Pseudometric space

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In mathematics, a pseudometric space or semi-metric space[1] is a generalized 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[edit]

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

  1. .
  2. (symmetry)
  3. (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[edit]

  • 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[edit]

The pseudometric topology is the topology induced by the open balls

which form a basis for the topology.[2] 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[edit]

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 and let

Then is a metric on and is a well-defined metric space, called the metric space induced by the pseudometric space .[3][4]

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 . The topological identification is the Kolmogorov quotient.

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

Notes[edit]

  1. ^ Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.
  2. ^ Pseudometric topology at PlanetMath.org.
  3. ^ Howes, Norman R. (1995). Modern Analysis and Topology. New York, NY: Springer. p. 27. ISBN 0-387-97986-7. Retrieved 10 September 2012. Let be a pseudo-metric space and define an equivalence relation in by if . Let be the quotient space and the canonical projection that maps each point of onto the equivalence class that contains it. Define the metric in by for each pair . It is easily shown that is indeed a metric and defines the quotient topology on . 
  4. ^ Simon, Barry (2015). A comprehensive course in analysis. Providence, Rhode Island: American Mathematical Society. ISBN 1470410990. 

References[edit]