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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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is_open_preimage (hf : quotient_map f) {s : set β} :
is_open (f ⁻¹' s) ↔ is_open s | ((quotient_map_iff.1 hf).2 s).symm | lemma | quotient_map.is_open_preimage | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_preimage (hf : quotient_map f) {s : set β} :
is_closed (f ⁻¹' s) ↔ is_closed s | ((quotient_map_iff_closed.1 hf).2 s).symm | lemma | quotient_map.is_closed_preimage | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_closed",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map [topological_space α] [topological_space β] (f : α → β) | ∀ U : set α, is_open U → is_open (f '' U) | def | is_open_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open",
"topological_space"
] | A map `f : α → β` is said to be an *open map*, if the image of any open `U : set α`
is open in `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : is_open_map (@id α) | assume s hs, by rwa [image_id] | lemma | is_open_map.id | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp
{g : β → γ} {f : α → β} (hg : is_open_map g) (hf : is_open_map f) : is_open_map (g ∘ f) | by intros s hs; rw [image_comp]; exact hg _ (hf _ hs) | lemma | is_open_map.comp | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_range (hf : is_open_map f) : is_open (range f) | by { rw ← image_univ, exact hf _ is_open_univ } | lemma | is_open_map.is_open_range | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open",
"is_open_map",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mem_nhds (hf : is_open_map f) {x : α} {s : set α} (hx : s ∈ 𝓝 x) :
f '' s ∈ 𝓝 (f x) | let ⟨t, hts, ht, hxt⟩ := mem_nhds_iff.1 hx in
mem_of_superset (is_open.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts) | lemma | is_open_map.image_mem_nhds | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open.mem_nhds",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_mem_nhds (hf : is_open_map f) (x : α) : range f ∈ 𝓝 (f x) | hf.is_open_range.mem_nhds $ mem_range_self _ | lemma | is_open_map.range_mem_nhds | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_interior (hf : is_open_map f) {s : set α} {t : set β} (h : maps_to f s t) :
maps_to f (interior s) (interior t) | maps_to'.2 $ interior_maximal (h.mono interior_subset subset.rfl).image_subset
(hf _ is_open_interior) | lemma | is_open_map.maps_to_interior | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"interior",
"interior_maximal",
"interior_subset",
"is_open_interior",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_interior_subset (hf : is_open_map f) (s : set α) :
f '' interior s ⊆ interior (f '' s) | (hf.maps_to_interior (maps_to_image f s)).image_subset | lemma | is_open_map.image_interior_subset | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"interior",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_le (hf : is_open_map f) (a : α) : 𝓝 (f a) ≤ (𝓝 a).map f | le_map $ λ s, hf.image_mem_nhds | lemma | is_open_map.nhds_le | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_nhds_le (hf : ∀ a, 𝓝 (f a) ≤ map f (𝓝 a)) : is_open_map f | λ s hs, is_open_iff_mem_nhds.2 $ λ b ⟨a, has, hab⟩,
hab ▸ hf _ (image_mem_map $ is_open.mem_nhds hs has) | lemma | is_open_map.of_nhds_le | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open.mem_nhds",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_sections {f : α → β}
(h : ∀ x, ∃ g : β → α, continuous_at g (f x) ∧ g (f x) = x ∧ right_inverse g f) :
is_open_map f | of_nhds_le $ λ x, let ⟨g, hgc, hgx, hgf⟩ := h x in
calc 𝓝 (f x) = map f (map g (𝓝 (f x))) : by rw [map_map, hgf.comp_eq_id, map_id]
... ≤ map f (𝓝 (g (f x))) : map_mono hgc
... = map f (𝓝 x) : by rw hgx | lemma | is_open_map.of_sections | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous_at",
"is_open_map",
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_inverse {f : α → β} {f' : β → α}
(h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') :
is_open_map f | of_sections $ λ x, ⟨f', h.continuous_at, r_inv _, l_inv⟩ | lemma | is_open_map.of_inverse | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_quotient_map {f : α → β}
(open_map : is_open_map f) (cont : continuous f) (surj : surjective f) :
quotient_map f | quotient_map_iff.2 ⟨surj, λ s, ⟨λ h, h.preimage cont, λ h, surj.image_preimage s ▸ open_map _ h⟩⟩ | lemma | is_open_map.to_quotient_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"cont",
"continuous",
"is_open_map",
"quotient_map"
] | A continuous surjective open map is a quotient map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interior_preimage_subset_preimage_interior (hf : is_open_map f) {s : set β} :
interior (f⁻¹' s) ⊆ f⁻¹' (interior s) | hf.maps_to_interior (maps_to_preimage _ _) | lemma | is_open_map.interior_preimage_subset_preimage_interior | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"interior",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_interior_eq_interior_preimage (hf₁ : is_open_map f) (hf₂ : continuous f)
(s : set β) :
f⁻¹' (interior s) = interior (f⁻¹' s) | subset.antisymm
(preimage_interior_subset_interior_preimage hf₂)
(interior_preimage_subset_preimage_interior hf₁) | lemma | is_open_map.preimage_interior_eq_interior_preimage | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"interior",
"is_open_map",
"preimage_interior_subset_interior_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closure_subset_closure_preimage (hf : is_open_map f) {s : set β} :
f ⁻¹' (closure s) ⊆ closure (f ⁻¹' s) | begin
rw ← compl_subset_compl,
simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior]
end | lemma | is_open_map.preimage_closure_subset_closure_preimage | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closure",
"interior_compl",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closure_eq_closure_preimage (hf : is_open_map f) (hfc : continuous f) (s : set β) :
f ⁻¹' (closure s) = closure (f ⁻¹' s) | hf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s) | lemma | is_open_map.preimage_closure_eq_closure_preimage | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closure",
"continuous",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_frontier_subset_frontier_preimage (hf : is_open_map f) {s : set β} :
f ⁻¹' (frontier s) ⊆ frontier (f ⁻¹' s) | by simpa only [frontier_eq_closure_inter_closure, preimage_inter]
using inter_subset_inter hf.preimage_closure_subset_closure_preimage
hf.preimage_closure_subset_closure_preimage | lemma | is_open_map.preimage_frontier_subset_frontier_preimage | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"frontier",
"frontier_eq_closure_inter_closure",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_frontier_eq_frontier_preimage (hf : is_open_map f) (hfc : continuous f) (s : set β) :
f ⁻¹' (frontier s) = frontier (f ⁻¹' s) | by simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl,
hf.preimage_closure_eq_closure_preimage hfc] | lemma | is_open_map.preimage_frontier_eq_frontier_preimage | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"frontier",
"frontier_eq_closure_inter_closure",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_iff_nhds_le [topological_space α] [topological_space β] {f : α → β} :
is_open_map f ↔ ∀(a:α), 𝓝 (f a) ≤ (𝓝 a).map f | ⟨λ hf, hf.nhds_le, is_open_map.of_nhds_le⟩ | lemma | is_open_map_iff_nhds_le | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_iff_interior [topological_space α] [topological_space β] {f : α → β} :
is_open_map f ↔ ∀ s, f '' (interior s) ⊆ interior (f '' s) | ⟨is_open_map.image_interior_subset, λ hs u hu, subset_interior_iff_is_open.mp $
calc f '' u = f '' (interior u) : by rw hu.interior_eq
... ⊆ interior (f '' u) : hs u⟩ | lemma | is_open_map_iff_interior | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"interior",
"is_open_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.is_open_map [topological_space α] [topological_space β] {f : α → β}
(hi : inducing f) (ho : is_open (range f)) :
is_open_map f | is_open_map.of_nhds_le $ λ x, (hi.map_nhds_of_mem _ $ is_open.mem_nhds ho $ mem_range_self _).ge | lemma | inducing.is_open_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"is_open",
"is_open.mem_nhds",
"is_open_map",
"is_open_map.of_nhds_le",
"topological_space"
] | An inducing map with an open range is an open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_map (f : α → β) | ∀ U : set α, is_closed U → is_closed (f '' U) | def | is_closed_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_closed"
] | A map `f : α → β` is said to be a *closed map*, if the image of any closed `U : set α`
is closed in `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : is_closed_map (@id α) | assume s hs, by rwa image_id | lemma | is_closed_map.id | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : β → γ} {f : α → β} (hg : is_closed_map g) (hf : is_closed_map f) :
is_closed_map (g ∘ f) | by { intros s hs, rw image_comp, exact hg _ (hf _ hs) } | lemma | is_closed_map.comp | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_image_subset {f : α → β} (hf : is_closed_map f) (s : set α) :
closure (f '' s) ⊆ f '' closure s | closure_minimal (image_subset _ subset_closure) (hf _ is_closed_closure) | lemma | is_closed_map.closure_image_subset | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closure",
"closure_minimal",
"is_closed_closure",
"is_closed_map",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_inverse {f : α → β} {f' : β → α}
(h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') :
is_closed_map f | assume s hs,
have f' ⁻¹' s = f '' s, by ext x; simp [mem_image_iff_of_inverse r_inv l_inv],
this ▸ hs.preimage h | lemma | is_closed_map.of_inverse | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_nonempty {f : α → β} (h : ∀ s, is_closed s → s.nonempty → is_closed (f '' s)) :
is_closed_map f | begin
intros s hs, cases eq_empty_or_nonempty s with h2s h2s,
{ simp_rw [h2s, image_empty, is_closed_empty] },
{ exact h s hs h2s }
end | lemma | is_closed_map.of_nonempty | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_closed",
"is_closed_empty",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_range {f : α → β} (hf : is_closed_map f) : is_closed (range f) | @image_univ _ _ f ▸ hf _ is_closed_univ | lemma | is_closed_map.closed_range | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_closed",
"is_closed_map",
"is_closed_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_quotient_map {f : α → β} (hcl : is_closed_map f) (hcont : continuous f)
(hsurj : surjective f) : quotient_map f | quotient_map_iff_closed.2
⟨hsurj, λ s, ⟨λ hs, hs.preimage hcont, λ hs, hsurj.image_preimage s ▸ hcl _ hs⟩⟩ | lemma | is_closed_map.to_quotient_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"is_closed_map",
"quotient_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.is_closed_map [topological_space α] [topological_space β]
{f : α → β} (hf : inducing f) (h : is_closed (range f)) : is_closed_map f | begin
intros s hs,
rcases hf.is_closed_iff.1 hs with ⟨t, ht, rfl⟩,
rw image_preimage_eq_inter_range,
exact ht.inter h
end | lemma | inducing.is_closed_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"inducing",
"is_closed",
"is_closed_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_iff_closure_image [topological_space α] [topological_space β] {f : α → β} :
is_closed_map f ↔ ∀ s, closure (f '' s) ⊆ f '' closure s | ⟨is_closed_map.closure_image_subset, λ hs c hc, is_closed_of_closure_subset $
calc closure (f '' c) ⊆ f '' (closure c) : hs c
... = f '' c : by rw hc.closure_eq⟩ | lemma | is_closed_map_iff_closure_image | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closure",
"is_closed_map",
"is_closed_of_closure_subset",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding (f : α → β) extends _root_.embedding f : Prop | (open_range : is_open $ range f) | structure | open_embedding | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open"
] | An open embedding is an embedding with open image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_embedding.is_open_map {f : α → β} (hf : open_embedding f) : is_open_map f | hf.to_embedding.to_inducing.is_open_map hf.open_range | lemma | open_embedding.is_open_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open_map",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.map_nhds_eq {f : α → β} (hf : open_embedding f) (a : α) :
map f (𝓝 a) = 𝓝 (f a) | hf.to_embedding.map_nhds_of_mem _ $ hf.open_range.mem_nhds $ mem_range_self _ | lemma | open_embedding.map_nhds_eq | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.open_iff_image_open {f : α → β} (hf : open_embedding f)
{s : set α} : is_open s ↔ is_open (f '' s) | ⟨hf.is_open_map s,
λ h, begin
convert ← h.preimage hf.to_embedding.continuous,
apply preimage_image_eq _ hf.inj
end⟩ | lemma | open_embedding.open_iff_image_open | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.tendsto_nhds_iff {ι : Type*}
{f : ι → β} {g : β → γ} {a : filter ι} {b : β} (hg : open_embedding g) :
tendsto f a (𝓝 b) ↔ tendsto (g ∘ f) a (𝓝 (g b)) | hg.to_embedding.tendsto_nhds_iff | lemma | open_embedding.tendsto_nhds_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"filter",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.continuous {f : α → β} (hf : open_embedding f) : continuous f | hf.to_embedding.continuous | lemma | open_embedding.continuous | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.open_iff_preimage_open {f : α → β} (hf : open_embedding f)
{s : set β} (hs : s ⊆ range f) : is_open s ↔ is_open (f ⁻¹' s) | begin
convert ←hf.open_iff_image_open.symm,
rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
end | lemma | open_embedding.open_iff_preimage_open | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_of_embedding_open {f : α → β} (h₁ : embedding f)
(h₂ : is_open_map f) : open_embedding f | ⟨h₁, h₂.is_open_range⟩ | lemma | open_embedding_of_embedding_open | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"embedding",
"is_open_map",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_iff_embedding_open {f : α → β} :
open_embedding f ↔ embedding f ∧ is_open_map f | ⟨λ h, ⟨h.1, h.is_open_map⟩, λ h, open_embedding_of_embedding_open h.1 h.2⟩ | lemma | open_embedding_iff_embedding_open | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"embedding",
"is_open_map",
"open_embedding",
"open_embedding_of_embedding_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_of_continuous_injective_open {f : α → β} (h₁ : continuous f)
(h₂ : injective f) (h₃ : is_open_map f) : open_embedding f | begin
simp only [open_embedding_iff_embedding_open, embedding_iff, inducing_iff_nhds, *, and_true],
exact λ a, le_antisymm (h₁.tendsto _).le_comap
(@comap_map _ _ (𝓝 a) _ h₂ ▸ comap_mono (h₃.nhds_le _))
end | lemma | open_embedding_of_continuous_injective_open | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"inducing_iff_nhds",
"is_open_map",
"open_embedding",
"open_embedding_iff_embedding_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_iff_continuous_injective_open {f : α → β} :
open_embedding f ↔ continuous f ∧ injective f ∧ is_open_map f | ⟨λ h, ⟨h.continuous, h.inj, h.is_open_map⟩,
λ h, open_embedding_of_continuous_injective_open h.1 h.2.1 h.2.2⟩ | lemma | open_embedding_iff_continuous_injective_open | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"continuous",
"is_open_map",
"open_embedding",
"open_embedding_of_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_id : open_embedding (@id α) | ⟨embedding_id, is_open_map.id.is_open_range⟩ | lemma | open_embedding_id | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.comp {g : β → γ} {f : α → β}
(hg : open_embedding g) (hf : open_embedding f) : open_embedding (g ∘ f) | ⟨hg.1.comp hf.1, (hg.is_open_map.comp hf.is_open_map).is_open_range⟩ | lemma | open_embedding.comp | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.is_open_map_iff {g : β → γ} {f : α → β} (hg : open_embedding g) :
is_open_map f ↔ is_open_map (g ∘ f) | by simp only [is_open_map_iff_nhds_le, ← @map_map _ _ _ _ f g, ← hg.map_nhds_eq,
map_le_map_iff hg.inj] | lemma | open_embedding.is_open_map_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_open_map",
"is_open_map_iff_nhds_le",
"open_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.of_comp_iff (f : α → β) {g : β → γ} (hg : open_embedding g) :
open_embedding (g ∘ f) ↔ open_embedding f | by simp only [open_embedding_iff_continuous_injective_open, ← hg.is_open_map_iff,
← hg.1.continuous_iff, hg.inj.of_comp_iff] | lemma | open_embedding.of_comp_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"open_embedding",
"open_embedding_iff_continuous_injective_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding.of_comp (f : α → β) {g : β → γ} (hg : open_embedding g)
(h : open_embedding (g ∘ f)) : open_embedding f | (open_embedding.of_comp_iff f hg).1 h | lemma | open_embedding.of_comp | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"open_embedding",
"open_embedding.of_comp_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding (f : α → β) extends _root_.embedding f : Prop | (closed_range : is_closed $ range f) | structure | closed_embedding | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"is_closed"
] | A closed embedding is an embedding with closed image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding.tendsto_nhds_iff {ι : Type*}
{g : ι → α} {a : filter ι} {b : α} (hf : closed_embedding f) :
tendsto g a (𝓝 b) ↔ tendsto (f ∘ g) a (𝓝 (f b)) | hf.to_embedding.tendsto_nhds_iff | lemma | closed_embedding.tendsto_nhds_iff | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.continuous (hf : closed_embedding f) : continuous f | hf.to_embedding.continuous | lemma | closed_embedding.continuous | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.is_closed_map (hf : closed_embedding f) : is_closed_map f | hf.to_embedding.to_inducing.is_closed_map hf.closed_range | lemma | closed_embedding.is_closed_map | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.closed_iff_image_closed (hf : closed_embedding f)
{s : set α} : is_closed s ↔ is_closed (f '' s) | ⟨hf.is_closed_map s,
λ h, begin
convert ←continuous_iff_is_closed.mp hf.continuous _ h,
apply preimage_image_eq _ hf.inj
end⟩ | lemma | closed_embedding.closed_iff_image_closed | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.closed_iff_preimage_closed (hf : closed_embedding f)
{s : set β} (hs : s ⊆ range f) : is_closed s ↔ is_closed (f ⁻¹' s) | begin
convert ←hf.closed_iff_image_closed.symm,
rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]
end | lemma | closed_embedding.closed_iff_preimage_closed | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_of_embedding_closed (h₁ : embedding f)
(h₂ : is_closed_map f) : closed_embedding f | ⟨h₁, by convert h₂ univ is_closed_univ; simp⟩ | lemma | closed_embedding_of_embedding_closed | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"embedding",
"is_closed_map",
"is_closed_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_of_continuous_injective_closed (h₁ : continuous f)
(h₂ : injective f) (h₃ : is_closed_map f) : closed_embedding f | begin
refine closed_embedding_of_embedding_closed ⟨⟨_⟩, h₂⟩ h₃,
apply le_antisymm (continuous_iff_le_induced.mp h₁) _,
intro s',
change is_open _ ≤ is_open _,
rw [←is_closed_compl_iff, ←is_closed_compl_iff],
generalize : s'ᶜ = s,
rw is_closed_induced_iff,
refine λ hs, ⟨f '' s, h₃ s hs, _⟩,
rw preimage... | lemma | closed_embedding_of_continuous_injective_closed | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"closed_embedding_of_embedding_closed",
"continuous",
"is_closed_induced_iff",
"is_closed_map",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_id : closed_embedding (@id α) | ⟨embedding_id, by convert is_closed_univ; apply range_id⟩ | lemma | closed_embedding_id | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"is_closed_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.comp {g : β → γ} {f : α → β}
(hg : closed_embedding g) (hf : closed_embedding f) : closed_embedding (g ∘ f) | ⟨hg.to_embedding.comp hf.to_embedding, show is_closed (range (g ∘ f)),
by rw [range_comp, ←hg.closed_iff_image_closed]; exact hf.closed_range⟩ | lemma | closed_embedding.comp | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding.closure_image_eq {f : α → β} (hf : closed_embedding f) (s : set α) :
closure (f '' s) = f '' closure s | (hf.is_closed_map.closure_image_subset _).antisymm
(image_closure_subset_closure_image hf.continuous) | lemma | closed_embedding.closure_image_eq | topology | src/topology/maps.lean | [
"topology.order",
"topology.nhds_set"
] | [
"closed_embedding",
"closure",
"image_closure_subset_closure_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set (s : set α) : filter α | Sup (nhds '' s) | def | nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"filter",
"nhds"
] | The filter of neighborhoods of a set in a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nhds_set_diagonal (α) [topological_space (α × α)] : 𝓝ˢ (diagonal α) = ⨆ x, 𝓝 (x, x) | by { rw [nhds_set, ← range_diag, ← range_comp], refl } | lemma | nhds_set_diagonal | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"nhds_set",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_set_iff_forall : s ∈ 𝓝ˢ t ↔ ∀ (x : α), x ∈ t → s ∈ 𝓝 x | by simp_rw [nhds_set, filter.mem_Sup, ball_image_iff] | lemma | mem_nhds_set_iff_forall | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"filter.mem_Sup",
"nhds_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bUnion_mem_nhds_set {t : α → set α} (h : ∀ x ∈ s, t x ∈ 𝓝 x) : (⋃ x ∈ s, t x) ∈ 𝓝ˢ s | mem_nhds_set_iff_forall.2 $ λ x hx, mem_of_superset (h x hx) (subset_Union₂ x hx) | lemma | bUnion_mem_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_iff_mem_nhds_set : s ⊆ interior t ↔ t ∈ 𝓝ˢ s | by simp_rw [mem_nhds_set_iff_forall, subset_interior_iff_nhds] | lemma | subset_interior_iff_mem_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"interior",
"mem_nhds_set_iff_forall",
"subset_interior_iff_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_set_iff_exists : s ∈ 𝓝ˢ t ↔ ∃ U : set α, is_open U ∧ t ⊆ U ∧ U ⊆ s | by { rw [← subset_interior_iff_mem_nhds_set, subset_interior_iff] } | lemma | mem_nhds_set_iff_exists | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"is_open",
"subset_interior_iff",
"subset_interior_iff_mem_nhds_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_basis_nhds_set (s : set α) : (𝓝ˢ s).has_basis (λ U, is_open U ∧ s ⊆ U) (λ U, U) | ⟨λ t, by simp [mem_nhds_set_iff_exists, and_assoc]⟩ | lemma | has_basis_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"is_open",
"mem_nhds_set_iff_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.mem_nhds_set (hU : is_open s) : s ∈ 𝓝ˢ t ↔ t ⊆ s | by rw [← subset_interior_iff_mem_nhds_set, interior_eq_iff_is_open.mpr hU] | lemma | is_open.mem_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"is_open",
"subset_interior_iff_mem_nhds_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_le_nhds_set : 𝓟 s ≤ 𝓝ˢ s | λ s hs, (subset_interior_iff_mem_nhds_set.mpr hs).trans interior_subset | lemma | principal_le_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"interior_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_eq_principal_iff : 𝓝ˢ s = 𝓟 s ↔ is_open s | by rw [← principal_le_nhds_set.le_iff_eq, le_principal_iff, mem_nhds_set_iff_forall,
is_open_iff_mem_nhds] | lemma | nhds_set_eq_principal_iff | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"is_open",
"is_open_iff_mem_nhds",
"mem_nhds_set_iff_forall"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_interior : 𝓝ˢ (interior s) = 𝓟 (interior s) | is_open_interior.nhds_set_eq | lemma | nhds_set_interior | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_singleton : 𝓝ˢ {x} = 𝓝 x | by { ext,
rw [← subset_interior_iff_mem_nhds_set, ← mem_interior_iff_mem_nhds, singleton_subset_iff] } | lemma | nhds_set_singleton | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"mem_interior_iff_mem_nhds",
"subset_interior_iff_mem_nhds_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_set_interior : s ∈ 𝓝ˢ (interior s) | subset_interior_iff_mem_nhds_set.mp subset.rfl | lemma | mem_nhds_set_interior | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_empty : 𝓝ˢ (∅ : set α) = ⊥ | by rw [is_open_empty.nhds_set_eq, principal_empty] | lemma | nhds_set_empty | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_set_empty : s ∈ 𝓝ˢ (∅ : set α) | by simp | lemma | mem_nhds_set_empty | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_univ : 𝓝ˢ (univ : set α) = ⊤ | by rw [is_open_univ.nhds_set_eq, principal_univ] | lemma | nhds_set_univ | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_mono (h : s ⊆ t) : 𝓝ˢ s ≤ 𝓝ˢ t | Sup_le_Sup $ image_subset _ h | lemma | nhds_set_mono | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"Sup_le_Sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_nhds_set : monotone (𝓝ˢ : set α → filter α) | λ s t, nhds_set_mono | lemma | monotone_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"filter",
"monotone",
"nhds_set_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_le_nhds_set (h : x ∈ s) : 𝓝 x ≤ 𝓝ˢ s | le_Sup $ mem_image_of_mem _ h | lemma | nhds_le_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"le_Sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_set_union (s t : set α) : 𝓝ˢ (s ∪ t) = 𝓝ˢ s ⊔ 𝓝ˢ t | by simp only [nhds_set, image_union, Sup_union] | lemma | nhds_set_union | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"Sup_union",
"nhds_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
union_mem_nhds_set (h₁ : s₁ ∈ 𝓝ˢ t₁) (h₂ : s₂ ∈ 𝓝ˢ t₂) : s₁ ∪ s₂ ∈ 𝓝ˢ (t₁ ∪ t₂) | by { rw nhds_set_union, exact union_mem_sup h₁ h₂ } | lemma | union_mem_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"nhds_set_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.tendsto_nhds_set {f : α → β} {t : set β} (hf : continuous f)
(hst : maps_to f s t) : tendsto f (𝓝ˢ s) (𝓝ˢ t) | ((has_basis_nhds_set s).tendsto_iff (has_basis_nhds_set t)).mpr $ λ U hU,
⟨f ⁻¹' U, ⟨hU.1.preimage hf, hst.mono subset.rfl hU.2⟩, λ x, id⟩ | lemma | continuous.tendsto_nhds_set | topology | src/topology/nhds_set.lean | [
"topology.basic"
] | [
"continuous",
"has_basis_nhds_set"
] | Preimage of a set neighborhood of `t` under a continuous map `f` is a set neighborhood of `s`
provided that `f` maps `s` to `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
noetherian_space : Prop | (well_founded : well_founded ((>) : opens α → opens α → Prop)) | class | topological_space.noetherian_space | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [] | Type class for noetherian spaces. It is defined to be spaces whose open sets satisfies ACC. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
noetherian_space_iff_opens :
noetherian_space α ↔ ∀ s : opens α, is_compact (s : set α) | begin
rw [noetherian_space_iff, complete_lattice.well_founded_iff_is_Sup_finite_compact,
complete_lattice.is_Sup_finite_compact_iff_all_elements_compact],
exact forall_congr opens.is_compact_element_iff,
end | lemma | topological_space.noetherian_space_iff_opens | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"complete_lattice.is_Sup_finite_compact_iff_all_elements_compact",
"complete_lattice.well_founded_iff_is_Sup_finite_compact",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.compact_space [h : noetherian_space α] : compact_space α | ⟨(noetherian_space_iff_opens α).mp h ⊤⟩ | instance | topological_space.noetherian_space.compact_space | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.is_compact [noetherian_space α] (s : set α) : is_compact s | begin
refine is_compact_iff_finite_subcover.2 (λ ι U hUo hs, _),
rcases ((noetherian_space_iff_opens α).mp ‹_›
⟨⋃ i, U i, is_open_Union hUo⟩).elim_finite_subcover U hUo set.subset.rfl with ⟨t, ht⟩,
exact ⟨t, hs.trans ht⟩
end | lemma | topological_space.noetherian_space.is_compact | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"is_compact",
"is_open_Union",
"set.subset.rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing.noetherian_space [noetherian_space α] {i : β → α} (hi : inducing i) :
noetherian_space β | (noetherian_space_iff_opens _).2 $ λ s, hi.is_compact_iff.1 (noetherian_space.is_compact _) | lemma | topological_space.inducing.noetherian_space | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.set [h : noetherian_space α] (s : set α) : noetherian_space s | inducing_coe.noetherian_space | instance | topological_space.noetherian_space.set | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space_tfae :
tfae [noetherian_space α,
well_founded (λ s t : closeds α, s < t),
∀ s : set α, is_compact s,
∀ s : opens α, is_compact (s : set α)] | begin
tfae_have : 1 ↔ 2,
{ refine (noetherian_space_iff _).trans (surjective.well_founded_iff opens.compl_bijective.2 _),
exact λ s t, (order_iso.compl (set α)).lt_iff_lt.symm },
tfae_have : 1 ↔ 4,
{ exact noetherian_space_iff_opens α },
tfae_have : 1 → 3,
{ exact @noetherian_space.is_compact _ _ },
t... | lemma | topological_space.noetherian_space_tfae | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"is_compact",
"order_iso.compl",
"surjective.well_founded_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space_of_surjective [noetherian_space α] (f : α → β)
(hf : continuous f) (hf' : function.surjective f) : noetherian_space β | begin
rw noetherian_space_iff_opens,
intro s,
obtain ⟨t, e⟩ := set.image_surjective.mpr hf' s,
exact e ▸ (noetherian_space.is_compact t).image hf,
end | lemma | topological_space.noetherian_space_of_surjective | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space_iff_of_homeomorph (f : α ≃ₜ β) :
noetherian_space α ↔ noetherian_space β | ⟨λ h, @@noetherian_space_of_surjective _ _ h f f.continuous f.surjective,
λ h, @@noetherian_space_of_surjective _ _ h f.symm f.symm.continuous f.symm.surjective⟩ | lemma | topological_space.noetherian_space_iff_of_homeomorph | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.range [noetherian_space α] (f : α → β) (hf : continuous f) :
noetherian_space (set.range f) | noetherian_space_of_surjective (set.cod_restrict f _ set.mem_range_self) (by continuity)
(λ ⟨a, b, h⟩, ⟨b, subtype.ext h⟩) | lemma | topological_space.noetherian_space.range | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"continuity",
"continuous",
"set.cod_restrict",
"set.mem_range_self",
"set.range",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space_set_iff (s : set α) :
noetherian_space s ↔ ∀ t ⊆ s, is_compact t | begin
rw (noetherian_space_tfae s).out 0 2,
split,
{ intros H t ht,
have := embedding_subtype_coe.is_compact_iff_is_compact_image.mp (H (coe ⁻¹' t)),
simpa [set.inter_eq_left_iff_subset.mpr ht] using this },
{ intros H t,
refine embedding_subtype_coe.is_compact_iff_is_compact_image.mpr (H (coe '' t)... | lemma | topological_space.noetherian_space_set_iff | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_univ_iff :
noetherian_space (set.univ : set α) ↔ noetherian_space α | noetherian_space_iff_of_homeomorph (homeomorph.set.univ α) | lemma | topological_space.noetherian_univ_iff | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"homeomorph.set.univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.Union {ι : Type*} (f : ι → set α) [finite ι]
[hf : ∀ i, noetherian_space (f i)] :
noetherian_space (⋃ i, f i) | begin
casesI nonempty_fintype ι,
simp_rw noetherian_space_set_iff at hf ⊢,
intros t ht,
rw [← set.inter_eq_left_iff_subset.mpr ht, set.inter_Union],
exact is_compact_Union (λ i, hf i _ (set.inter_subset_right _ _))
end | lemma | topological_space.noetherian_space.Union | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"finite",
"is_compact_Union",
"nonempty_fintype",
"set.inter_Union",
"set.inter_subset_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.discrete [noetherian_space α] [t2_space α] : discrete_topology α | ⟨eq_bot_iff.mpr (λ U _, is_closed_compl_iff.mp (noetherian_space.is_compact _).is_closed)⟩ | lemma | topological_space.noetherian_space.discrete | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"discrete_topology",
"is_closed",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.finite [noetherian_space α] [t2_space α] : finite α | begin
letI : fintype α :=
set.fintype_of_finite_univ (noetherian_space.is_compact set.univ).finite_of_discrete,
apply_instance
end | lemma | topological_space.noetherian_space.finite | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"finite",
"fintype",
"set.fintype_of_finite_univ",
"t2_space"
] | Spaces that are both Noetherian and Hausdorff is finite. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite.to_noetherian_space [finite α] : noetherian_space α | ⟨finite.well_founded_of_trans_of_irrefl _⟩ | instance | topological_space.finite.to_noetherian_space | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"finite"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
noetherian_space.exists_finset_irreducible [noetherian_space α] (s : closeds α) :
∃ S : finset (closeds α), (∀ k : S, is_irreducible (k : set α)) ∧ s = S.sup id | begin
classical,
have := ((noetherian_space_tfae α).out 0 1).mp infer_instance,
apply well_founded.induction this s, clear s,
intros s H,
by_cases h₁ : is_preirreducible s.1,
cases h₂ : s.1.eq_empty_or_nonempty,
{ use ∅, refine ⟨λ k, k.2.elim, _⟩, rw finset.sup_empty, ext1, exact h },
{ use {s},
sim... | lemma | topological_space.noetherian_space.exists_finset_irreducible | topology | src/topology/noetherian_space.lean | [
"order.compactly_generated",
"topology.sets.closeds"
] | [
"coe_coe",
"finset",
"finset.sup_empty",
"finset.sup_singleton",
"finset.sup_union",
"inf_sup_left",
"is_irreducible",
"is_preirreducible",
"is_preirreducible_iff_closed_union_closed",
"left_eq_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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