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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simps.symm_apply (h : α ≃ₜ β) : β → α | h.symm
initialize_simps_projections homeomorph
(to_equiv_to_fun → apply, to_equiv_inv_fun → symm_apply, -to_equiv) | def | homeomorph.simps.symm_apply | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"homeomorph"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_equiv (h : α ≃ₜ β) : ⇑h.to_equiv = h | rfl | lemma | homeomorph.coe_to_equiv | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_symm_to_equiv (h : α ≃ₜ β) : ⇑h.to_equiv.symm = h.symm | rfl | lemma | homeomorph.coe_symm_to_equiv | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_injective : function.injective (to_equiv : α ≃ₜ β → α ≃ β) | | ⟨e, h₁, h₂⟩ ⟨e', h₁', h₂'⟩ rfl := rfl | lemma | homeomorph.to_equiv_injective | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {h h' : α ≃ₜ β} (H : ∀ x, h x = h' x) : h = h' | to_equiv_injective $ equiv.ext H | lemma | homeomorph.ext | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm (h : α ≃ₜ β) : h.symm.symm = h | ext $ λ _, rfl | lemma | homeomorph.symm_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (α : Type*) [topological_space α] : α ≃ₜ α | { continuous_to_fun := continuous_id,
continuous_inv_fun := continuous_id,
to_equiv := equiv.refl α } | def | homeomorph.refl | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_id",
"equiv.refl",
"topological_space"
] | Identity map as a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans (h₁ : α ≃ₜ β) (h₂ : β ≃ₜ γ) : α ≃ₜ γ | { continuous_to_fun := h₂.continuous_to_fun.comp h₁.continuous_to_fun,
continuous_inv_fun := h₁.continuous_inv_fun.comp h₂.continuous_inv_fun,
to_equiv := equiv.trans h₁.to_equiv h₂.to_equiv } | def | homeomorph.trans | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"equiv.trans"
] | Composition of two homeomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (h₁ : α ≃ₜ β) (h₂ : β ≃ₜ γ) (a : α) : h₁.trans h₂ a = h₂ (h₁ a) | rfl | lemma | homeomorph.trans_apply | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph_mk_coe_symm (a : equiv α β) (b c) :
((homeomorph.mk a b c).symm : β → α) = a.symm | rfl | lemma | homeomorph.homeomorph_mk_coe_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm : (homeomorph.refl α).symm = homeomorph.refl α | rfl | lemma | homeomorph.refl_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (h : α ≃ₜ β) : continuous h | h.continuous_to_fun | lemma | homeomorph.continuous | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_symm (h : α ≃ₜ β) : continuous (h.symm) | h.continuous_inv_fun | lemma | homeomorph.continuous_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply (h : α ≃ₜ β) (x : β) : h (h.symm x) = x | h.to_equiv.apply_symm_apply x | lemma | homeomorph.apply_symm_apply | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (h : α ≃ₜ β) (x : α) : h.symm (h x) = x | h.to_equiv.symm_apply_apply x | lemma | homeomorph.symm_apply_apply | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_trans_symm (h : α ≃ₜ β) : h.trans h.symm = homeomorph.refl α | by { ext, apply symm_apply_apply } | lemma | homeomorph.self_trans_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self (h : α ≃ₜ β) : h.symm.trans h = homeomorph.refl β | by { ext, apply apply_symm_apply } | lemma | homeomorph.symm_trans_self | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"homeomorph.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bijective (h : α ≃ₜ β) : function.bijective h | h.to_equiv.bijective | lemma | homeomorph.bijective | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (h : α ≃ₜ β) : function.injective h | h.to_equiv.injective | lemma | homeomorph.injective | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective (h : α ≃ₜ β) : function.surjective h | h.to_equiv.surjective | lemma | homeomorph.surjective | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
change_inv (f : α ≃ₜ β) (g : β → α) (hg : function.right_inverse g f) : α ≃ₜ β | have g = f.symm, from funext (λ x, calc g x = f.symm (f (g x)) : (f.left_inv (g x)).symm
... = f.symm x : by rw hg x),
{ to_fun := f,
inv_fun := g,
left_inv := by convert f.left_inv,
right_inv := by convert f.right_inv,
continuous_to_fun := f.continuous,
continuous_inv_... | def | homeomorph.change_inv | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"inv_fun"
] | Change the homeomorphism `f` to make the inverse function definitionally equal to `g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_comp_self (h : α ≃ₜ β) : ⇑h.symm ∘ ⇑h = id | funext h.symm_apply_apply | lemma | homeomorph.symm_comp_self | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_comp_symm (h : α ≃ₜ β) : ⇑h ∘ ⇑h.symm = id | funext h.apply_symm_apply | lemma | homeomorph.self_comp_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_coe (h : α ≃ₜ β) : range h = univ | h.surjective.range_eq | lemma | homeomorph.range_coe | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_symm (h : α ≃ₜ β) : image h.symm = preimage h | funext h.symm.to_equiv.image_eq_preimage | lemma | homeomorph.image_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_symm (h : α ≃ₜ β) : preimage h.symm = image h | (funext h.to_equiv.image_eq_preimage).symm | lemma | homeomorph.preimage_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_preimage (h : α ≃ₜ β) (s : set β) : h '' (h ⁻¹' s) = s | h.to_equiv.image_preimage s | lemma | homeomorph.image_preimage | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_image (h : α ≃ₜ β) (s : set α) : h ⁻¹' (h '' s) = s | h.to_equiv.preimage_image s | lemma | homeomorph.preimage_image | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing (h : α ≃ₜ β) : inducing h | inducing_of_inducing_compose h.continuous h.symm.continuous $
by simp only [symm_comp_self, inducing_id] | lemma | homeomorph.inducing | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"inducing",
"inducing_id",
"inducing_of_inducing_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_eq (h : α ≃ₜ β) : topological_space.induced h ‹_› = ‹_› | h.inducing.1.symm | lemma | homeomorph.induced_eq | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map (h : α ≃ₜ β) : quotient_map h | quotient_map.of_quotient_map_compose h.symm.continuous h.continuous $
by simp only [self_comp_symm, quotient_map.id] | lemma | homeomorph.quotient_map | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"quotient_map",
"quotient_map.id",
"quotient_map.of_quotient_map_compose"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_eq (h : α ≃ₜ β) : topological_space.coinduced h ‹_› = ‹_› | h.quotient_map.2.symm | lemma | homeomorph.coinduced_eq | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"topological_space.coinduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding (h : α ≃ₜ β) : embedding h | ⟨h.inducing, h.injective⟩ | lemma | homeomorph.embedding | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_embedding (f : α → β) (hf : embedding f) : α ≃ₜ (set.range f) | { continuous_to_fun := hf.continuous.subtype_mk _,
continuous_inv_fun := by simp [hf.continuous_iff, continuous_subtype_coe],
to_equiv := equiv.of_injective f hf.inj } | def | homeomorph.of_embedding | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_subtype_coe",
"embedding",
"equiv.of_injective",
"set.range"
] | Homeomorphism given an embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_countable_topology [topological_space.second_countable_topology β]
(h : α ≃ₜ β) :
topological_space.second_countable_topology α | h.inducing.second_countable_topology | lemma | homeomorph.second_countable_topology | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"topological_space.second_countable_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_image {s : set α} (h : α ≃ₜ β) : is_compact (h '' s) ↔ is_compact s | h.embedding.is_compact_iff_is_compact_image.symm | lemma | homeomorph.is_compact_image | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact_preimage {s : set β} (h : α ≃ₜ β) : is_compact (h ⁻¹' s) ↔ is_compact s | by rw ← image_symm; exact h.symm.is_compact_image | lemma | homeomorph.is_compact_preimage | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_cocompact (h : α ≃ₜ β) : comap h (cocompact β) = cocompact α | (comap_cocompact_le h.continuous).antisymm $
(has_basis_cocompact.le_basis_iff (has_basis_cocompact.comap h)).2 $ λ K hK,
⟨h ⁻¹' K, h.is_compact_preimage.2 hK, subset.rfl⟩ | lemma | homeomorph.comap_cocompact | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_cocompact (h : α ≃ₜ β) : map h (cocompact α) = cocompact β | by rw [← h.comap_cocompact, map_comap_of_surjective h.surjective] | lemma | homeomorph.map_cocompact | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_space [compact_space α] (h : α ≃ₜ β) : compact_space β | { is_compact_univ := by { rw [← image_univ_of_surjective h.surjective, h.is_compact_image],
apply compact_space.is_compact_univ } } | lemma | homeomorph.compact_space | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"compact_space",
"is_compact_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t0_space [t0_space α] (h : α ≃ₜ β) : t0_space β | h.symm.embedding.t0_space | lemma | homeomorph.t0_space | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t1_space [t1_space α] (h : α ≃ₜ β) : t1_space β | h.symm.embedding.t1_space | lemma | homeomorph.t1_space | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"t1_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t2_space [t2_space α] (h : α ≃ₜ β) : t2_space β | h.symm.embedding.t2_space | lemma | homeomorph.t2_space | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
t3_space [t3_space α] (h : α ≃ₜ β) : t3_space β | h.symm.embedding.t3_space | lemma | homeomorph.t3_space | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_embedding (h : α ≃ₜ β) : dense_embedding h | { dense := h.surjective.dense_range,
.. h.embedding } | lemma | homeomorph.dense_embedding | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"dense",
"dense_embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_preimage (h : α ≃ₜ β) {s : set β} : is_open (h ⁻¹' s) ↔ is_open s | h.quotient_map.is_open_preimage | lemma | homeomorph.is_open_preimage | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_image (h : α ≃ₜ β) {s : set α} : is_open (h '' s) ↔ is_open s | by rw [← preimage_symm, is_open_preimage] | lemma | homeomorph.is_open_image | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map (h : α ≃ₜ β) : is_open_map h | λ s, h.is_open_image.2 | lemma | homeomorph.is_open_map | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_preimage (h : α ≃ₜ β) {s : set β} : is_closed (h ⁻¹' s) ↔ is_closed s | by simp only [← is_open_compl_iff, ← preimage_compl, is_open_preimage] | lemma | homeomorph.is_closed_preimage | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_closed",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_image (h : α ≃ₜ β) {s : set α} : is_closed (h '' s) ↔ is_closed s | by rw [← preimage_symm, is_closed_preimage] | lemma | homeomorph.is_closed_image | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map (h : α ≃ₜ β) : is_closed_map h | λ s, h.is_closed_image.2 | lemma | homeomorph.is_closed_map | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding (h : α ≃ₜ β) : open_embedding h | open_embedding_of_embedding_open h.embedding h.is_open_map | lemma | homeomorph.open_embedding | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"open_embedding",
"open_embedding_of_embedding_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding (h : α ≃ₜ β) : closed_embedding h | closed_embedding_of_embedding_closed h.embedding h.is_closed_map | lemma | homeomorph.closed_embedding | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"closed_embedding",
"closed_embedding_of_embedding_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normal_space [normal_space α] (h : α ≃ₜ β) : normal_space β | h.symm.closed_embedding.normal_space | lemma | homeomorph.normal_space | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"normal_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_closure (h : α ≃ₜ β) (s : set β) : h ⁻¹' (closure s) = closure (h ⁻¹' s) | h.is_open_map.preimage_closure_eq_closure_preimage h.continuous _ | lemma | homeomorph.preimage_closure | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_closure (h : α ≃ₜ β) (s : set α) : h '' (closure s) = closure (h '' s) | by rw [← preimage_symm, preimage_closure] | lemma | homeomorph.image_closure | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_interior (h : α ≃ₜ β) (s : set β) : h⁻¹' (interior s) = interior (h ⁻¹' s) | h.is_open_map.preimage_interior_eq_interior_preimage h.continuous _ | lemma | homeomorph.preimage_interior | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_interior (h : α ≃ₜ β) (s : set α) : h '' (interior s) = interior (h '' s) | by rw [← preimage_symm, preimage_interior] | lemma | homeomorph.image_interior | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_frontier (h : α ≃ₜ β) (s : set β) : h ⁻¹' (frontier s) = frontier (h ⁻¹' s) | h.is_open_map.preimage_frontier_eq_frontier_preimage h.continuous _ | lemma | homeomorph.preimage_frontier | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_frontier (h : α ≃ₜ β) (s : set α) : h '' frontier s = frontier (h '' s) | by rw [←preimage_symm, preimage_frontier] | lemma | homeomorph.image_frontier | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"frontier"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.has_compact_mul_support.comp_homeomorph {M} [has_one M] {f : β → M}
(hf : has_compact_mul_support f) (φ : α ≃ₜ β) : has_compact_mul_support (f ∘ φ) | hf.comp_closed_embedding φ.closed_embedding | lemma | has_compact_mul_support.comp_homeomorph | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"has_compact_mul_support"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_eq (h : α ≃ₜ β) (x : α) : map h (𝓝 x) = 𝓝 (h x) | h.embedding.map_nhds_of_mem _ (by simp) | lemma | homeomorph.map_nhds_eq | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_map_nhds_eq (h : α ≃ₜ β) (x : α) : map h.symm (𝓝 (h x)) = 𝓝 x | by rw [h.symm.map_nhds_eq, h.symm_apply_apply] | lemma | homeomorph.symm_map_nhds_eq | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_eq_comap (h : α ≃ₜ β) (x : α) : 𝓝 x = comap h (𝓝 (h x)) | h.embedding.to_inducing.nhds_eq_comap x | lemma | homeomorph.nhds_eq_comap | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_nhds_eq (h : α ≃ₜ β) (y : β) : comap h (𝓝 y) = 𝓝 (h.symm y) | by rw [h.nhds_eq_comap, h.apply_symm_apply] | lemma | homeomorph.comap_nhds_eq | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph_of_continuous_open (e : α ≃ β) (h₁ : continuous e) (h₂ : is_open_map e) :
α ≃ₜ β | { continuous_to_fun := h₁,
continuous_inv_fun := begin
rw continuous_def,
intros s hs,
convert ← h₂ s hs using 1,
apply e.image_eq_preimage
end,
to_equiv := e } | def | homeomorph.homeomorph_of_continuous_open | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous",
"continuous_def",
"is_open_map"
] | If an bijective map `e : α ≃ β` is continuous and open, then it is a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_continuous_on_iff (h : α ≃ₜ β) (f : γ → α) (s : set γ) :
continuous_on (h ∘ f) s ↔ continuous_on f s | h.inducing.continuous_on_iff.symm | lemma | homeomorph.comp_continuous_on_iff | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff (h : α ≃ₜ β) {f : γ → α} :
continuous (h ∘ f) ↔ continuous f | h.inducing.continuous_iff.symm | lemma | homeomorph.comp_continuous_iff | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_iff' (h : α ≃ₜ β) {f : β → γ} :
continuous (f ∘ h) ↔ continuous f | h.quotient_map.continuous_iff.symm | lemma | homeomorph.comp_continuous_iff' | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_at_iff (h : α ≃ₜ β) (f : γ → α) (x : γ) :
continuous_at (h ∘ f) x ↔ continuous_at f x | h.inducing.continuous_at_iff.symm | lemma | homeomorph.comp_continuous_at_iff | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_at_iff' (h : α ≃ₜ β) (f : β → γ) (x : α) :
continuous_at (f ∘ h) x ↔ continuous_at f (h x) | h.inducing.continuous_at_iff' (by simp) | lemma | homeomorph.comp_continuous_at_iff' | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_continuous_within_at_iff (h : α ≃ₜ β) (f : γ → α) (s : set γ) (x : γ) :
continuous_within_at f s x ↔ continuous_within_at (h ∘ f) s x | h.inducing.continuous_within_at_iff | lemma | homeomorph.comp_continuous_within_at_iff | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_is_open_map_iff (h : α ≃ₜ β) {f : γ → α} :
is_open_map (h ∘ f) ↔ is_open_map f | begin
refine ⟨_, λ hf, h.is_open_map.comp hf⟩,
intros hf,
rw [← function.comp.left_id f, ← h.symm_comp_self, function.comp.assoc],
exact h.symm.is_open_map.comp hf,
end | lemma | homeomorph.comp_is_open_map_iff | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_is_open_map_iff' (h : α ≃ₜ β) {f : β → γ} :
is_open_map (f ∘ h) ↔ is_open_map f | begin
refine ⟨_, λ hf, hf.comp h.is_open_map⟩,
intros hf,
rw [← function.comp.right_id f, ← h.self_comp_symm, ← function.comp.assoc],
exact hf.comp h.symm.is_open_map,
end | lemma | homeomorph.comp_is_open_map_iff' | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_congr {s t : set α} (h : s = t) : s ≃ₜ t | { continuous_to_fun := continuous_inclusion h.subset,
continuous_inv_fun := continuous_inclusion h.symm.subset,
to_equiv := equiv.set_congr h } | def | homeomorph.set_congr | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_inclusion",
"equiv.set_congr"
] | If two sets are equal, then they are homeomorphic. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_congr (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) : α ⊕ γ ≃ₜ β ⊕ δ | { continuous_to_fun := h₁.continuous.sum_map h₂.continuous,
continuous_inv_fun := h₁.symm.continuous.sum_map h₂.symm.continuous,
to_equiv := h₁.to_equiv.sum_congr h₂.to_equiv } | def | homeomorph.sum_congr | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | Sum of two homeomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_congr (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) : α × γ ≃ₜ β × δ | { continuous_to_fun := (h₁.continuous.comp continuous_fst).prod_mk
(h₂.continuous.comp continuous_snd),
continuous_inv_fun := (h₁.symm.continuous.comp continuous_fst).prod_mk
(h₂.symm.continuous.comp continuous_snd),
to_equiv := h₁.to_equiv.prod_congr h₂.to_equiv } | def | homeomorph.prod_congr | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_fst",
"continuous_snd"
] | Product of two homeomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_congr_symm (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) :
(h₁.prod_congr h₂).symm = h₁.symm.prod_congr h₂.symm | rfl | lemma | homeomorph.prod_congr_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod_congr (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) :
⇑(h₁.prod_congr h₂) = prod.map h₁ h₂ | rfl | lemma | homeomorph.coe_prod_congr | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comm : α × β ≃ₜ β × α | { continuous_to_fun := continuous_snd.prod_mk continuous_fst,
continuous_inv_fun := continuous_snd.prod_mk continuous_fst,
to_equiv := equiv.prod_comm α β } | def | homeomorph.prod_comm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_fst",
"equiv.prod_comm"
] | `α × β` is homeomorphic to `β × α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comm_symm : (prod_comm α β).symm = prod_comm β α | rfl | lemma | homeomorph.prod_comm_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod_comm : ⇑(prod_comm α β) = prod.swap | rfl | lemma | homeomorph.coe_prod_comm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_assoc : (α × β) × γ ≃ₜ α × (β × γ) | { continuous_to_fun := (continuous_fst.comp continuous_fst).prod_mk
((continuous_snd.comp continuous_fst).prod_mk continuous_snd),
continuous_inv_fun := (continuous_fst.prod_mk (continuous_fst.comp continuous_snd)).prod_mk
(continuous_snd.comp continuous_snd),
to_equiv := equiv.prod_assoc α β γ } | def | homeomorph.prod_assoc | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_fst",
"continuous_snd",
"equiv.prod_assoc"
] | `(α × β) × γ` is homeomorphic to `α × (β × γ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_punit : α × punit ≃ₜ α | { to_equiv := equiv.prod_punit α,
continuous_to_fun := continuous_fst,
continuous_inv_fun := continuous_id.prod_mk continuous_const } | def | homeomorph.prod_punit | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_const",
"continuous_fst",
"equiv.prod_punit"
] | `α × {*}` is homeomorphic to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_prod : punit × α ≃ₜ α | (prod_comm _ _).trans (prod_punit _) | def | homeomorph.punit_prod | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | `{*} × α` is homeomorphic to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_punit_prod : ⇑(punit_prod α) = prod.snd | rfl | lemma | homeomorph.coe_punit_prod | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.homeomorph.homeomorph_of_unique [unique α] [unique β] : α ≃ₜ β | { continuous_to_fun := @continuous_const α β _ _ default,
continuous_inv_fun := @continuous_const β α _ _ default,
.. equiv.equiv_of_unique α β } | def | homeomorph.homeomorph_of_unique | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_const",
"equiv.equiv_of_unique",
"unique"
] | If both `α` and `β` have a unique element, then `α ≃ₜ β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi_congr_right {ι : Type*} {β₁ β₂ : ι → Type*}
[Π i, topological_space (β₁ i)] [Π i, topological_space (β₂ i)] (F : Π i, β₁ i ≃ₜ β₂ i) :
(Π i, β₁ i) ≃ₜ (Π i, β₂ i) | { continuous_to_fun := continuous_pi (λ i, (F i).continuous.comp $ continuous_apply i),
continuous_inv_fun := continuous_pi (λ i, (F i).symm.continuous.comp $ continuous_apply i),
to_equiv := equiv.Pi_congr_right (λ i, (F i).to_equiv) } | def | homeomorph.Pi_congr_right | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous.comp",
"continuous_apply",
"continuous_pi",
"equiv.Pi_congr_right",
"topological_space"
] | If each `β₁ i` is homeomorphic to `β₂ i`, then `Π i, β₁ i` is homeomorphic to `Π i, β₂ i`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi_congr_right_symm {ι : Type*} {β₁ β₂ : ι → Type*} [Π i, topological_space (β₁ i)]
[Π i, topological_space (β₂ i)] (F : Π i, β₁ i ≃ₜ β₂ i) :
(Pi_congr_right F).symm = Pi_congr_right (λ i, (F i).symm) | rfl | lemma | homeomorph.Pi_congr_right_symm | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
{u v} ulift {α : Type u} [topological_space α] : ulift.{v u} α ≃ₜ α | { continuous_to_fun := continuous_ulift_down,
continuous_inv_fun := continuous_ulift_up,
to_equiv := equiv.ulift } | def | homeomorph.ulift | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_ulift_down",
"continuous_ulift_up",
"equiv.ulift",
"topological_space"
] | `ulift α` is homeomorphic to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sum_prod_distrib : (α ⊕ β) × γ ≃ₜ α × γ ⊕ β × γ | homeomorph.symm $ homeomorph_of_continuous_open (equiv.sum_prod_distrib α β γ).symm
((continuous_inl.prod_map continuous_id).sum_elim (continuous_inr.prod_map continuous_id)) $
(is_open_map_inl.prod is_open_map.id).sum_elim (is_open_map_inr.prod is_open_map.id) | def | homeomorph.sum_prod_distrib | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_id",
"equiv.sum_prod_distrib",
"homeomorph.symm",
"is_open_map.id"
] | `(α ⊕ β) × γ` is homeomorphic to `α × γ ⊕ β × γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_sum_distrib : α × (β ⊕ γ) ≃ₜ α × β ⊕ α × γ | (prod_comm _ _).trans $
sum_prod_distrib.trans $
sum_congr (prod_comm _ _) (prod_comm _ _) | def | homeomorph.prod_sum_distrib | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | `α × (β ⊕ γ)` is homeomorphic to `α × β ⊕ α × γ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_prod_distrib : ((Σ i, σ i) × β) ≃ₜ (Σ i, (σ i × β)) | homeomorph.symm $ homeomorph_of_continuous_open (equiv.sigma_prod_distrib σ β).symm
(continuous_sigma $ λ i, continuous_sigma_mk.fst'.prod_mk continuous_snd)
(is_open_map_sigma.2 $ λ i, is_open_map_sigma_mk.prod is_open_map.id) | def | homeomorph.sigma_prod_distrib | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_sigma",
"continuous_snd",
"equiv.sigma_prod_distrib",
"homeomorph.symm",
"is_open_map.id"
] | `(Σ i, σ i) × β` is homeomorphic to `Σ i, (σ i × β)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_unique (ι α : Type*) [unique ι] [topological_space α] : (ι → α) ≃ₜ α | { to_equiv := equiv.fun_unique ι α,
continuous_to_fun := continuous_apply _,
continuous_inv_fun := continuous_pi (λ _, continuous_id) } | def | homeomorph.fun_unique | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_apply",
"continuous_id",
"continuous_pi",
"equiv.fun_unique",
"topological_space",
"unique"
] | If `ι` has a unique element, then `ι → α` is homeomorphic to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
{u} pi_fin_two (α : fin 2 → Type u) [Π i, topological_space (α i)] : (Π i, α i) ≃ₜ α 0 × α 1 | { to_equiv := pi_fin_two_equiv α,
continuous_to_fun := (continuous_apply 0).prod_mk (continuous_apply 1),
continuous_inv_fun := continuous_pi $ fin.forall_fin_two.2 ⟨continuous_fst, continuous_snd⟩ } | def | homeomorph.pi_fin_two | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_apply",
"continuous_pi",
"pi_fin_two_equiv",
"topological_space"
] | Homeomorphism between dependent functions `Π i : fin 2, α i` and `α 0 × α 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_two_arrow : (fin 2 → α) ≃ₜ α × α | { to_equiv := fin_two_arrow_equiv α, .. pi_fin_two (λ _, α) } | def | homeomorph.fin_two_arrow | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"fin_two_arrow_equiv"
] | Homeomorphism between `α² = fin 2 → α` and `α × α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image (e : α ≃ₜ β) (s : set α) : s ≃ₜ e '' s | { continuous_to_fun := by continuity!,
continuous_inv_fun := by continuity!,
to_equiv := e.to_equiv.image s, } | def | homeomorph.image | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [] | A subset of a topological space is homeomorphic to its image under a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.univ (α : Type*) [topological_space α] : (univ : set α) ≃ₜ α | { to_equiv := equiv.set.univ α,
continuous_to_fun := continuous_subtype_coe,
continuous_inv_fun := continuous_id.subtype_mk _ } | def | homeomorph.set.univ | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous_subtype_coe",
"equiv.set.univ",
"topological_space"
] | `set.univ α` is homeomorphic to `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.prod (s : set α) (t : set β) : ↥(s ×ˢ t) ≃ₜ s × t | { to_equiv := equiv.set.prod s t,
continuous_to_fun := (continuous_subtype_coe.fst.subtype_mk _).prod_mk
(continuous_subtype_coe.snd.subtype_mk _),
continuous_inv_fun := (continuous_subtype_coe.fst'.prod_mk
continuous_subtype_coe.snd').subtype_mk _ } | def | homeomorph.set.prod | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"equiv.set.prod",
"set.prod"
] | `s ×ˢ t` is homeomorphic to `s × t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_equiv_pi_subtype_prod (p : ι → Prop) (β : ι → Type*) [Π i, topological_space (β i)]
[decidable_pred p] : (Π i, β i) ≃ₜ (Π i : {x // p x}, β i) × Π i : {x // ¬p x}, β i | { to_equiv := equiv.pi_equiv_pi_subtype_prod p β,
continuous_to_fun := by apply continuous.prod_mk; exact continuous_pi (λ j, continuous_apply j),
continuous_inv_fun := continuous_pi $ λ j, begin
dsimp only [equiv.pi_equiv_pi_subtype_prod], split_ifs,
exacts [(continuous_apply _).comp continuous_fst, (conti... | def | homeomorph.pi_equiv_pi_subtype_prod | topology | src/topology/homeomorph.lean | [
"logic.equiv.fin",
"topology.dense_embedding",
"topology.support"
] | [
"continuous.prod_mk",
"continuous_apply",
"continuous_fst",
"continuous_pi",
"continuous_snd",
"equiv.pi_equiv_pi_subtype_prod",
"topological_space"
] | The topological space `Π i, β i` can be split as a product by separating the indices in ι
depending on whether they satisfy a predicate p or not. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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