statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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is_open.prod {s : set α} {t : set β} (hs : is_open s) (ht : is_open t) :
is_open (s ×ˢ t) | (hs.preimage continuous_fst).inter (ht.preimage continuous_snd) | lemma | is_open.prod | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_fst",
"continuous_snd",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_prod_eq {a : α} {b : β} : 𝓝 (a, b) = 𝓝 a ×ᶠ 𝓝 b | by rw [filter.prod, prod.topological_space, nhds_inf, nhds_induced, nhds_induced] | lemma | nhds_prod_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter.prod",
"nhds_induced",
"nhds_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_uncurry_of_discrete_topology [discrete_topology α]
{f : α → β → γ} (hf : ∀ a, continuous (f a)) : continuous (uncurry f) | begin
apply continuous_iff_continuous_at.2,
rintros ⟨a, x⟩,
change map _ _ ≤ _,
rw [nhds_prod_eq, nhds_discrete, filter.map_pure_prod],
exact (hf a).continuous_at
end | lemma | continuous_uncurry_of_discrete_topology | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_at",
"discrete_topology",
"filter.map_pure_prod",
"nhds_discrete",
"nhds_prod_eq"
] | If a function `f x y` is such that `y ↦ f x y` is continuous for all `x`, and `x` lives in a
discrete space, then `f` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_nhds_prod_iff {a : α} {b : β} {s : set (α × β)} :
s ∈ 𝓝 (a, b) ↔ ∃ (u ∈ 𝓝 a) (v ∈ 𝓝 b), u ×ˢ v ⊆ s | by rw [nhds_prod_eq, mem_prod_iff] | lemma | mem_nhds_prod_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_prod_iff' {a : α} {b : β} {s : set (α × β)} :
s ∈ 𝓝 (a, b) ↔ ∃ (u : set α) (v : set β), is_open u ∧ a ∈ u ∧ is_open v ∧ b ∈ v ∧ u ×ˢ v ⊆ s | begin
rw mem_nhds_prod_iff,
split,
{ rintros ⟨u, Hu, v, Hv, h⟩,
rcases mem_nhds_iff.1 Hu with ⟨u', u'u, u'_open, Hu'⟩,
rcases mem_nhds_iff.1 Hv with ⟨v', v'v, v'_open, Hv'⟩,
exact ⟨u', v', u'_open, Hu', v'_open, Hv', (set.prod_mono u'u v'v).trans h⟩ },
{ rintros ⟨u, v, u_open, au, v_open, bv, huv⟩,
... | lemma | mem_nhds_prod_iff' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"mem_nhds_prod_iff",
"set.prod_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.prod.tendsto_iff {α} (seq : α → β × γ) {f : filter α} (x : β × γ) :
tendsto seq f (𝓝 x)
↔ tendsto (λ n, (seq n).fst) f (𝓝 x.fst) ∧ tendsto (λ n, (seq n).snd) f (𝓝 x.snd) | by { cases x, rw [nhds_prod_eq, filter.tendsto_prod_iff'], } | lemma | prod.tendsto_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter",
"filter.tendsto_prod_iff'",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.prod_nhds {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
{sa : ιa → set α} {sb : ιb → set β} {a : α} {b : β} (ha : (𝓝 a).has_basis pa sa)
(hb : (𝓝 b).has_basis pb sb) :
(𝓝 (a, b)).has_basis (λ i : ιa × ιb, pa i.1 ∧ pb i.2) (λ i, sa i.1 ×ˢ sb i.2) | by { rw nhds_prod_eq, exact ha.prod hb } | lemma | filter.has_basis.prod_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.prod_nhds' {ιa ιb : Type*} {pa : ιa → Prop} {pb : ιb → Prop}
{sa : ιa → set α} {sb : ιb → set β} {ab : α × β} (ha : (𝓝 ab.1).has_basis pa sa)
(hb : (𝓝 ab.2).has_basis pb sb) :
(𝓝 ab).has_basis (λ i : ιa × ιb, pa i.1 ∧ pb i.2) (λ i, sa i.1 ×ˢ sb i.2) | by { cases ab, exact ha.prod_nhds hb } | lemma | filter.has_basis.prod_nhds' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mem_nhds_iff {s : set α} {t : set β} {a : α} {b : β} :
s ×ˢ t ∈ 𝓝 (a, b) ↔ s ∈ 𝓝 a ∧ t ∈ 𝓝 b | by rw [nhds_prod_eq, prod_mem_prod_iff] | lemma | prod_mem_nhds_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mem_nhds {s : set α} {t : set β} {a : α} {b : β}
(ha : s ∈ 𝓝 a) (hb : t ∈ 𝓝 b) : s ×ˢ t ∈ 𝓝 (a, b) | prod_mem_nhds_iff.2 ⟨ha, hb⟩ | lemma | prod_mem_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.prod_nhds {p : α → Prop} {q : β → Prop} {a : α} {b : β}
(ha : ∀ᶠ x in 𝓝 a, p x) (hb : ∀ᶠ y in 𝓝 b, q y) :
∀ᶠ z : α × β in 𝓝 (a, b), p z.1 ∧ q z.2 | prod_mem_nhds ha hb | lemma | filter.eventually.prod_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"prod_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_swap (a : α) (b : β) : 𝓝 (a, b) = (𝓝 (b, a)).map prod.swap | by rw [nhds_prod_eq, filter.prod_comm, nhds_prod_eq]; refl | lemma | nhds_swap | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter.prod_comm",
"nhds_prod_eq",
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.prod_mk_nhds {γ} {a : α} {b : β} {f : filter γ} {ma : γ → α} {mb : γ → β}
(ha : tendsto ma f (𝓝 a)) (hb : tendsto mb f (𝓝 b)) :
tendsto (λc, (ma c, mb c)) f (𝓝 (a, b)) | by rw [nhds_prod_eq]; exact filter.tendsto.prod_mk ha hb | lemma | filter.tendsto.prod_mk_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter",
"filter.tendsto.prod_mk",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually.curry_nhds {p : α × β → Prop} {x : α} {y : β} (h : ∀ᶠ x in 𝓝 (x, y), p x) :
∀ᶠ x' in 𝓝 x, ∀ᶠ y' in 𝓝 y, p (x', y') | by { rw [nhds_prod_eq] at h, exact h.curry } | lemma | filter.eventually.curry_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.prod {f : α → β} {g : α → γ} {x : α}
(hf : continuous_at f x) (hg : continuous_at g x) : continuous_at (λx, (f x, g x)) x | hf.prod_mk_nhds hg | lemma | continuous_at.prod | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.prod_map {f : α → γ} {g : β → δ} {p : α × β}
(hf : continuous_at f p.fst) (hg : continuous_at g p.snd) :
continuous_at (λ p : α × β, (f p.1, g p.2)) p | hf.fst''.prod hg.snd'' | lemma | continuous_at.prod_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.prod_map' {f : α → γ} {g : β → δ} {x : α} {y : β}
(hf : continuous_at f x) (hg : continuous_at g y) :
continuous_at (λ p : α × β, (f p.1, g p.2)) (x, y) | hf.fst'.prod hg.snd' | lemma | continuous_at.prod_map' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_generate_from_generate_from_eq {α β : Type*} {s : set (set α)} {t : set (set β)}
(hs : ⋃₀ s = univ) (ht : ⋃₀ t = univ) :
@prod.topological_space α β (generate_from s) (generate_from t) =
generate_from {g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v} | let G := generate_from {g | ∃ u ∈ s, ∃ v ∈ t, g = u ×ˢ v} in
le_antisymm
(le_generate_from $ λ g ⟨u, hu, v, hv, g_eq⟩, g_eq.symm ▸
@is_open.prod _ _ (generate_from s) (generate_from t) _ _
(generate_open.basic _ hu) (generate_open.basic _ hv))
(le_inf
(coinduced_le_iff_le_induced.mp $ le_generate_from... | lemma | prod_generate_from_generate_from_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open.prod",
"is_open_Union",
"le_generate_from",
"le_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_eq_generate_from :
prod.topological_space =
generate_from {g | ∃(s:set α) (t:set β), is_open s ∧ is_open t ∧ g = s ×ˢ t} | le_antisymm
(le_generate_from $ λ g ⟨s, t, hs, ht, g_eq⟩, g_eq.symm ▸ hs.prod ht)
(le_inf
(ball_image_of_ball $ λt ht, generate_open.basic _ ⟨t, univ, by simpa [set.prod_eq] using ht⟩)
(ball_image_of_ball $ λt ht, generate_open.basic _ ⟨univ, t, by simpa [set.prod_eq] using ht⟩)) | lemma | prod_eq_generate_from | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"le_generate_from",
"le_inf",
"set.prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_prod_iff {s : set (α × β)} : is_open s ↔
(∀a b, (a, b) ∈ s →
∃ (u : set α) (v : set β), is_open u ∧ is_open v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s) | begin
rw [is_open_iff_nhds],
simp_rw [le_principal_iff, prod.forall,
((nhds_basis_opens _).prod_nhds (nhds_basis_opens _)).mem_iff, prod.exists, exists_prop],
simp only [and_assoc, and.left_comm]
end | lemma | is_open_prod_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"exists_prop",
"is_open",
"is_open_iff_nhds",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_induced_induced {α γ : Type*} (f : α → β) (g : γ → δ) :
@prod.topological_space α γ (induced f ‹_›) (induced g ‹_›) =
induced (λ p, (f p.1, g p.2)) prod.topological_space | by simp_rw [prod.topological_space, induced_inf, induced_compose] | lemma | prod_induced_induced | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"induced_compose",
"induced_inf"
] | A product of induced topologies is induced by the product map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_uncurry_of_discrete_topology_left [discrete_topology α]
{f : α → β → γ} (h : ∀ a, continuous (f a)) : continuous (uncurry f) | continuous_iff_continuous_at.2 $ λ ⟨a, b⟩,
by simp only [continuous_at, nhds_prod_eq, nhds_discrete α, pure_prod, tendsto_map'_iff, (∘),
uncurry, (h a).tendsto] | lemma | continuous_uncurry_of_discrete_topology_left | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_at",
"discrete_topology",
"nhds_discrete",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_nhds_square {s : set (α × α)} {x : α} (hx : s ∈ 𝓝 (x, x)) :
∃ U : set α, is_open U ∧ x ∈ U ∧ U ×ˢ U ⊆ s | by simpa [nhds_prod_eq, (nhds_basis_opens x).prod_self.mem_iff, and.assoc, and.left_comm] using hx | lemma | exists_nhds_square | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"nhds_basis_opens",
"nhds_prod_eq"
] | Given a neighborhood `s` of `(x, x)`, then `(x, x)` has a square open neighborhood
that is a subset of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_fst_nhds_within (x : α × β) : map prod.fst (𝓝[prod.snd ⁻¹' {x.2}] x) = 𝓝 x.1 | begin
refine le_antisymm (continuous_at_fst.mono_left inf_le_left) (λ s hs, _),
rcases x with ⟨x, y⟩,
rw [mem_map, nhds_within, mem_inf_principal, mem_nhds_prod_iff] at hs,
rcases hs with ⟨u, hu, v, hv, H⟩,
simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H,
exact mem_of_supers... | lemma | map_fst_nhds_within | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"inf_le_left",
"mem_map",
"mem_nhds_prod_iff",
"mem_of_mem_nhds",
"nhds_within"
] | `prod.fst` maps neighborhood of `x : α × β` within the section `prod.snd ⁻¹' {x.2}`
to `𝓝 x.1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_fst_nhds (x : α × β) : map prod.fst (𝓝 x) = 𝓝 x.1 | le_antisymm continuous_at_fst $ (map_fst_nhds_within x).symm.trans_le (map_mono inf_le_left) | lemma | map_fst_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at_fst",
"inf_le_left",
"map_fst_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_fst : is_open_map (@prod.fst α β) | is_open_map_iff_nhds_le.2 $ λ x, (map_fst_nhds x).ge | lemma | is_open_map_fst | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"map_fst_nhds"
] | The first projection in a product of topological spaces sends open sets to open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_snd_nhds_within (x : α × β) : map prod.snd (𝓝[prod.fst ⁻¹' {x.1}] x) = 𝓝 x.2 | begin
refine le_antisymm (continuous_at_snd.mono_left inf_le_left) (λ s hs, _),
rcases x with ⟨x, y⟩,
rw [mem_map, nhds_within, mem_inf_principal, mem_nhds_prod_iff] at hs,
rcases hs with ⟨u, hu, v, hv, H⟩,
simp only [prod_subset_iff, mem_singleton_iff, mem_set_of_eq, mem_preimage] at H,
exact mem_of_supers... | lemma | map_snd_nhds_within | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"inf_le_left",
"mem_map",
"mem_nhds_prod_iff",
"mem_of_mem_nhds",
"nhds_within"
] | `prod.snd` maps neighborhood of `x : α × β` within the section `prod.fst ⁻¹' {x.1}`
to `𝓝 x.2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_snd_nhds (x : α × β) : map prod.snd (𝓝 x) = 𝓝 x.2 | le_antisymm continuous_at_snd $ (map_snd_nhds_within x).symm.trans_le (map_mono inf_le_left) | lemma | map_snd_nhds | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at_snd",
"inf_le_left",
"map_snd_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_snd : is_open_map (@prod.snd α β) | is_open_map_iff_nhds_le.2 $ λ x, (map_snd_nhds x).ge | lemma | is_open_map_snd | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"map_snd_nhds"
] | The second projection in a product of topological spaces sends open sets to open sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_prod_iff' {s : set α} {t : set β} :
is_open (s ×ˢ t) ↔ (is_open s ∧ is_open t) ∨ (s = ∅) ∨ (t = ∅) | begin
cases (s ×ˢ t).eq_empty_or_nonempty with h h,
{ simp [h, prod_eq_empty_iff.1 h] },
{ have st : s.nonempty ∧ t.nonempty, from prod_nonempty_iff.1 h,
split,
{ assume H : is_open (s ×ˢ t),
refine or.inl ⟨_, _⟩,
show is_open s,
{ rw ← fst_image_prod s st.2,
exact is_open_map_fs... | lemma | is_open_prod_iff' | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"is_open_map_fst",
"is_open_map_snd"
] | A product set is open in a product space if and only if each factor is open, or one of them is
empty | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closure_prod_eq {s : set α} {t : set β} :
closure (s ×ˢ t) = closure s ×ˢ closure t | set.ext $ assume ⟨a, b⟩,
have (𝓝 a ×ᶠ 𝓝 b) ⊓ 𝓟 (s ×ˢ t) = (𝓝 a ⊓ 𝓟 s) ×ᶠ (𝓝 b ⊓ 𝓟 t),
by rw [←prod_inf_prod, prod_principal_principal],
by simp [closure_eq_cluster_pts, cluster_pt, nhds_prod_eq, this]; exact prod_ne_bot | lemma | closure_prod_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closure",
"closure_eq_cluster_pts",
"cluster_pt",
"nhds_prod_eq",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_prod_eq (s : set α) (t : set β) :
interior (s ×ˢ t) = interior s ×ˢ interior t | set.ext $ λ ⟨a, b⟩, by simp only [mem_interior_iff_mem_nhds, mem_prod, prod_mem_nhds_iff] | lemma | interior_prod_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"interior",
"mem_interior_iff_mem_nhds",
"prod_mem_nhds_iff",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_prod_eq (s : set α) (t : set β) :
frontier (s ×ˢ t) = closure s ×ˢ frontier t ∪ frontier s ×ˢ closure t | by simp only [frontier, closure_prod_eq, interior_prod_eq, prod_diff_prod] | lemma | frontier_prod_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closure",
"closure_prod_eq",
"frontier",
"interior_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_prod_univ_eq (s : set α) :
frontier (s ×ˢ (univ : set β)) = frontier s ×ˢ univ | by simp [frontier_prod_eq] | lemma | frontier_prod_univ_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"frontier",
"frontier_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_univ_prod_eq (s : set β) :
frontier ((univ : set α) ×ˢ s) = univ ×ˢ frontier s | by simp [frontier_prod_eq] | lemma | frontier_univ_prod_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"frontier",
"frontier_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mem_closure₂ {f : α → β → γ} {a : α} {b : β} {s : set α} {t : set β} {u : set γ}
(hf : continuous (uncurry f)) (ha : a ∈ closure s) (hb : b ∈ closure t)
(h : ∀ (a ∈ s) (b ∈ t), f a b ∈ u) :
f a b ∈ closure u | have H₁ : (a, b) ∈ closure (s ×ˢ t), by simpa only [closure_prod_eq] using mk_mem_prod ha hb,
have H₂ : maps_to (uncurry f) (s ×ˢ t) u, from forall_prod_set.2 h,
H₂.closure hf H₁ | lemma | map_mem_closure₂ | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closure",
"closure_prod_eq",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.prod {s₁ : set α} {s₂ : set β} (h₁ : is_closed s₁) (h₂ : is_closed s₂) :
is_closed (s₁ ×ˢ s₂) | closure_eq_iff_is_closed.mp $ by simp only [h₁.closure_eq, h₂.closure_eq, closure_prod_eq] | lemma | is_closed.prod | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closure_prod_eq",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.prod {s : set α} {t : set β} (hs : dense s) (ht : dense t) :
dense (s ×ˢ t) | λ x, by { rw closure_prod_eq, exact ⟨hs x.1, ht x.2⟩ } | lemma | dense.prod | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closure_prod_eq",
"dense"
] | The product of two dense sets is a dense set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_range.prod_map {ι : Type*} {κ : Type*} {f : ι → β} {g : κ → γ}
(hf : dense_range f) (hg : dense_range g) : dense_range (prod.map f g) | by simpa only [dense_range, prod_range_range_eq] using hf.prod hg | lemma | dense_range.prod_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"dense_range"
] | If `f` and `g` are maps with dense range, then `prod.map f g` has dense range. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inducing.prod_mk {f : α → β} {g : γ → δ} (hf : inducing f) (hg : inducing g) :
inducing (λx:α×γ, (f x.1, g x.2)) | ⟨by rw [prod.topological_space, prod.topological_space, hf.induced, hg.induced,
induced_compose, induced_compose, induced_inf, induced_compose, induced_compose]⟩ | lemma | inducing.prod_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"induced_compose",
"induced_inf",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_const_prod {a : α} {f : β → γ} : inducing (λ x, (a, f x)) ↔ inducing f | by simp_rw [inducing_iff, prod.topological_space, induced_inf, induced_compose, function.comp,
induced_const, top_inf_eq] | lemma | inducing_const_prod | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"induced_compose",
"induced_const",
"induced_inf",
"inducing",
"top_inf_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_prod_const {b : β} {f : α → γ} : inducing (λ x, (f x, b)) ↔ inducing f | by simp_rw [inducing_iff, prod.topological_space, induced_inf, induced_compose, function.comp,
induced_const, inf_top_eq] | lemma | inducing_prod_const | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"induced_compose",
"induced_const",
"induced_inf",
"inducing",
"inf_top_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.prod_mk {f : α → β} {g : γ → δ} (hf : embedding f) (hg : embedding g) :
embedding (λx:α×γ, (f x.1, g x.2)) | { inj := assume ⟨x₁, x₂⟩ ⟨y₁, y₂⟩, by simp; exact assume h₁ h₂, ⟨hf.inj h₁, hg.inj h₂⟩,
..hf.to_inducing.prod_mk hg.to_inducing } | lemma | embedding.prod_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map.prod {f : α → β} {g : γ → δ} (hf : is_open_map f) (hg : is_open_map g) :
is_open_map (λ p : α × γ, (f p.1, g p.2)) | begin
rw [is_open_map_iff_nhds_le],
rintros ⟨a, b⟩,
rw [nhds_prod_eq, nhds_prod_eq, ← filter.prod_map_map_eq],
exact filter.prod_mono (hf.nhds_le a) (hg.nhds_le b)
end | lemma | is_open_map.prod | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter.prod_map_map_eq",
"filter.prod_mono",
"is_open_map",
"is_open_map_iff_nhds_le",
"nhds_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.prod {f : α → β} {g : γ → δ}
(hf : open_embedding f) (hg : open_embedding g) : open_embedding (λ x : α × γ, (f x.1, g x.2)) | open_embedding_of_embedding_open (hf.1.prod_mk hg.1)
(hf.is_open_map.prod hg.is_open_map) | lemma | open_embedding.prod | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"open_embedding",
"open_embedding_of_embedding_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_graph {f : α → β} (hf : continuous f) : embedding (λ x, (x, f x)) | embedding_of_embedding_compose (continuous_id.prod_mk hf) continuous_fst embedding_id | lemma | embedding_graph | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_fst",
"embedding",
"embedding_id",
"embedding_of_embedding_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inl : continuous (@inl α β) | continuous_sup_rng_left continuous_coinduced_rng | lemma | continuous_inl | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_coinduced_rng",
"continuous_sup_rng_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inr : continuous (@inr α β) | continuous_sup_rng_right continuous_coinduced_rng | lemma | continuous_inr | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_coinduced_rng",
"continuous_sup_rng_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_sum_iff {s : set (α ⊕ β)} :
is_open s ↔ is_open (inl ⁻¹' s) ∧ is_open (inr ⁻¹' s) | iff.rfl | lemma | is_open_sum_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_inl : is_open_map (@inl α β) | λ u hu, by simpa [is_open_sum_iff, preimage_image_eq u sum.inl_injective] | lemma | is_open_map_inl | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"is_open_sum_iff",
"sum.inl_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_inr : is_open_map (@inr α β) | λ u hu, by simpa [is_open_sum_iff, preimage_image_eq u sum.inr_injective] | lemma | is_open_map_inr | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"is_open_sum_iff",
"sum.inr_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_inl : open_embedding (@inl α β) | open_embedding_of_continuous_injective_open continuous_inl inl_injective is_open_map_inl | lemma | open_embedding_inl | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_inl",
"is_open_map_inl",
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_inr : open_embedding (@inr α β) | open_embedding_of_continuous_injective_open continuous_inr inr_injective is_open_map_inr | lemma | open_embedding_inr | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_inr",
"is_open_map_inr",
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_inl : embedding (@inl α β) | open_embedding_inl.1 | lemma | embedding_inl | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_inr : embedding (@inr α β) | open_embedding_inr.1 | lemma | embedding_inr | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_range_inl : is_open (range (inl : α → α ⊕ β)) | open_embedding_inl.2 | lemma | is_open_range_inl | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_range_inr : is_open (range (inr : β → α ⊕ β)) | open_embedding_inr.2 | lemma | is_open_range_inr | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_range_inl : is_closed (range (inl : α → α ⊕ β)) | by { rw [← is_open_compl_iff, compl_range_inl], exact is_open_range_inr } | lemma | is_closed_range_inl | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_closed",
"is_open_compl_iff",
"is_open_range_inr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_range_inr : is_closed (range (inr : β → α ⊕ β)) | by { rw [← is_open_compl_iff, compl_range_inr], exact is_open_range_inl } | lemma | is_closed_range_inr | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_closed",
"is_open_compl_iff",
"is_open_range_inl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_inl : closed_embedding (inl : α → α ⊕ β) | ⟨embedding_inl, is_closed_range_inl⟩ | lemma | closed_embedding_inl | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closed_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_inr : closed_embedding (inr : β → α ⊕ β) | ⟨embedding_inr, is_closed_range_inr⟩ | lemma | closed_embedding_inr | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closed_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_inl (x : α) : 𝓝 (inl x : α ⊕ β) = map inl (𝓝 x) | (open_embedding_inl.map_nhds_eq _).symm | lemma | nhds_inl | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_inr (x : β) : 𝓝 (inr x : α ⊕ β) = map inr (𝓝 x) | (open_embedding_inr.map_nhds_eq _).symm | lemma | nhds_inr | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sum_dom {f : α ⊕ β → γ} :
continuous f ↔ continuous (f ∘ sum.inl) ∧ continuous (f ∘ sum.inr) | by simp only [continuous_sup_dom, continuous_coinduced_dom] | theorem | continuous_sum_dom | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_coinduced_dom",
"continuous_sup_dom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sum_elim {f : α → γ} {g : β → γ} :
continuous (sum.elim f g) ↔ continuous f ∧ continuous g | continuous_sum_dom | lemma | continuous_sum_elim | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_sum_dom",
"sum.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.sum_elim {f : α → γ} {g : β → γ}
(hf : continuous f) (hg : continuous g) : continuous (sum.elim f g) | continuous_sum_elim.2 ⟨hf, hg⟩ | lemma | continuous.sum_elim | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"sum.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sum_map {f : α → β} {g : γ → δ} :
continuous (sum.map f g) ↔ continuous f ∧ continuous g | continuous_sum_elim.trans $ embedding_inl.continuous_iff.symm.and embedding_inr.continuous_iff.symm | lemma | continuous_sum_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"sum.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.sum_map {f : α → β} {g : γ → δ} (hf : continuous f)
(hg : continuous g) : continuous (sum.map f g) | continuous_sum_map.2 ⟨hf, hg⟩ | lemma | continuous.sum_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"sum.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_sum {f : α ⊕ β → γ} :
is_open_map f ↔ is_open_map (λ a, f (inl a)) ∧ is_open_map (λ b, f (inr b)) | by simp only [is_open_map_iff_nhds_le, sum.forall, nhds_inl, nhds_inr, filter.map_map] | lemma | is_open_map_sum | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter.map_map",
"is_open_map",
"is_open_map_iff_nhds_le",
"nhds_inl",
"nhds_inr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_sum_elim {f : α → γ} {g : β → γ} :
is_open_map (sum.elim f g) ↔ is_open_map f ∧ is_open_map g | by simp only [is_open_map_sum, elim_inl, elim_inr] | lemma | is_open_map_sum_elim | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"is_open_map_sum",
"sum.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map.sum_elim {f : α → γ} {g : β → γ} (hf : is_open_map f) (hg : is_open_map g) :
is_open_map (sum.elim f g) | is_open_map_sum_elim.2 ⟨hf, hg⟩ | lemma | is_open_map.sum_elim | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open_map",
"sum.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_coe {b : set β} : inducing (coe : b → β) | ⟨rfl⟩ | lemma | inducing_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.of_cod_restrict {f : α → β} {b : set β} (hb : ∀ a, f a ∈ b)
(h : inducing (b.cod_restrict f hb)) : inducing f | inducing_coe.comp h | lemma | inducing.of_cod_restrict | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_subtype_coe : embedding (coe : subtype p → α) | ⟨⟨rfl⟩, subtype.coe_injective⟩ | lemma | embedding_subtype_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_subtype_coe (h : is_closed {a | p a}) :
closed_embedding (coe : subtype p → α) | ⟨embedding_subtype_coe, by rwa [subtype.range_coe_subtype]⟩ | lemma | closed_embedding_subtype_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closed_embedding",
"is_closed",
"subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_subtype_val : continuous (@subtype.val α p) | continuous_induced_dom | lemma | continuous_subtype_val | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_induced_dom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_subtype_coe : continuous (coe : subtype p → α) | continuous_subtype_val | lemma | continuous_subtype_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_subtype_val"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.subtype_coe {f : β → subtype p} (hf : continuous f) :
continuous (λ x, (f x : α)) | continuous_subtype_coe.comp hf | lemma | continuous.subtype_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.open_embedding_subtype_coe {s : set α} (hs : is_open s) :
open_embedding (coe : s → α) | { induced := rfl,
inj := subtype.coe_injective,
open_range := (subtype.range_coe : range coe = s).symm ▸ hs } | lemma | is_open.open_embedding_subtype_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"open_embedding",
"subtype.coe_injective",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.is_open_map_subtype_coe {s : set α} (hs : is_open s) :
is_open_map (coe : s → α) | hs.open_embedding_subtype_coe.is_open_map | lemma | is_open.is_open_map_subtype_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map.restrict {f : α → β} (hf : is_open_map f) {s : set α} (hs : is_open s) :
is_open_map (s.restrict f) | hf.comp hs.is_open_map_subtype_coe | lemma | is_open_map.restrict | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"is_open",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.closed_embedding_subtype_coe {s : set α} (hs : is_closed s) :
closed_embedding (coe : {x // x ∈ s} → α) | { induced := rfl,
inj := subtype.coe_injective,
closed_range := (subtype.range_coe : range coe = s).symm ▸ hs } | lemma | is_closed.closed_embedding_subtype_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closed_embedding",
"is_closed",
"subtype.coe_injective",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.subtype_mk {f : β → α} (h : continuous f)
(hp : ∀x, p (f x)) : continuous (λx, (⟨f x, hp x⟩ : subtype p)) | continuous_induced_rng.2 h | lemma | continuous.subtype_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.subtype_map {f : α → β} (h : continuous f) {q : β → Prop}
(hpq : ∀ x, p x → q (f x)) : continuous (subtype.map f hpq) | (h.comp continuous_subtype_coe).subtype_mk _ | lemma | continuous.subtype_map | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous",
"continuous_subtype_coe",
"subtype.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_inclusion {s t : set α} (h : s ⊆ t) : continuous (inclusion h) | continuous_id.subtype_map h | lemma | continuous_inclusion | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_subtype_coe {p : α → Prop} {a : subtype p} :
continuous_at (coe : subtype p → α) a | continuous_iff_continuous_at.mp continuous_subtype_coe _ | lemma | continuous_at_subtype_coe | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"continuous_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.dense_iff {s : set α} {t : set s} : dense t ↔ s ⊆ closure (coe '' t) | by { rw [inducing_coe.dense_iff, set_coe.forall], refl } | lemma | subtype.dense_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closure",
"dense",
"set_coe.forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_subtype_coe_eq {a : α} (ha : p a) (h : {a | p a} ∈ 𝓝 a) :
map (coe : subtype p → α) (𝓝 ⟨a, ha⟩) = 𝓝 a | map_nhds_induced_of_mem $ by simpa only [subtype.coe_mk, subtype.range_coe] using h | lemma | map_nhds_subtype_coe_eq | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"map_nhds_induced_of_mem",
"subtype.coe_mk",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_subtype_eq_comap {a : α} {h : p a} :
𝓝 (⟨a, h⟩ : subtype p) = comap coe (𝓝 a) | nhds_induced _ _ | lemma | nhds_subtype_eq_comap | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"nhds_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_subtype_rng {β : Type*} {p : α → Prop} {b : filter β} {f : β → subtype p} :
∀{a:subtype p}, tendsto f b (𝓝 a) ↔ tendsto (λx, (f x : α)) b (𝓝 (a : α)) | | ⟨a, ha⟩ := by rw [nhds_subtype_eq_comap, tendsto_comap_iff, subtype.coe_mk] | lemma | tendsto_subtype_rng | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"filter",
"nhds_subtype_eq_comap",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_subtype {x : {a // p a}} {s : set {a // p a}}:
x ∈ closure s ↔ (x : α) ∈ closure ((coe : _ → α) '' s) | closure_induced | lemma | closure_subtype | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"closure",
"closure_induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_cod_restrict_iff {f : α → β} {t : set β} (h1 : ∀ x, f x ∈ t) {x : α} :
continuous_at (cod_restrict f t h1) x ↔ continuous_at f x | by simp_rw [inducing_coe.continuous_at_iff, function.comp, coe_cod_restrict_apply] | lemma | continuous_at_cod_restrict_iff | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.restrict {f : α → β} {s : set α} {t : set β} (h1 : maps_to f s t) {x : s}
(h2 : continuous_at f x) : continuous_at (h1.restrict f s t) x | (h2.comp continuous_at_subtype_coe).cod_restrict _ | lemma | continuous_at.restrict | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at",
"continuous_at_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.restrict_preimage {f : α → β} {s : set β} {x : f ⁻¹' s}
(h : continuous_at f x) : continuous_at (s.restrict_preimage f) x | h.restrict _ | lemma | continuous_at.restrict_preimage | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.cod_restrict {f : α → β} {s : set β} (hf : continuous f)
(hs : ∀ a, f a ∈ s) : continuous (s.cod_restrict f hs) | hf.subtype_mk hs | lemma | continuous.cod_restrict | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.cod_restrict {e : α → β} (he : inducing e) {s : set β} (hs : ∀ x, e x ∈ s) :
inducing (cod_restrict e s hs) | inducing_of_inducing_compose (he.continuous.cod_restrict hs) continuous_subtype_coe he | lemma | inducing.cod_restrict | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_subtype_coe",
"inducing",
"inducing_of_inducing_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.cod_restrict {e : α → β} (he : embedding e) (s : set β) (hs : ∀ x, e x ∈ s) :
embedding (cod_restrict e s hs) | embedding_of_embedding_compose (he.continuous.cod_restrict hs) continuous_subtype_coe he | lemma | embedding.cod_restrict | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"continuous_subtype_coe",
"embedding",
"embedding_of_embedding_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_inclusion {s t : set α} (h : s ⊆ t) : embedding (set.inclusion h) | embedding_subtype_coe.cod_restrict _ _ | lemma | embedding_inclusion | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"embedding",
"set.inclusion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.of_subset {X : Type*} [topological_space X] {s t : set X}
(ds : discrete_topology s) (ts : t ⊆ s) :
discrete_topology t | (embedding_inclusion ts).discrete_topology | lemma | discrete_topology.of_subset | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"discrete_topology",
"embedding_inclusion",
"topological_space"
] | Let `s, t ⊆ X` be two subsets of a topological space `X`. If `t ⊆ s` and the topology induced
by `X`on `s` is discrete, then also the topology induces on `t` is discrete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_quot_mk : quotient_map (@quot.mk α r) | ⟨quot.exists_rep, rfl⟩ | lemma | quotient_map_quot_mk | topology | src/topology/constructions.lean | [
"topology.maps",
"order.filter.pi"
] | [
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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