Pseudometric space

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In mathematics, a pseudometric space 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 (X,d) is a set X together with a non-negative real-valued function d: X \times X \longrightarrow \mathbb{R}_{\geq 0} (called a pseudometric) such that, for every x,y,z \in X,

  1. \,\!d(x,x) = 0.
  2. \,\!d(x,y) = d(y,x) (symmetry)
  3. \,\!d(x,z) \leq d(x,y) + d(y,z) (subadditivity/triangle inequality)

Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have d(x,y)=0 for distinct values x\ne y.

Examples[edit]

  • Pseudometrics arise naturally in functional analysis. Consider the space \mathcal{F}(X) of real-valued functions f:X\to\mathbb{R} together with a special point x_0\in X. This point then induces a pseudometric on the space of functions, given by
\,\!d(f,g) = |f(x_0)-g(x_0)|\;
for f,g\in \mathcal{F}(X)
  • For vector spaces V, a seminorm p induces a pseudometric on V, as
\,\!d(x,y)=p(x-y).
Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.
  • Every measure space (\Omega,\mathcal{A},\mu) can be viewed as a complete pseudometric space by defining
d(A,B) := \mu(A\Delta B)
for all A,B\in\mathcal{A}.

Topology[edit]

The pseudometric topology is the topology induced by the open balls

B_r(p)=\{ x\in X\mid d(p,x)<r \},

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[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 x\sim y if d(x,y)=0. Let X^*=X/{\sim} and let

d^*([x],[y])=d(x,y)

Then d^* is a metric on X^* and (X^*,d^*) is a well-defined metric space.[2]

The metric identification preserves the induced topologies. That is, a subset A\subset X is open (or closed) in (X,d) if and only if \pi(A)=[A] is open (or closed) in (X^*,d^*).

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

Notes[edit]

  1. ^ Pseudometric topology at PlanetMath.org.
  2. ^ Howes, Norman R. (1995). Modern Analysis and Topology. New York, NY: Springer. p. 27. ISBN 0-387-97986-7. Retrieved 10 September 2012. "Let (X,d) be a pseudo-metric space and define an equivalence relation \sim in X by x \sim y if d(x,y)=0. Let Y be the quotient space X/\sim and p:X\to Y the canonical projection that maps each point of X onto the equivalence class that contains it. Define the metric \rho in Y by \rho(a,b) = d(p^{-1}(a),p^{-1}(b)) for each pair a,b \in Y. It is easily shown that \rho is indeed a metric and \rho defines the quotient topology on Y." 

References[edit]