“Pseudometric space”的意思、由来-开放百科全书

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Pseudometric space

释义

Definition
Examples
Topology
Metric identification
Notes
References

In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis.

When a topology is generated using a family of pseudometrics, the space is called a gauge space.

Definition

A pseudometric space is a set together with a non-negative real-valued function (called a pseudometric) such that, for every ,

.
(symmetry)
(subadditivity/triangle inequality)

Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have for distinct values .

Examples

Pseudometrics arise naturally in functional analysis. Consider the space of real-valued functions together with a special point . This point then induces a pseudometric on the space of functions, given by

for

For vector spaces , a seminorm induces a pseudometric on , as

Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.

Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric.
Every measure space can be viewed as a complete pseudometric space by defining

for all , where the triangle denotes symmetric difference.

If is a function and d₂ is a pseudometric on X₂, then gives a pseudometric on X₁. If d₂ is a metric and f is injective, then d₁ is a metric.

Topology

The pseudometric topology is the topology induced by the open balls

which form a basis for the topology.^[1] A topological space is said to be a pseudometrizable topological space if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.

The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is T₀ (i.e. distinct points are topologically distinguishable).

Metric identification

The vanishing of the pseudometric induces an equivalence relation, called the metric identification, that converts the pseudometric space into a full-fledged metric space. This is done by defining if . Let be the quotient space of {{Var|X}} by this equivalence relation and define

Then is a metric on and is a well-defined metric space, called the metric space induced by the pseudometric space .^[2]^[3]

The metric identification preserves the induced topologies. That is, a subset is open (or closed) in if and only if is open (or closed) in and A is saturated. The topological identification is the Kolmogorov quotient.

An example of this construction is the completion of a metric space by its Cauchy sequences.

Notes

1. ^{{planetmath reference|id=6284|title=Pseudometric topology}}
2. ^{{cite book|last=Howes|first=Norman R.|title=Modern Analysis and Topology|year=1995|publisher=Springer|location=New York, NY|isbn=0-387-97986-7|url=https://www.springer.com/mathematics/analysis/book/978-0-387-97986-1|accessdate=10 September 2012|page=27|quote=Let

be a pseudo-metric space and define an equivalence relation

. Let

be the quotient space

and

the canonical projection that maps each point of

onto the equivalence class that contains it. Define the metric

for each pair

. It is easily shown that

is indeed a metric and

defines the quotient topology on

.}}
3. ^{{cite book|title=A comprehensive course in analysis|last=Simon|first=Barry|publisher=American Mathematical Society|year=2015|isbn=1470410990|location=Providence, Rhode Island}}

References

{{cite book | title=General Topology I: Basic Concepts and Constructions Dimension Theory | last=Arkhangel'skii | first=A.V. |author2=Pontryagin, L.S. | year=1990 | isbn=3-540-18178-4 | publisher=Springer | series=Encyclopaedia of Mathematical Sciences}}
{{cite book | title=Counterexamples in Topology | last=Steen | first=Lynn Arthur |author2=Seebach, Arthur | year=1995 | origyear=1970 | isbn=0-486-68735-X | publisher=Dover Publications | edition=new }}
{{PlanetMath attribution|id=6273|title=Pseudometric space}}
{{planetmath reference|id=6275|title=Example of pseudometric space}}

2 : Properties of topological spaces|Metric geometry

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