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.

Contents

Definition

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,

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: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

for f,g\in \mathcal{F}(X)

For vector spaces V, a seminorm p induces a pseudometric on V, as

Conversely, a homogeneous, translation invariant pseudometric induces a seminorm.

Topology

The pseudometric topology is the topology induced by the open balls

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).

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