Pseudometric space
Template:Short description In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa<ref>Template:Cite journal</ref><ref>Template:Cite book</ref> in 1934. 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 <math>(X,d)</math> is a set <math>X</math> together with a non-negative real-valued function <math>d : X \times X \longrightarrow \R_{\geq 0},</math> called a Template:Visible anchor, such that for every <math>x, y, z \in X,</math>
- <math>d(x,x) = 0.</math>
- Symmetry: <math>d(x,y) = d(y,x)</math>
- Subadditivity/Triangle inequality: <math>d(x,z) \leq d(x,y) + d(y,z)</math>
Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have <math>d(x, y) = 0</math> for distinct values <math>x \neq y.</math>
ExamplesEdit
Any metric space is a pseudometric space. Pseudometrics arise naturally in functional analysis. Consider the space <math>\mathcal{F}(X)</math> of real-valued functions <math>f : X \to \R</math> together with a special point <math>x_0 \in X.</math> This point then induces a pseudometric on the space of functions, given by <math display=block>d(f,g) = \left|f(x_0) - g(x_0)\right|</math> for <math>f, g \in \mathcal{F}(X)</math>
A seminorm <math>p</math> induces the pseudometric <math>d(x, y) = p(x - y)</math>. This is a convex function of an affine function of <math>x</math> (in particular, a translation), and therefore convex in <math>x</math>. (Likewise for <math>y</math>.)
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 <math>(\Omega,\mathcal{A},\mu)</math> can be viewed as a complete pseudometric space by defining <math display=block>d(A,B) := \mu(A \vartriangle B)</math> for all <math>A, B \in \mathcal{A},</math> where the triangle denotes symmetric difference.
If <math>f : X_1 \to X_2</math> is a function and d2 is a pseudometric on X2, then <math>d_1(x, y) := d_2(f(x), f(y))</math> gives a pseudometric on X1. If d2 is a metric and f is injective, then d1 is a metric.
TopologyEdit
The Template:Visible anchor is the topology generated by the open balls <math display=block>B_r(p) = \{x \in X : d(p, x) < r\},</math> which form a basis for the topology.<ref>Template:Planetmath reference</ref> A topological space is said to be a Template:Visible anchor<ref>Willard, p. 23</ref> 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 (that is, distinct points are topologically distinguishable).
The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
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 <math>x\sim y</math> if <math>d(x,y)=0</math>. Let <math>X^* = X/{\sim}</math> be the quotient space of <math>X</math> by this equivalence relation and define <math display=block>\begin{align}
d^*:(X/\sim)&\times (X/\sim) \longrightarrow \R_{\geq 0} \\
d^*([x],[y])&=d(x,y)
\end{align}</math> This is well defined because for any <math>x' \in [x]</math> we have that <math>d(x, x') = 0</math> and so <math>d(x', y) \leq d(x, x') + d(x, y) = d(x, y)</math> and vice versa. Then <math>d^*</math> is a metric on <math>X^*</math> and <math>(X^*,d^*)</math> is a well-defined metric space, called the metric space induced by the pseudometric space <math>(X, d)</math>.<ref>Template:Cite book</ref><ref>Template:Cite book</ref>
The metric identification preserves the induced topologies. That is, a subset <math>A \subseteq X</math> is open (or closed) in <math>(X, d)</math> if and only if <math>\pi(A) = [A]</math> is open (or closed) in <math>\left(X^*, d^*\right)</math> and <math>A</math> 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
ReferencesEdit
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