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

A pseudometric space (X,d) is a set X together with a non-negative real-valued function d: X \times X \longrightarrow \mathbb{R} (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

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.

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^[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

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.

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^*).

Notes

1. ↑

References

it:Spazio pseudometrico nl:Pseudometriek sv:Pseudometriskt rum zh:???