Metric Spaces¶
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sage.categories.metric_spaces.
MetricSpaces
¶ The category of metric spaces.
A metric on a set \(S\) is a function \(d : S \times S \to \RR\) such that:
- \(d(a, b) \geq 0\),
- \(d(a, b) = 0\) if and only if \(a = b\).
A metric space is a set \(S\) with a distinguished metric.
Implementation
Objects in this category must implement either a
dist
on the parent or the elements ormetric
on the parent; otherwise this will cause an infinite recursion.Todo
- Implement a general geodesics class.
- Implement a category for metric additive groups and move the generic distance \(d(a, b) = |a - b|\) there.
- Incorporate the length of a geodesic as part of the default distance cycle.
EXAMPLES:
sage: from sage.categories.metric_spaces import MetricSpaces sage: C = MetricSpaces() sage: C Category of metric spaces sage: TestSuite(C).run()
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class
sage.categories.metric_spaces.
MetricSpacesCategory
(category, *args)¶ Bases:
sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory
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classmethod
default_super_categories
(category)¶ Return the default super categories of
category.Metric()
.Mathematical meaning: if \(A\) is a metric space in the category \(C\), then \(A\) is also a topological space.
INPUT:
cls
– the classMetricSpaces
category
– a category \(Cat\)
OUTPUT:
A (join) category
In practice, this returns
category.Metric()
, joined together with the result of the methodRegressiveCovariantConstructionCategory.default_super_categories()
(that is the join ofcategory
andcat.Metric()
for eachcat
in the super categories ofcategory
).EXAMPLES:
Consider
category=Groups()
. Then, a group \(G\) with a metric is simultaneously a topological group by itself, and a metric space:sage: Groups().Metric().super_categories() [Category of topological groups, Category of metric spaces]
This resulted from the following call:
sage: sage.categories.metric_spaces.MetricSpacesCategory.default_super_categories(Groups()) Join of Category of topological groups and Category of metric spaces
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classmethod