Metric space gluing #
Gluing two metric spaces along a common subset. Formally, we are given
Φ
Z > X

Ψ
v
Y
where hΦ : Isometry Φ
and hΨ : Isometry Ψ
.
We want to complete the square by a space GlueSpacescan hΦ hΨ
and two isometries
toGlueL hΦ hΨ
and toGlueR hΦ hΨ
that make the square commute.
We start by defining a predistance on the disjoint union X ⊕ Y
, for which
points Φ p
and Ψ p
are at distance 0. The (quotient) metric space associated
to this predistance is the desired space.
This is an instance of a more general construction, where Φ
and Ψ
do not have to be isometries,
but the distances in the image almost coincide, up to 2ε
say. Then one can almost glue the two
spaces so that the images of a point under Φ
and Ψ
are ε
close. If ε > 0
, this yields a
metric space structure on X ⊕ Y
, without the need to take a quotient. In particular,
this gives a natural metric space structure on X ⊕ Y
, where the basepoints
are at distance 1, say, and the distances between other points are obtained by going through the two
basepoints.
(We also register the same metric space structure on a general disjoint union Σ i, E i
).
We also define the inductive limit of metric spaces. Given
f 0 f 1 f 2 f 3
X 0 > X 1 > X 2 > X 3 > ...
where the X n
are metric spaces and f n
isometric embeddings, we define the inductive
limit of the X n
, also known as the increasing union of the X n
in this context, if we
identify X n
and X (n+1)
through f n
. This is a metric space in which all X n
embed
isometrically and in a way compatible with f n
.
Define a predistance on X ⊕ Y
, for which Φ p
and Ψ p
are at distance ε
Equations
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Instances For
Given two maps Φ
and Ψ
intro metric spaces X
and Y
such that the distances between
Φ p
and Φ q
, and between Ψ p
and Ψ q
, coincide up to 2 ε
where ε > 0
, one can almost
glue the two spaces X
and Y
along the images of Φ
and Ψ
, so that Φ p
and Ψ p
are
at distance ε
.
Equations
 Metric.glueMetricApprox Φ Ψ ε ε0 H = MetricSpace.mk ⋯
Instances For
Metric on X ⊕ Y
#
A particular case of the previous construction is when one uses basepoints in X
and Y
and one
glues only along the basepoints, putting them at distance 1. We give a direct definition of
the distance, without iInf
, as it is easier to use in applications, and show that it is equal to
the gluing distance defined above to take advantage of the lemmas we have already proved.
Distance on a disjoint union. There are many (noncanonical) ways to put a distance compatible
with each factor.
If the two spaces are bounded, one can say for instance that each point in the first is at distance
diam X + diam Y + 1
of each point in the second.
Instead, we choose a construction that works for unbounded spaces, but requires basepoints,
chosen arbitrarily.
We embed isometrically each factor, set the basepoints at distance 1,
arbitrarily, and say that the distance from a
to b
is the sum of the distances of a
and b
to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an arbitrary choice in this construction, it is not an instance by default.
Equations
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Instances For
The distance on the disjoint union indeed defines a metric space. All the distance properties follow from our choice of the distance. The harder work is to show that the uniform structure defined by the distance coincides with the disjoint union uniform structure.
Equations
 Metric.metricSpaceSum = MetricSpace.mk ⋯
Instances For
The left injection of a space in a disjoint union is an isometry
The right injection of a space in a disjoint union is an isometry
Distance on a disjoint union. There are many (noncanonical) ways to put a distance compatible
with each factor.
We choose a construction that works for unbounded spaces, but requires basepoints,
chosen arbitrarily.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the distance from a
to b
is the sum of the distances of a
and b
to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an arbitrary choice in this construction, it is not an instance by default.
Equations
Instances For
A Dist
instance on the disjoint union Σ i, E i
.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the distance from a
to b
is the sum of the distances of a
and b
to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an arbitrary choice in this construction, it is not an instance by default.
Equations
 Metric.Sigma.instDist = { dist := Metric.Sigma.dist }
Instances For
A metric space structure on the disjoint union Σ i, E i
.
We embed isometrically each factor, set the basepoints at distance 1, arbitrarily,
and say that the distance from a
to b
is the sum of the distances of a
and b
to
their respective basepoints, plus the distance 1 between the basepoints.
Since there is an arbitrary choice in this construction, it is not an instance by default.
Equations
 Metric.Sigma.metricSpace = MetricSpace.ofDistTopology Metric.Sigma.dist ⋯ ⋯ ⋯ ⋯ ⋯
Instances For
The injection of a space in a disjoint union is an isometry
A disjoint union of complete metric spaces is complete.
Given two isometric embeddings Φ : Z → X
and Ψ : Z → Y
, we define a pseudo metric space
structure on X ⊕ Y
by declaring that Φ x
and Ψ x
are at distance 0
.
Equations
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Instances For
Given two isometric embeddings Φ : Z → X
and Ψ : Z → Y
, we define a
space GlueSpace hΦ hΨ
by identifying in X ⊕ Y
the points Φ x
and Ψ x
.
Equations
 Metric.GlueSpace hΦ hΨ = SeparationQuotient (X ⊕ Y)
Instances For
Equations
 Metric.instMetricSpaceGlueSpace hΦ hΨ = inferInstanceAs (MetricSpace (SeparationQuotient (X ⊕ Y)))
The canonical map from X
to the space obtained by gluing isometric subsets in X
and Y
.
Equations
 Metric.toGlueL hΦ hΨ x = Quotient.mk'' (Sum.inl x)
Instances For
The canonical map from Y
to the space obtained by gluing isometric subsets in X
and Y
.
Equations
 Metric.toGlueR hΦ hΨ y = Quotient.mk'' (Sum.inr y)
Instances For
Equations
 Metric.inhabitedLeft hΦ hΨ = { default := Metric.toGlueL hΦ hΨ default }
Equations
 Metric.inhabitedRight hΦ hΨ = { default := Metric.toGlueR hΦ hΨ default }
Inductive limit of metric spaces #
In this section, we define the inductive limit of
f 0 f 1 f 2 f 3
X 0 > X 1 > X 2 > X 3 > ...
where the X n
are metric spaces and f n isometric embeddings. We do it by defining a premetric
space structure on Σ n, X n
, where the predistance dist x y
is obtained by pushing x
and y
in a common X k
using composition by the f n
, and taking the distance there. This does not
depend on the choice of k
as the f n
are isometries. The metric space associated to this
premetric space is the desired inductive limit.
Predistance on the disjoint union Σ n, X n
.
Equations
 Metric.inductiveLimitDist f x y = dist (Nat.leRecOn ⋯ (fun {k : ℕ} => f k) x.snd) (Nat.leRecOn ⋯ (fun {k : ℕ} => f k) y.snd)
Instances For
The predistance on the disjoint union Σ n, X n
can be computed in any X k
for large
enough k
.
Premetric space structure on Σ n, X n
.
Equations
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Instances For
The type giving the inductive limit in a metric space context.
Equations
 Metric.InductiveLimit I = SeparationQuotient ((n : ℕ) × X n)
Instances For
Equations
 Metric.instMetricSpaceInductiveLimit = inferInstanceAs (MetricSpace (SeparationQuotient ((n : ℕ) × X n)))
Mapping each X n
to the inductive limit.
Equations
 Metric.toInductiveLimit I n x = Quotient.mk'' ⟨n, x⟩
Instances For
Equations
 Metric.instInhabitedInductiveLimitOfOfNatNat I = { default := Metric.toInductiveLimit I 0 default }
The map toInductiveLimit n
mapping X n
to the inductive limit is an isometry.
The maps toInductiveLimit n
are compatible with the maps f n
.