mathlib documentation

topology.homeomorph

Homeomorphisms #

This file defines homeomorphisms between two topological spaces. They are bijections with both directions continuous. We denote homeomorphisms with the notation ≃ₜ.

Main definitions #

Main results #

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@[nolint]
structure homeomorph (α : Type u_5) (β : Type u_6) [topological_space α] [topological_space β] :
Type (max u_5 u_6)

Homeomorphism between α and β, also called topological isomorphism

Instances for homeomorph
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@[protected, instance]
def homeomorph.has_coe_to_fun {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] :
has_coe_to_fun (α ≃ₜ β) (λ (_x : α ≃ₜ β), α → β)
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@[simp]
theorem homeomorph.homeomorph_mk_coe {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (a : α ≃ β) (b : continuous a.to_fun . "continuity'") (c : continuous a.inv_fun . "continuity'") :
⇑{to_equiv := a, continuous_to_fun := b, continuous_inv_fun := c} = ⇑a
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@[protected]
def homeomorph.symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
β ≃ₜ α

Inverse of a homeomorphism.

Equations
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def homeomorph.simps.apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
α → β

See Note [custom simps projection]. We need to specify this projection explicitly in this case, because it is a composition of multiple projections.

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def homeomorph.simps.symm_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
β → α

See Note [custom simps projection]

Equations
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@[simp]
theorem homeomorph.coe_to_equiv {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
⇑(h.to_equiv) = ⇑h
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@[simp]
theorem homeomorph.coe_symm_to_equiv {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
⇑(h.to_equiv.symm) = ⇑(h.symm)
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theorem homeomorph.to_equiv_injective {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] :
function.injective homeomorph.to_equiv
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@[ext]
theorem homeomorph.ext {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {h h' : α ≃ₜ β} (H : ∀ (x : α), ⇑h x = ⇑h' x) :
h = h'
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@[protected]
def homeomorph.refl (α : Type u_1) [topological_space α] :
α ≃ₜ α

Identity map as a homeomorphism.

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@[simp]
theorem homeomorph.refl_apply (α : Type u_1) [topological_space α] :
⇑(homeomorph.refl α) = id
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@[protected]
def homeomorph.trans {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h₁ : α ≃ₜ β) (h₂ : β ≃ₜ γ) :
α ≃ₜ γ

Composition of two homeomorphisms.

Equations
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@[simp]
theorem homeomorph.trans_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h₁ : α ≃ₜ β) (h₂ : β ≃ₜ γ) (a : α) :
⇑(h₁.trans h₂) a = ⇑h₂ (⇑h₁ a)
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@[simp]
theorem homeomorph.homeomorph_mk_coe_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (a : α ≃ β) (b : continuous a.to_fun . "continuity'") (c : continuous a.inv_fun . "continuity'") :
⇑({to_equiv := a, continuous_to_fun := b, continuous_inv_fun := c}.symm) = ⇑(a.symm)
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@[simp]
theorem homeomorph.refl_symm {α : Type u_1} [topological_space α] :
(homeomorph.refl α).symm = homeomorph.refl α
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@[protected, continuity]
theorem homeomorph.continuous {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
continuous ⇑h
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@[protected, continuity]
theorem homeomorph.continuous_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
continuous ⇑(h.symm)
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@[simp]
theorem homeomorph.apply_symm_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : β) :
⇑h (⇑(h.symm) x) = x
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@[simp]
theorem homeomorph.symm_apply_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : α) :
⇑(h.symm) (⇑h x) = x
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@[protected]
theorem homeomorph.bijective {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
function.bijective ⇑h
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@[protected]
theorem homeomorph.injective {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
function.injective ⇑h
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@[protected]
theorem homeomorph.surjective {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
function.surjective ⇑h
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def homeomorph.change_inv {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α ≃ₜ β) (g : β → α) (hg : function.right_inverse g ⇑f) :
α ≃ₜ β

Change the homeomorphism f to make the inverse function definitionally equal to g.

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@[simp]
theorem homeomorph.symm_comp_self {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
⇑(h.symm) ∘ ⇑h = id
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@[simp]
theorem homeomorph.self_comp_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
⇑h ∘ ⇑(h.symm) = id
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@[simp]
theorem homeomorph.range_coe {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
set.range ⇑h = set.univ
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theorem homeomorph.image_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
set.image ⇑(h.symm) = set.preimage ⇑h
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theorem homeomorph.preimage_symm {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
set.preimage ⇑(h.symm) = set.image ⇑h
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@[simp]
theorem homeomorph.image_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set β) :
⇑h '' (⇑h ⁻¹' s) = s
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@[simp]
theorem homeomorph.preimage_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set α) :
⇑h ⁻¹' (⇑h '' s) = s
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@[protected]
theorem homeomorph.inducing {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
inducing ⇑h
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theorem homeomorph.induced_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
topological_space.induced ⇑h _inst_2 = _inst_1
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@[protected]
theorem homeomorph.quotient_map {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
quotient_map ⇑h
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theorem homeomorph.coinduced_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
topological_space.coinduced ⇑h _inst_1 = _inst_2
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@[protected]
theorem homeomorph.embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
embedding ⇑h
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noncomputable def homeomorph.of_embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α → β) (hf : embedding f) :
α ≃ₜ ↥(set.range f)

Homeomorphism given an embedding.

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@[protected]
theorem homeomorph.second_countable_topology {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [topological_space.second_countable_topology β] (h : α ≃ₜ β) :
topological_space.second_countable_topology α
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theorem homeomorph.compact_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {s : set α} (h : α ≃ₜ β) :
is_compact (⇑h '' s) ↔ is_compact s
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theorem homeomorph.compact_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {s : set β} (h : α ≃ₜ β) :
is_compact (⇑h ⁻¹' s) ↔ is_compact s
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@[simp]
theorem homeomorph.comap_cocompact {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
filter.comap ⇑h (filter.cocompact β) = filter.cocompact α
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@[simp]
theorem homeomorph.map_cocompact {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
filter.map ⇑h (filter.cocompact α) = filter.cocompact β
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@[protected]
theorem homeomorph.compact_space {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [compact_space α] (h : α ≃ₜ β) :
compact_space β
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@[protected]
theorem homeomorph.t0_space {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [t0_space α] (h : α ≃ₜ β) :
t0_space β
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@[protected]
theorem homeomorph.t1_space {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [t1_space α] (h : α ≃ₜ β) :
t1_space β
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@[protected]
theorem homeomorph.t2_space {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [t2_space α] (h : α ≃ₜ β) :
t2_space β
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@[protected]
theorem homeomorph.regular_space {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [regular_space α] (h : α ≃ₜ β) :
regular_space β
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@[protected]
theorem homeomorph.dense_embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
dense_embedding ⇑h
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@[simp]
theorem homeomorph.is_open_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) {s : set β} :
is_open (⇑h ⁻¹' s) ↔ is_open s
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@[simp]
theorem homeomorph.is_open_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) {s : set α} :
is_open (⇑h '' s) ↔ is_open s
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@[protected]
theorem homeomorph.is_open_map {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
is_open_map ⇑h
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@[simp]
theorem homeomorph.is_closed_preimage {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) {s : set β} :
is_closed (⇑h ⁻¹' s) ↔ is_closed s
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@[simp]
theorem homeomorph.is_closed_image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) {s : set α} :
is_closed (⇑h '' s) ↔ is_closed s
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@[protected]
theorem homeomorph.is_closed_map {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
is_closed_map ⇑h
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@[protected]
theorem homeomorph.open_embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
open_embedding ⇑h
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@[protected]
theorem homeomorph.closed_embedding {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) :
closed_embedding ⇑h
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@[protected]
theorem homeomorph.normal_space {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [normal_space α] (h : α ≃ₜ β) :
normal_space β
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theorem homeomorph.preimage_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set β) :
⇑h ⁻¹' closure s = closure (⇑h ⁻¹' s)
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theorem homeomorph.image_closure {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set α) :
⇑h '' closure s = closure (⇑h '' s)
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theorem homeomorph.preimage_interior {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set β) :
⇑h ⁻¹' interior s = interior (⇑h ⁻¹' s)
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theorem homeomorph.image_interior {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set α) :
⇑h '' interior s = interior (⇑h '' s)
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theorem homeomorph.preimage_frontier {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (s : set β) :
⇑h ⁻¹' frontier s = frontier (⇑h ⁻¹' s)
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theorem has_compact_support.comp_homeomorph {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {M : Type u_3} [has_zero M] {f : β → M} (hf : has_compact_support f) (φ : α ≃ₜ β) :
has_compact_support (f ∘ ⇑φ)
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theorem has_compact_mul_support.comp_homeomorph {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] {M : Type u_3} [has_one M] {f : β → M} (hf : has_compact_mul_support f) (φ : α ≃ₜ β) :
has_compact_mul_support (f ∘ ⇑φ)
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@[simp]
theorem homeomorph.map_nhds_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : α) :
filter.map ⇑h (nhds x) = nhds (⇑h x)
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theorem homeomorph.symm_map_nhds_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : α) :
filter.map ⇑(h.symm) (nhds (⇑h x)) = nhds x
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theorem homeomorph.nhds_eq_comap {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (x : α) :
nhds x = filter.comap ⇑h (nhds (⇑h x))
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@[simp]
theorem homeomorph.comap_nhds_eq {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (h : α ≃ₜ β) (y : β) :
filter.comap ⇑h (nhds y) = nhds (⇑(h.symm) y)
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def homeomorph.homeomorph_of_continuous_open {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (e : α ≃ β) (h₁ : continuous ⇑e) (h₂ : is_open_map ⇑e) :
α ≃ₜ β

If an bijective map e : α ≃ β is continuous and open, then it is a homeomorphism.

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@[simp]
theorem homeomorph.comp_continuous_on_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) (f : γ → α) (s : set γ) :
continuous_on (⇑h ∘ f) s ↔ continuous_on f s
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@[simp]
theorem homeomorph.comp_continuous_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) {f : γ → α} :
continuous (⇑h ∘ f) ↔ continuous f
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@[simp]
theorem homeomorph.comp_continuous_iff' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) {f : β → γ} :
continuous (f ∘ ⇑h) ↔ continuous f
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theorem homeomorph.comp_continuous_at_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) (f : γ → α) (x : γ) :
continuous_at (⇑h ∘ f) x ↔ continuous_at f x
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theorem homeomorph.comp_continuous_at_iff' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) (f : β → γ) (x : α) :
continuous_at (f ∘ ⇑h) x ↔ continuous_at f (⇑h x)
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theorem homeomorph.comp_continuous_within_at_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (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
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@[simp]
theorem homeomorph.comp_is_open_map_iff {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) {f : γ → α} :
is_open_map (⇑h ∘ f) ↔ is_open_map f
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@[simp]
theorem homeomorph.comp_is_open_map_iff' {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] (h : α ≃ₜ β) {f : β → γ} :
is_open_map (f ∘ ⇑h) ↔ is_open_map f
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def homeomorph.set_congr {α : Type u_1} [topological_space α] {s t : set α} (h : s = t) :
↥s ≃ₜ ↥t

If two sets are equal, then they are homeomorphic.

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def homeomorph.sum_congr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) :
α ⊕ γ ≃ₜ β ⊕ δ

Sum of two homeomorphisms.

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def homeomorph.prod_congr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) :
α × γ ≃ₜ β × δ

Product of two homeomorphisms.

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@[simp]
theorem homeomorph.prod_congr_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) :
(h₁.prod_congr h₂).symm = h₁.symm.prod_congr h₂.symm
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@[simp]
theorem homeomorph.coe_prod_congr {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} [topological_space α] [topological_space β] [topological_space γ] [topological_space δ] (h₁ : α ≃ₜ β) (h₂ : γ ≃ₜ δ) :
⇑(h₁.prod_congr h₂) = prod.map ⇑h₁ ⇑h₂
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def homeomorph.prod_comm (α : Type u_1) (β : Type u_2) [topological_space α] [topological_space β] :
α × β ≃ₜ β × α

α × β is homeomorphic to β × α.

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@[simp]
theorem homeomorph.prod_comm_symm (α : Type u_1) (β : Type u_2) [topological_space α] [topological_space β] :
(homeomorph.prod_comm α β).symm = homeomorph.prod_comm β α
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@[simp]
theorem homeomorph.coe_prod_comm (α : Type u_1) (β : Type u_2) [topological_space α] [topological_space β] :
⇑(homeomorph.prod_comm α β) = prod.swap
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def homeomorph.prod_assoc (α : Type u_1) (β : Type u_2) (γ : Type u_3) [topological_space α] [topological_space β] [topological_space γ] :
(α × β) × γ ≃ₜ α × β × γ

(α × β) × γ is homeomorphic to α × (β × γ).

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@[simp]
theorem homeomorph.prod_punit_apply (α : Type u_1) [topological_space α] :
⇑(homeomorph.prod_punit α) = prod.fst
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def homeomorph.prod_punit (α : Type u_1) [topological_space α] :
α × punit ≃ₜ α

α × {*} is homeomorphic to α.

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def homeomorph.punit_prod (α : Type u_1) [topological_space α] :
punit × α ≃ₜ α

{*} × α is homeomorphic to α.

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@[simp]
theorem homeomorph.coe_punit_prod (α : Type u_1) [topological_space α] :
⇑(homeomorph.punit_prod α) = prod.snd
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def homeomorph.ulift {α : Type u} [topological_space α] :
ulift α ≃ₜ α

ulift α is homeomorphic to α.

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def homeomorph.sum_prod_distrib {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] :
(α ⊕ β) × γ ≃ₜ α × γ ⊕ β × γ

(α ⊕ β) × γ is homeomorphic to α × γ ⊕ β × γ.

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def homeomorph.prod_sum_distrib {α : Type u_1} {β : Type u_2} {γ : Type u_3} [topological_space α] [topological_space β] [topological_space γ] :
α × (β ⊕ γ) ≃ₜ α × β ⊕ α × γ

α × (β ⊕ γ) is homeomorphic to α × β ⊕ α × γ.

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def homeomorph.sigma_prod_distrib {β : Type u_2} [topological_space β] {ι : Type u_5} {σ : ι → Type u_6} [Π (i : ι), topological_space (σ i)] :
(Σ (i : ι), σ i) × β ≃ₜ Σ (i : ι), σ i × β

(Σ i, σ i) × β is homeomorphic to Σ i, (σ i × β).

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def homeomorph.fun_unique (ι : Type u_1) (α : Type u_2) [unique ι] [topological_space α] :
(ι → α) ≃ₜ α

If ι has a unique element, then ι → α is homeomorphic to α.

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@[simp]
theorem homeomorph.fun_unique_apply (ι : Type u_1) (α : Type u_2) [unique ι] [topological_space α] :
⇑(homeomorph.fun_unique ι α) = function.eval inhabited.default
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@[simp]
theorem homeomorph.fun_unique_symm_apply (ι : Type u_1) (α : Type u_2) [unique ι] [topological_space α] :
⇑((homeomorph.fun_unique ι α).symm) = λ (x : α) (b : ι), x
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@[simp]
theorem homeomorph.pi_fin_two_symm_apply (α : fin 2 → Type u) [Π (i : fin 2), topological_space (α i)] :
⇑((homeomorph.pi_fin_two α).symm) = λ (p : α 0 × α 1), fin.cons p.fst (fin.cons p.snd fin_zero_elim)
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def homeomorph.pi_fin_two (α : fin 2 → Type u) [Π (i : fin 2), topological_space (α i)] :
(Π (i : fin 2), α i) ≃ₜ α 0 × α 1

Homeomorphism between dependent functions Π i : fin 2, α i and α 0 × α 1.

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@[simp]
theorem homeomorph.pi_fin_two_apply (α : fin 2 → Type u) [Π (i : fin 2), topological_space (α i)] :
⇑(homeomorph.pi_fin_two α) = λ (f : Π (i : fin 2), α i), (f 0, f 1)
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@[simp]
theorem homeomorph.fin_two_arrow_apply {α : Type u_1} [topological_space α] :
⇑homeomorph.fin_two_arrow = λ (f : fin 2 → α), (f 0, f 1)
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@[simp]
theorem homeomorph.fin_two_arrow_symm_apply {α : Type u_1} [topological_space α] :
⇑(homeomorph.fin_two_arrow.symm) = λ (x : α × α), ![x.fst, x.snd]
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def homeomorph.fin_two_arrow {α : Type u_1} [topological_space α] :
(fin 2 → α) ≃ₜ α × α

Homeomorphism between α² = fin 2 → α and α × α.

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def homeomorph.image {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (e : α ≃ₜ β) (s : set α) :
↥s ≃ₜ ↥(⇑e '' s)

A subset of a topological space is homeomorphic to its image under a homeomorphism.

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@[simp]
theorem equiv.to_homeomorph_of_inducing_symm_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α ≃ β) (hf : inducing ⇑f) (ᾰ : β) :
⇑((f.to_homeomorph_of_inducing hf).symm) ᾰ = f.inv_fun ᾰ
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@[simp]
theorem equiv.to_homeomorph_of_inducing_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α ≃ β) (hf : inducing ⇑f) (ᾰ : α) :
⇑(f.to_homeomorph_of_inducing hf) ᾰ = f.to_fun ᾰ
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def equiv.to_homeomorph_of_inducing {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] (f : α ≃ β) (hf : inducing ⇑f) :
α ≃ₜ β

An inducing equiv between topological spaces is a homeomorphism.

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theorem continuous.continuous_symm_of_equiv_compact_to_t2 {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [compact_space α] [t2_space β] {f : α ≃ β} (hf : continuous ⇑f) :
continuous ⇑(f.symm)
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def continuous.homeo_of_equiv_compact_to_t2 {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [compact_space α] [t2_space β] {f : α ≃ β} (hf : continuous ⇑f) :
α ≃ₜ β

Continuous equivalences from a compact space to a T2 space are homeomorphisms.

This is not true when T2 is weakened to T1 (see continuous.homeo_of_equiv_compact_to_t2.t1_counterexample).

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@[simp]
theorem continuous.homeo_of_equiv_compact_to_t2_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [compact_space α] [t2_space β] {f : α ≃ β} (hf : continuous ⇑f) (ᾰ : α) :
⇑(hf.homeo_of_equiv_compact_to_t2) ᾰ = f.to_fun ᾰ
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@[simp]
theorem continuous.homeo_of_equiv_compact_to_t2_symm_apply {α : Type u_1} {β : Type u_2} [topological_space α] [topological_space β] [compact_space α] [t2_space β] {f : α ≃ β} (hf : continuous ⇑f) (ᾰ : β) :
⇑(hf.homeo_of_equiv_compact_to_t2.symm) ᾰ = f.inv_fun ᾰ