Pseudometric space

In mathematics, a pseudometric 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

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

$d(x,y) \geq 0$ .
$d(x,y) = d(y,x)$ (symmetry)
$d(x,z) \leq d(x,y) + d(y,z)$ (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

Pseudometrics arise naturally in functional analysis. Consider the space $\mathcal{F}(X)$ of real-valued functions $f\colon X\to\mathbb{R}$ 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.

Pseudometrics also arise in the theory of hyperbolic complex manifolds: see Kobayashi metric.
Every measure space $(\Omega,\mathcal{A},\mu)$ can be viewed as a complete pseudometric space by defining

d(A,B) := \mu(A\vartriangle B)

for all

A,B\in\mathcal{A}

, where the triangle denotes symmetric difference.

If $f:X_1 \rightarrow X_2$ is a function and d₂ is a pseudometric on X₂, then $d_1(x,y) := d_2(f(x),f(y))$ 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

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

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

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.^[3]

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 $(X^*,d^*)$ . The topological identification is the Kolmogorov quotient.

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

Notes

↑ Dmitri Burago, Yu D Burago, Sergei Ivanov, A Course in Metric Geometry, American Mathematical Society, 2001, ISBN 0-8218-2129-6.
↑ Pseudometric topology at PlanetMath.org.
↑ 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 $p\colon X\to Y$ 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

Arkhangel'skii, A.V.; Pontryagin, L.S. (1990). General Topology I: Basic Concepts and Constructions Dimension Theory. Encyclopaedia of Mathematical Sciences. Springer. ISBN 3-540-18178-4.
Steen, Lynn Arthur; Seebach, Arthur (1995) [1970]. Counterexamples in Topology (new ed.). Dover Publications. ISBN 0-486-68735-X.
This article incorporates material from Pseudometric space on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Example of pseudometric space at PlanetMath.org.

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