Pseudometric space

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.

DefinitionEdit

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  .

ExamplesEdit

  • Any metric space is a pseudometric space.
  • 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.

TopologyEdit

The pseudometric topology is the topology generated by the open balls

 

which form a basis for the topology.[1] A topological space is said to be a pseudometrizable space[2] 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).

The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged.[3]

Metric identificationEdit

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 X by this equivalence relation and define

 

This is well defined because for any   we have that   and so   and vice versa. Then   is a metric on   and   is a well-defined metric space, called the metric space induced by the pseudometric space  .[4][5]

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.

See alsoEdit

NotesEdit

  1. ^ "Pseudometric topology". PlanetMath.
  2. ^ Willard, p. 23
  3. ^ Cain, George (Summer 2000). "Chapter 7: Complete pseudometric spaces" (PDF). Archived from the original on 7 October 2020. Retrieved 7 October 2020.
  4. ^ 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  .
  5. ^ Simon, Barry (2015). A comprehensive course in analysis. Providence, Rhode Island: American Mathematical Society. ISBN 978-1470410995.

ReferencesEdit