Pseudometric space

Definition

A pseudometric space ( X , d ) is a set X together with a non-negative real-valued function d : X × X ⟶ R ≥ 0 (called a pseudometric) such that, for every x , y , z ∈ X ,

1. d ( x , y ) ⩾ 0 and d ( x , x ) = 0 .
2. d ( x , y ) = d ( y , x ) (symmetry)
3. d ( x , z ) ⩽ 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 ≠ y .

Examples

Pseudometrics arise naturally in functional analysis. Consider the space F ( X ) of real-valued functions f : X → R together with a special point x 0 ∈ X . This point then induces a pseudometric on the space of functions, given by

for f , g ∈ F ( X )

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

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 ( Ω , A , μ ) can be viewed as a complete pseudometric space by defining

for all A , B ∈ A , where the triangle denotes symmetric difference.

If f : X 1 → 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 ∈ X ∣ d ( p , x ) < r } ,

which form a basis for the topology. 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 ∼ y if d ( x , y ) = 0 . Let X ∗ = X / ∼ and let

d ∗ ( [ x ] , [ y ] ) = d ( x , y )

Then d ∗ is a metric on X ∗ and ( X ∗ , d ∗ ) is a well-defined metric space, called the metric space induced by the pseudometric space ( X , d ) .

The metric identification preserves the induced topologies. That is, a subset A ⊂ X is open (or closed) in ( X , d ) if and only if π ( 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.