :
e₁.prod e₂ = e₁'.prod e₂' ↔ e₁ = e₁' ∧ e₂ = e₂' | begin
obtain ⟨⟨x, y⟩, -⟩ := id h,
haveI : nonempty α := ⟨x⟩,
haveI : nonempty β := ⟨e₁ x⟩,
haveI : nonempty γ := ⟨y⟩,
haveI : nonempty δ := ⟨e₂ y⟩,
simp_rw [local_homeomorph.ext_iff, prod_apply, prod_symm_apply, prod_source, prod.ext_iff,
set.prod_eq_prod_iff_of_nonempty h,
forall_and_distrib, prod... | lemma | local_homeomorph.prod_eq_prod_of_nonempty | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"forall_and_distrib",
"forall_const",
"forall_forall_const",
"local_homeomorph",
"local_homeomorph.ext_iff",
"prod.ext_iff",
"set.prod_eq_prod_iff_of_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_prod_of_nonempty' {e₁ e₁' : local_homeomorph α β} {e₂ e₂' : local_homeomorph γ δ}
(h : (e₁'.prod e₂').source.nonempty) :
e₁.prod e₂ = e₁'.prod e₂' ↔ e₁ = e₁' ∧ e₂ = e₂' | by rw [eq_comm, prod_eq_prod_of_nonempty h, eq_comm, @eq_comm _ e₂'] | lemma | local_homeomorph.prod_eq_prod_of_nonempty' | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
piecewise (e e' : local_homeomorph α β) (s : set α) (t : set β)
[∀ x, decidable (x ∈ s)] [∀ y, decidable (y ∈ t)] (H : e.is_image s t) (H' : e'.is_image s t)
(Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
(Heq : eq_on e e' (e.source ∩ frontier s)) :
local_homeomorph α β | { to_local_equiv := e.to_local_equiv.piecewise e'.to_local_equiv s t H H',
open_source := e.open_source.ite e'.open_source Hs,
open_target := e.open_target.ite e'.open_target $
H.frontier.inter_eq_of_inter_eq_of_eq_on H'.frontier Hs Heq,
continuous_to_fun := continuous_on_piecewise_ite e.continuous_on e'.cont... | def | local_homeomorph.piecewise | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_on_piecewise_ite",
"frontier",
"local_homeomorph"
] | Combine two `local_homeomorph`s using `set.piecewise`. The source of the new `local_homeomorph`
is `s.ite e.source e'.source = e.source ∩ s ∪ e'.source \ s`, and similarly for target. The
function sends `e.source ∩ s` to `e.target ∩ t` using `e` and `e'.source \ s` to `e'.target \ t`
using `e'`, and similarly for the ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_piecewise (e e' : local_homeomorph α β) {s : set α} {t : set β}
[∀ x, decidable (x ∈ s)] [∀ y, decidable (y ∈ t)] (H : e.is_image s t) (H' : e'.is_image s t)
(Hs : e.source ∩ frontier s = e'.source ∩ frontier s)
(Heq : eq_on e e' (e.source ∩ frontier s)) :
(e.piecewise e' s t H H' Hs Heq).symm =
e.symm... | rfl | lemma | local_homeomorph.symm_piecewise | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"frontier",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint_union (e e' : local_homeomorph α β)
[∀ x, decidable (x ∈ e.source)] [∀ y, decidable (y ∈ e.target)]
(Hs : disjoint e.source e'.source) (Ht : disjoint e.target e'.target) :
local_homeomorph α β | (e.piecewise e' e.source e.target e.is_image_source_target
(e'.is_image_source_target_of_disjoint e Hs.symm Ht.symm)
(by rw [e.open_source.inter_frontier_eq, (Hs.symm.frontier_right e'.open_source).inter_eq])
(by { rw e.open_source.inter_frontier_eq, exact eq_on_empty _ _ })).replace_equiv
(e.to_local_equiv.d... | def | local_homeomorph.disjoint_union | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"disjoint",
"local_equiv.disjoint_union_eq_piecewise",
"local_homeomorph"
] | Combine two `local_homeomorph`s with disjoint sources and disjoint targets. We reuse
`local_homeomorph.piecewise` then override `to_local_equiv` to `local_equiv.disjoint_union`.
This way we have better definitional equalities for `source` and `target`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi : local_homeomorph (Π i, Xi i) (Π i, Yi i) | { to_local_equiv := local_equiv.pi (λ i, (ei i).to_local_equiv),
open_source := is_open_set_pi finite_univ $ λ i hi, (ei i).open_source,
open_target := is_open_set_pi finite_univ $ λ i hi, (ei i).open_target,
continuous_to_fun := continuous_on_pi.2 $ λ i, (ei i).continuous_on.comp
(continuous_apply _).continu... | def | local_homeomorph.pi | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_apply",
"continuous_on",
"continuous_on.comp",
"is_open_set_pi",
"local_equiv.pi",
"local_homeomorph"
] | The product of a finite family of `local_homeomorph`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_within_at_iff_continuous_within_at_comp_right
{f : β → γ} {s : set β} {x : β} (h : x ∈ e.target) :
continuous_within_at f s x ↔ continuous_within_at (f ∘ e) (e ⁻¹' s) (e.symm x) | by simp_rw [continuous_within_at, ← @tendsto_map'_iff _ _ _ _ e,
e.map_nhds_within_preimage_eq (e.map_target h), (∘), e.right_inv h] | lemma | local_homeomorph.continuous_within_at_iff_continuous_within_at_comp_right | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_within_at"
] | Continuity within a set at a point can be read under right composition with a local
homeomorphism, if the point is in its target | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_iff_continuous_at_comp_right
{f : β → γ} {x : β} (h : x ∈ e.target) :
continuous_at f x ↔ continuous_at (f ∘ e) (e.symm x) | by rw [← continuous_within_at_univ, e.continuous_within_at_iff_continuous_within_at_comp_right h,
preimage_univ, continuous_within_at_univ] | lemma | local_homeomorph.continuous_at_iff_continuous_at_comp_right | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_at",
"continuous_within_at_univ"
] | Continuity at a point can be read under right composition with a local homeomorphism, if the
point is in its target | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_iff_continuous_on_comp_right {f : β → γ} {s : set β} (h : s ⊆ e.target) :
continuous_on f s ↔ continuous_on (f ∘ e) (e.source ∩ e ⁻¹' s) | begin
simp only [← e.symm_image_eq_source_inter_preimage h, continuous_on, ball_image_iff],
refine forall₂_congr (λ x hx, _),
rw [e.continuous_within_at_iff_continuous_within_at_comp_right (h hx),
e.symm_image_eq_source_inter_preimage h, inter_comm, continuous_within_at_inter],
exact is_open.mem_nhds e.open... | lemma | local_homeomorph.continuous_on_iff_continuous_on_comp_right | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_on",
"continuous_within_at_inter",
"forall₂_congr",
"is_open.mem_nhds"
] | A function is continuous on a set if and only if its composition with a local homeomorphism
on the right is continuous on the corresponding set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_within_at_iff_continuous_within_at_comp_left
{f : γ → α} {s : set γ} {x : γ} (hx : f x ∈ e.source) (h : f ⁻¹' e.source ∈ 𝓝[s] x) :
continuous_within_at f s x ↔ continuous_within_at (e ∘ f) s x | begin
refine ⟨(e.continuous_at hx).comp_continuous_within_at, λ fe_cont, _⟩,
rw [← continuous_within_at_inter' h] at fe_cont ⊢,
have : continuous_within_at (e.symm ∘ (e ∘ f)) (s ∩ f ⁻¹' e.source) x,
{ have : continuous_within_at e.symm univ (e (f x))
:= (e.continuous_at_symm (e.map_source hx)).continuous_... | lemma | local_homeomorph.continuous_within_at_iff_continuous_within_at_comp_left | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_within_at",
"continuous_within_at.comp",
"continuous_within_at_inter'"
] | Continuity within a set at a point can be read under left composition with a local
homeomorphism if a neighborhood of the initial point is sent to the source of the local
homeomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_iff_continuous_at_comp_left {f : γ → α} {x : γ} (h : f ⁻¹' e.source ∈ 𝓝 x) :
continuous_at f x ↔ continuous_at (e ∘ f) x | begin
have hx : f x ∈ e.source := (mem_of_mem_nhds h : _),
have h' : f ⁻¹' e.source ∈ 𝓝[univ] x, by rwa nhds_within_univ,
rw [← continuous_within_at_univ, ← continuous_within_at_univ,
e.continuous_within_at_iff_continuous_within_at_comp_left hx h']
end | lemma | local_homeomorph.continuous_at_iff_continuous_at_comp_left | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_at",
"continuous_within_at_univ",
"mem_of_mem_nhds",
"nhds_within_univ"
] | Continuity at a point can be read under left composition with a local homeomorphism if a
neighborhood of the initial point is sent to the source of the local homeomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_iff_continuous_on_comp_left {f : γ → α} {s : set γ} (h : s ⊆ f ⁻¹' e.source) :
continuous_on f s ↔ continuous_on (e ∘ f) s | forall₂_congr $ λ x hx, e.continuous_within_at_iff_continuous_within_at_comp_left
(h hx) (mem_of_superset self_mem_nhds_within h) | lemma | local_homeomorph.continuous_on_iff_continuous_on_comp_left | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_on",
"forall₂_congr",
"self_mem_nhds_within"
] | A function is continuous on a set if and only if its composition with a local homeomorphism
on the left is continuous on the corresponding set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_iff_continuous_comp_left {f : γ → α} (h : f ⁻¹' e.source = univ) :
continuous f ↔ continuous (e ∘ f) | begin
simp only [continuous_iff_continuous_on_univ],
exact e.continuous_on_iff_continuous_on_comp_left (eq.symm h).subset,
end | lemma | local_homeomorph.continuous_iff_continuous_comp_left | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous",
"continuous_iff_continuous_on_univ"
] | A function is continuous if and only if its composition with a local homeomorphism
on the left is continuous and its image is contained in the source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph_of_image_subset_source
{s : set α} {t : set β} (hs : s ⊆ e.source) (ht : e '' s = t) : s ≃ₜ t | { to_fun := λ a, ⟨e a, (congr_arg ((∈) (e a)) ht).mp ⟨a, a.2, rfl⟩⟩,
inv_fun := λ b, ⟨e.symm b, let ⟨a, ha1, ha2⟩ := (congr_arg ((∈) ↑b) ht).mpr b.2 in
ha2 ▸ (e.left_inv (hs ha1)).symm ▸ ha1⟩,
left_inv := λ a, subtype.ext (e.left_inv (hs a.2)),
right_inv := λ b, let ⟨a, ha1, ha2⟩ := (congr_arg ((∈) ↑b) ht).mp... | def | local_homeomorph.homeomorph_of_image_subset_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"inv_fun",
"subtype.ext"
] | The homeomorphism obtained by restricting a `local_homeomorph` to a subset of the source. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_homeomorph_source_target : e.source ≃ₜ e.target | e.homeomorph_of_image_subset_source subset_rfl e.image_source_eq_target | def | local_homeomorph.to_homeomorph_source_target | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"subset_rfl"
] | A local homeomrphism defines a homeomorphism between its source and target. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_countable_topology_source [second_countable_topology β]
(e : local_homeomorph α β) :
second_countable_topology e.source | e.to_homeomorph_source_target.second_countable_topology | lemma | local_homeomorph.second_countable_topology_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph_of_source_eq_univ_target_eq_univ (h : e.source = (univ : set α))
(h' : e.target = univ) : α ≃ₜ β | { to_fun := e,
inv_fun := e.symm,
left_inv := λx, e.left_inv $ by { rw h, exact mem_univ _ },
right_inv := λx, e.right_inv $ by { rw h', exact mem_univ _ },
continuous_to_fun := begin
rw [continuous_iff_continuous_on_univ],
convert e.continuous_to_fun,
rw h
end,
continuous_inv_fun := begin
r... | def | local_homeomorph.to_homeomorph_of_source_eq_univ_target_eq_univ | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_iff_continuous_on_univ",
"inv_fun"
] | If a local homeomorphism has source and target equal to univ, then it induces a homeomorphism
between the whole spaces, expressed in this definition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_open_embedding (h : e.source = set.univ) : open_embedding e | begin
apply open_embedding_of_continuous_injective_open,
{ apply continuous_iff_continuous_on_univ.mpr,
rw ← h,
exact e.continuous_to_fun },
{ apply set.injective_iff_inj_on_univ.mpr,
rw ← h,
exact e.inj_on },
{ intros U hU,
simpa only [h, subset_univ] with mfld_simps using e.image_open_of_o... | lemma | local_homeomorph.to_open_embedding | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | A local homeomorphism whose source is all of `α` defines an open embedding of `α` into `β`. The
converse is also true; see `open_embedding.to_local_homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_to_local_homeomorph :
(homeomorph.refl α).to_local_homeomorph = local_homeomorph.refl α | rfl | lemma | homeomorph.refl_to_local_homeomorph | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"homeomorph.refl",
"local_homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_to_local_homeomorph :
e.symm.to_local_homeomorph = e.to_local_homeomorph.symm | rfl | lemma | homeomorph.symm_to_local_homeomorph | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_to_local_homeomorph :
(e.trans e').to_local_homeomorph = e.to_local_homeomorph.trans e'.to_local_homeomorph | local_homeomorph.eq_of_local_equiv_eq $ equiv.trans_to_local_equiv _ _ | lemma | homeomorph.trans_to_local_homeomorph | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"equiv.trans_to_local_equiv",
"local_homeomorph.eq_of_local_equiv_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_homeomorph [nonempty α] : local_homeomorph α β | local_homeomorph.of_continuous_open
((h.to_embedding.inj.inj_on univ).to_local_equiv _ _)
h.continuous.continuous_on h.is_open_map is_open_univ | def | open_embedding.to_local_homeomorph | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open_univ",
"local_homeomorph",
"local_homeomorph.of_continuous_open"
] | An open embedding of `α` into `β`, with `α` nonempty, defines a local homeomorphism whose source
is all of `α`. The converse is also true; see `local_homeomorph.to_open_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_iff
{f : α → β} {g : β → γ} (hf : open_embedding f) {x : α} :
continuous_at (g ∘ f) x ↔ continuous_at g (f x) | begin
haveI : nonempty α := ⟨x⟩,
convert (((hf.to_local_homeomorph f).continuous_at_iff_continuous_at_comp_right) _).symm,
{ apply (local_homeomorph.left_inv _ _).symm,
simp, },
{ simp, },
end | lemma | open_embedding.continuous_at_iff | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"continuous_at",
"local_homeomorph.left_inv",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_homeomorph_subtype_coe : local_homeomorph s α | open_embedding.to_local_homeomorph _ s.2.open_embedding_subtype_coe | def | topological_space.opens.local_homeomorph_subtype_coe | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph",
"open_embedding.to_local_homeomorph"
] | The inclusion of an open subset `s` of a space `α` into `α` is a local homeomorphism from the
subtype `s` to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_homeomorph_subtype_coe_coe :
(s.local_homeomorph_subtype_coe : s → α) = coe | rfl | lemma | topological_space.opens.local_homeomorph_subtype_coe_coe | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_homeomorph_subtype_coe_source :
s.local_homeomorph_subtype_coe.source = set.univ | rfl | lemma | topological_space.opens.local_homeomorph_subtype_coe_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_homeomorph_subtype_coe_target :
s.local_homeomorph_subtype_coe.target = s | by { simp only [local_homeomorph_subtype_coe, subtype.range_coe_subtype] with mfld_simps, refl } | lemma | topological_space.opens.local_homeomorph_subtype_coe_target | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_restr : local_homeomorph s β | s.local_homeomorph_subtype_coe.trans e | def | local_homeomorph.subtype_restr | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph"
] | The restriction of a local homeomorphism `e` to an open subset `s` of the domain type produces a
local homeomorphism whose domain is the subtype `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype_restr_def : e.subtype_restr s = s.local_homeomorph_subtype_coe.trans e | rfl | lemma | local_homeomorph.subtype_restr_def | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_restr_coe : ((e.subtype_restr s : local_homeomorph s β) : s → β)
= set.restrict ↑s (e : α → β) | rfl | lemma | local_homeomorph.subtype_restr_coe | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph",
"set.restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_restr_source : (e.subtype_restr s).source = coe ⁻¹' e.source | by simp only [subtype_restr_def] with mfld_simps | lemma | local_homeomorph.subtype_restr_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_subtype_source {x : s} (hxe : (x:α) ∈ e.source) : e x ∈ (e.subtype_restr s).target | begin
refine ⟨e.map_source hxe, _⟩,
rw [s.local_homeomorph_subtype_coe_target, mem_preimage, e.left_inv_on hxe],
exact x.prop
end | lemma | local_homeomorph.map_subtype_source | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_restr_symm_trans_subtype_restr (f f' : local_homeomorph α β) :
(f.subtype_restr s).symm.trans (f'.subtype_restr s)
≈ (f.symm.trans f').restr (f.target ∩ (f.symm) ⁻¹' s) | begin
simp only [subtype_restr_def, trans_symm_eq_symm_trans_symm],
have openness₁ : is_open (f.target ∩ f.symm ⁻¹' s) := f.preimage_open_of_open_symm s.2,
rw [← of_set_trans _ openness₁, ← trans_assoc, ← trans_assoc],
refine eq_on_source.trans' _ (eq_on_source_refl _),
-- f' has been eliminated !!!
have se... | lemma | local_homeomorph.subtype_restr_symm_trans_subtype_restr | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"is_open",
"local_homeomorph"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype_restr_symm_eq_on_of_le {U V : opens α} [nonempty U] [nonempty V] (hUV : U ≤ V) :
eq_on (e.subtype_restr V).symm (set.inclusion hUV ∘ (e.subtype_restr U).symm)
(e.subtype_restr U).target | begin
set i := set.inclusion hUV,
intros y hy,
dsimp [local_homeomorph.subtype_restr_def] at ⊢ hy,
have hyV : e.symm y ∈ V.local_homeomorph_subtype_coe.target,
{ rw opens.local_homeomorph_subtype_coe_target at ⊢ hy,
exact hUV hy.2 },
refine V.local_homeomorph_subtype_coe.inj_on _ trivial _,
{ rw ←loca... | lemma | local_homeomorph.subtype_restr_symm_eq_on_of_le | topology | src/topology/local_homeomorph.lean | [
"logic.equiv.local_equiv",
"topology.sets.opens"
] | [
"local_homeomorph.map_source",
"local_homeomorph.subtype_restr_def",
"local_homeomorph.symm_source",
"set.inclusion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing [tα : topological_space α] [tβ : topological_space β] (f : α → β) : Prop | (induced : tα = tβ.induced f) | structure | inducing | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"topological_space"
] | A function `f : α → β` between topological spaces is inducing if the topology on `α` is induced
by the topology on `β` through `f`, meaning that a set `s : set α` is open iff it is the preimage
under `f` of some open set `t : set β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inducing_id : inducing (@id α) | ⟨induced_id.symm⟩ | lemma | inducing_id | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.comp {g : β → γ} {f : α → β} (hg : inducing g) (hf : inducing f) :
inducing (g ∘ f) | ⟨by rw [hf.induced, hg.induced, induced_compose]⟩ | lemma | inducing.comp | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"induced_compose",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_of_inducing_compose {f : α → β} {g : β → γ} (hf : continuous f) (hg : continuous g)
(hgf : inducing (g ∘ f)) : inducing f | ⟨le_antisymm
(by rwa ← continuous_iff_le_induced)
(by { rw [hgf.induced, ← continuous_iff_le_induced], apply hg.comp continuous_induced_dom })⟩ | lemma | inducing_of_inducing_compose | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"continuous_iff_le_induced",
"continuous_induced_dom",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_iff_nhds {f : α → β} : inducing f ↔ ∀ a, 𝓝 a = comap f (𝓝 (f a)) | (inducing_iff _).trans (induced_iff_nhds_eq f) | lemma | inducing_iff_nhds | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"induced_iff_nhds_eq",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.nhds_eq_comap {f : α → β} (hf : inducing f) :
∀ (a : α), 𝓝 a = comap f (𝓝 $ f a) | inducing_iff_nhds.1 hf | lemma | inducing.nhds_eq_comap | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.nhds_set_eq_comap {f : α → β} (hf : inducing f) (s : set α) :
𝓝ˢ s = comap f (𝓝ˢ (f '' s)) | by simp only [nhds_set, Sup_image, comap_supr, hf.nhds_eq_comap, supr_image] | lemma | inducing.nhds_set_eq_comap | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"Sup_image",
"inducing",
"nhds_set",
"supr_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.map_nhds_eq {f : α → β} (hf : inducing f) (a : α) :
(𝓝 a).map f = 𝓝[range f] (f a) | hf.induced.symm ▸ map_nhds_induced_eq a | lemma | inducing.map_nhds_eq | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"map_nhds_induced_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.map_nhds_of_mem {f : α → β} (hf : inducing f) (a : α) (h : range f ∈ 𝓝 (f a)) :
(𝓝 a).map f = 𝓝 (f a) | hf.induced.symm ▸ map_nhds_induced_of_mem h | lemma | inducing.map_nhds_of_mem | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"map_nhds_induced_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.image_mem_nhds_within {f : α → β} (hf : inducing f) {a : α} {s : set α}
(hs : s ∈ 𝓝 a) : f '' s ∈ 𝓝[range f] (f a) | hf.map_nhds_eq a ▸ image_mem_map hs | lemma | inducing.image_mem_nhds_within | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.tendsto_nhds_iff {ι : Type*}
{f : ι → β} {g : β → γ} {a : filter ι} {b : β} (hg : inducing g) :
tendsto f a (𝓝 b) ↔ tendsto (g ∘ f) a (𝓝 (g b)) | by rw [hg.nhds_eq_comap, tendsto_comap_iff] | lemma | inducing.tendsto_nhds_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"filter",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.continuous_at_iff {f : α → β} {g : β → γ} (hg : inducing g) {x : α} :
continuous_at f x ↔ continuous_at (g ∘ f) x | by simp_rw [continuous_at, inducing.tendsto_nhds_iff hg] | lemma | inducing.continuous_at_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous_at",
"inducing",
"inducing.tendsto_nhds_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.continuous_iff {f : α → β} {g : β → γ} (hg : inducing g) :
continuous f ↔ continuous (g ∘ f) | by simp_rw [continuous_iff_continuous_at, hg.continuous_at_iff] | lemma | inducing.continuous_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"continuous_iff_continuous_at",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.continuous_at_iff' {f : α → β} {g : β → γ} (hf : inducing f) {x : α}
(h : range f ∈ 𝓝 (f x)) : continuous_at (g ∘ f) x ↔ continuous_at g (f x) | by { simp_rw [continuous_at, filter.tendsto, ← hf.map_nhds_of_mem _ h, filter.map_map] } | lemma | inducing.continuous_at_iff' | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous_at",
"filter.map_map",
"filter.tendsto",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.continuous {f : α → β} (hf : inducing f) : continuous f | hf.continuous_iff.mp continuous_id | lemma | inducing.continuous | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"continuous_id",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.inducing_iff {f : α → β} {g : β → γ} (hg : inducing g) :
inducing f ↔ inducing (g ∘ f) | begin
refine ⟨λ h, hg.comp h, λ hgf, inducing_of_inducing_compose _ hg.continuous hgf⟩,
rw hg.continuous_iff,
exact hgf.continuous
end | lemma | inducing.inducing_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"inducing_of_inducing_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.closure_eq_preimage_closure_image {f : α → β} (hf : inducing f) (s : set α) :
closure s = f ⁻¹' closure (f '' s) | by { ext x, rw [set.mem_preimage, ← closure_induced, hf.induced] } | lemma | inducing.closure_eq_preimage_closure_image | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closure",
"closure_induced",
"inducing",
"set.mem_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.is_closed_iff {f : α → β} (hf : inducing f) {s : set α} :
is_closed s ↔ ∃ t, is_closed t ∧ f ⁻¹' t = s | by rw [hf.induced, is_closed_induced_iff] | lemma | inducing.is_closed_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"is_closed",
"is_closed_induced_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.is_closed_iff' {f : α → β} (hf : inducing f) {s : set α} :
is_closed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s | by rw [hf.induced, is_closed_induced_iff'] | lemma | inducing.is_closed_iff' | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closure",
"inducing",
"is_closed",
"is_closed_induced_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.is_closed_preimage {f : α → β} (h : inducing f) (s : set β) (hs : is_closed s) :
is_closed (f ⁻¹' s) | (inducing.is_closed_iff h).mpr ⟨s, hs, rfl⟩ | lemma | inducing.is_closed_preimage | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"inducing.is_closed_iff",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.is_open_iff {f : α → β} (hf : inducing f) {s : set α} :
is_open s ↔ ∃ t, is_open t ∧ f ⁻¹' t = s | by rw [hf.induced, is_open_induced_iff] | lemma | inducing.is_open_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"is_open",
"is_open_induced_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.dense_iff {f : α → β} (hf : inducing f) {s : set α} :
dense s ↔ ∀ x, f x ∈ closure (f '' s) | by simp only [dense, hf.closure_eq_preimage_closure_image, mem_preimage] | lemma | inducing.dense_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closure",
"dense",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding [tα : topological_space α] [tβ : topological_space β] (f : α → β)
extends inducing f : Prop | (inj : injective f) | structure | embedding | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"topological_space"
] | A function between topological spaces is an embedding if it is injective,
and for all `s : set α`, `s` is open iff it is the preimage of an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
function.injective.embedding_induced [t : topological_space β]
{f : α → β} (hf : injective f) :
@_root_.embedding α β (t.induced f) t f | { induced := rfl,
inj := hf } | lemma | function.injective.embedding_induced | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.mk' (f : α → β) (inj : injective f)
(induced : ∀ a, comap f (𝓝 (f a)) = 𝓝 a) : embedding f | ⟨inducing_iff_nhds.2 (λ a, (induced a).symm), inj⟩ | lemma | embedding.mk' | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_id : embedding (@id α) | ⟨inducing_id, assume a₁ a₂ h, h⟩ | lemma | embedding_id | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.comp {g : β → γ} {f : α → β} (hg : embedding g) (hf : embedding f) :
embedding (g ∘ f) | { inj:= assume a₁ a₂ h, hf.inj $ hg.inj h,
..hg.to_inducing.comp hf.to_inducing } | lemma | embedding.comp | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_of_embedding_compose {f : α → β} {g : β → γ} (hf : continuous f) (hg : continuous g)
(hgf : embedding (g ∘ f)) : embedding f | { induced := (inducing_of_inducing_compose hf hg hgf.to_inducing).induced,
inj := assume a₁ a₂ h, hgf.inj $ by simp [h, (∘)] } | lemma | embedding_of_embedding_compose | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"embedding",
"inducing_of_inducing_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.left_inverse.embedding {f : α → β} {g : β → α}
(h : left_inverse f g) (hf : continuous f) (hg : continuous g) :
embedding g | embedding_of_embedding_compose hg hf $ h.comp_eq_id.symm ▸ embedding_id | lemma | function.left_inverse.embedding | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"embedding",
"embedding_id",
"embedding_of_embedding_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.map_nhds_eq {f : α → β} (hf : embedding f) (a : α) :
(𝓝 a).map f = 𝓝[range f] (f a) | hf.1.map_nhds_eq a | lemma | embedding.map_nhds_eq | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.map_nhds_of_mem {f : α → β}
(hf : embedding f) (a : α) (h : range f ∈ 𝓝 (f a)) : (𝓝 a).map f = 𝓝 (f a) | hf.1.map_nhds_of_mem a h | lemma | embedding.map_nhds_of_mem | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.tendsto_nhds_iff {ι : Type*}
{f : ι → β} {g : β → γ} {a : filter ι} {b : β} (hg : embedding g) :
tendsto f a (𝓝 b) ↔ tendsto (g ∘ f) a (𝓝 (g b)) | hg.to_inducing.tendsto_nhds_iff | lemma | embedding.tendsto_nhds_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"embedding",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.continuous_iff {f : α → β} {g : β → γ} (hg : embedding g) :
continuous f ↔ continuous (g ∘ f) | inducing.continuous_iff hg.1 | lemma | embedding.continuous_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"embedding",
"inducing.continuous_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.continuous {f : α → β} (hf : embedding f) : continuous f | inducing.continuous hf.1 | lemma | embedding.continuous | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"embedding",
"inducing.continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.closure_eq_preimage_closure_image {e : α → β} (he : embedding e) (s : set α) :
closure s = e ⁻¹' closure (e '' s) | he.1.closure_eq_preimage_closure_image s | lemma | embedding.closure_eq_preimage_closure_image | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closure",
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.discrete_topology {X Y : Type*} [topological_space X] [tY : topological_space Y]
[discrete_topology Y] {f : X → Y} (hf : embedding f) : discrete_topology X | discrete_topology_iff_nhds.2 $ λ x, by rw [hf.nhds_eq_comap, nhds_discrete, comap_pure,
← image_singleton, hf.inj.preimage_image, principal_singleton] | lemma | embedding.discrete_topology | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"discrete_topology",
"embedding",
"nhds_discrete",
"topological_space"
] | The topology induced under an inclusion `f : X → Y` from the discrete topological space `Y`
is the discrete topology on `X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map {α : Type*} {β : Type*} [tα : topological_space α] [tβ : topological_space β]
(f : α → β) : Prop | surjective f ∧ tβ = tα.coinduced f | def | quotient_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"topological_space"
] | A function between topological spaces is a quotient map if it is surjective,
and for all `s : set β`, `s` is open iff its preimage is an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_iff [topological_space α] [topological_space β] {f : α → β} :
quotient_map f ↔ surjective f ∧ ∀ s : set β, is_open s ↔ is_open (f ⁻¹' s) | and_congr iff.rfl topological_space_eq_iff | lemma | quotient_map_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open",
"quotient_map",
"topological_space",
"topological_space_eq_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map_iff_closed [topological_space α] [topological_space β] {f : α → β} :
quotient_map f ↔ surjective f ∧ ∀ s : set β, is_closed s ↔ is_closed (f ⁻¹' s) | quotient_map_iff.trans $ iff.rfl.and $ compl_surjective.forall.trans $
by simp only [is_open_compl_iff, preimage_compl] | lemma | quotient_map_iff_closed | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_closed",
"is_open_compl_iff",
"quotient_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : quotient_map (@id α) | ⟨assume a, ⟨a, rfl⟩, coinduced_id.symm⟩ | lemma | quotient_map.id | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (hg : quotient_map g) (hf : quotient_map f) :
quotient_map (g ∘ f) | ⟨hg.left.comp hf.left, by rw [hg.right, hf.right, coinduced_compose]⟩ | lemma | quotient_map.comp | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"coinduced_compose",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_quotient_map_compose (hf : continuous f) (hg : continuous g)
(hgf : quotient_map (g ∘ f)) : quotient_map g | ⟨hgf.1.of_comp,
le_antisymm
(by { rw [hgf.right, ← continuous_iff_coinduced_le], apply continuous_coinduced_rng.comp hf })
(by rwa ← continuous_iff_coinduced_le)⟩ | lemma | quotient_map.of_quotient_map_compose | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"continuous_iff_coinduced_le",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_inverse {g : β → α} (hf : continuous f) (hg : continuous g) (h : left_inverse g f) :
quotient_map g | quotient_map.of_quotient_map_compose hf hg $ h.comp_eq_id.symm ▸ quotient_map.id | lemma | quotient_map.of_inverse | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"quotient_map",
"quotient_map.id",
"quotient_map.of_quotient_map_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff (hf : quotient_map f) :
continuous g ↔ continuous (g ∘ f) | by rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.right, coinduced_compose] | lemma | quotient_map.continuous_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"coinduced_compose",
"continuous",
"continuous_iff_coinduced_le",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (hf : quotient_map f) : continuous f | hf.continuous_iff.mp continuous_id | lemma | quotient_map.continuous | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"continuous_id",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective (hf : quotient_map f) : surjective f | hf.1 | lemma | quotient_map.surjective | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.