Further properties of homeomorphisms #
This file proves further properties of homeomorphisms between topological spaces. Pretty much every topological property is preserved under homeomorphisms.
Homeomorphism given an embedding.
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Alias of Topology.IsEmbedding.toHomeomorph
.
Homeomorphism given an embedding.
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Alias of Topology.IsEmbedding.toHomeomorph
.
Homeomorphism given an embedding.
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A surjective embedding is a homeomorphism.
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Alias of Topology.IsEmbedding.toHomeomorphOfSurjective
.
A surjective embedding is a homeomorphism.
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Alias of Topology.IsEmbedding.toHomeomorphOfSurjective
.
A surjective embedding is a homeomorphism.
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If h : X → Y
is a homeomorphism, h(s)
is compact iff s
is.
If h : X → Y
is a homeomorphism, h⁻¹(s)
is compact iff s
is.
If h : X → Y
is a homeomorphism, s
is σ-compact iff h(s)
is.
If h : X → Y
is a homeomorphism, h⁻¹(s)
is σ-compact iff s
is.
If the codomain of a homeomorphism is a locally connected space, then the domain is also a locally connected space.
The codomain of a homeomorphism is a locally compact space if and only if the domain is a locally compact space.
A homeomorphism h : X ≃ₜ Y
lifts to a homeomorphism between subtypes corresponding to
predicates p : X → Prop
and q : Y → Prop
so long as p = q ∘ h
.
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A homeomorphism h : X ≃ₜ Y
lifts to a homeomorphism between sets s : Set X
and t : Set Y
whenever h
maps s
onto t
.
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If two sets are equal, then they are homeomorphic.
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The product over S ⊕ T
of a family of topological spaces
is homeomorphic to the product of (the product over S
) and (the product over T
).
This is Equiv.sumPiEquivProdPi
as a Homeomorph
.
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The product Π t : α, f t
of a family of topological spaces is homeomorphic to the
space f ⬝
when α
only contains ⬝
.
This is Equiv.piUnique
as a Homeomorph
.
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Equiv.piCongrLeft
as a homeomorphism: this is the natural homeomorphism
Π i, Y (e i) ≃ₜ Π j, Y j
obtained from a bijection ι ≃ ι'
.
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Equiv.piCongrRight
as a homeomorphism: this is the natural homeomorphism
Π i, Y₁ i ≃ₜ Π j, Y₂ i
obtained from homeomorphisms Y₁ i ≃ₜ Y₂ i
for each i
.
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Equiv.piCongr
as a homeomorphism: this is the natural homeomorphism
Π i₁, Y₁ i ≃ₜ Π i₂, Y₂ i₂
obtained from a bijection ι₁ ≃ ι₂
and homeomorphisms
Y₁ i₁ ≃ₜ Y₂ (e i₁)
for each i₁ : ι₁
.
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The natural homeomorphism (ι ⊕ ι' → X) ≃ₜ (ι → X) × (ι' → X)
.
Equiv.sumArrowEquivProdArrow
as a homeomorphism.
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The natural homeomorphism between (Fin m → X) × (Fin n → X)
and Fin (m + n) → X
.
Fin.appendEquiv
as a homeomorphism
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(Σ i, X i) × Y
is homeomorphic to Σ i, (X i × Y)
.
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If ι
has a unique element, then ι → X
is homeomorphic to X
.
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Homeomorphism between dependent functions Π i : Fin 2, X i
and X 0 × X 1
.
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A subset of a topological space is homeomorphic to its image under a homeomorphism.
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s ×ˢ t
is homeomorphic to s × t
.
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The topological space Π i, Y i
can be split as a product by separating the indices in ι
depending on whether they satisfy a predicate p or not.
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A product of topological spaces can be split as the binary product of one of the spaces and the product of all the remaining spaces.
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A product of copies of a topological space can be split as the binary product of one copy and the product of all the remaining copies.
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Continuous equivalences from a compact space to a T2 space are homeomorphisms.
This is not true when T2 is weakened to T1
(see Continuous.homeoOfEquivCompactToT2.t1_counterexample
).
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Bundled homeomorphism constructed from a map that is a homeomorphism.
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Alias of IsHomeomorph.isInducing
.
Alias of IsHomeomorph.isEmbedding
.
Alias of IsHomeomorph.isQuotientMap
.
A map is a homeomorphism iff it is the map underlying a bundled homeomorphism h : X ≃ₜ Y
.
A map is a homeomorphism iff it is continuous and has a continuous inverse.
A map is a homeomorphism iff it is a surjective embedding.
Alias of isHomeomorph_iff_isEmbedding_surjective
.
A map is a homeomorphism iff it is a surjective embedding.
A map is a homeomorphism iff it is continuous, closed and bijective.
A map from a compact space to a T2 space is a homeomorphism iff it is continuous and bijective.