Pseudometric space

In

.

When a topology is generated using a family of pseudometrics, the space is called a

gauge space

.

Definition

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

1. ${\displaystyle d(x,x)=0.}$
2. Symmetry: ${\displaystyle d(x,y)=d(y,x)}$
3. Subadditivity/Triangle inequality: ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$

Unlike a metric space, points in a pseudometric space need not be distinguishable; that is, one may have ${\displaystyle d(x,y)=0}$ for distinct values ${\displaystyle x\neq y.}$

Examples

Any metric space is a pseudometric space. Pseudometrics arise naturally in functional analysis. Consider the space ${\displaystyle {\mathcal {F}}(X)}$ of real-valued functions ${\displaystyle f:X\to \mathbb {R} }$ together with a special point ${\displaystyle x_{0}\in X.}$ This point then induces a pseudometric on the space of functions, given by

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

A seminorm ${\displaystyle p}$ induces the pseudometric ${\displaystyle d(x,y)=p(x-y)}$. This is a

of ${\displaystyle x}$ (in particular, a translation), and therefore convex in ${\displaystyle x}$. (Likewise for ${\displaystyle y}$.)

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 ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ can be viewed as a complete pseudometric space by defining

for all ${\displaystyle A,B\in {\mathcal {A}},}$ where the triangle denotes symmetric difference.

If ${\displaystyle f:X_{1}\to X_{2}}$ is a function and d₂ is a pseudometric on X₂, then ${\displaystyle d_{1}(x,y):=d_{2}(f(x),f(y))}$ 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

which form a 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

topologically distinguishable

).

The definitions of

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 ${\displaystyle x\sim y}$ if ${\displaystyle d(x,y)=0}$. Let ${\displaystyle X^{*}=X/{\sim }}$ be the quotient space of ${\displaystyle X}$ by this equivalence relation and define

This is well defined because for any ${\displaystyle x'\in [x]}$ we have that ${\displaystyle d(x,x')=0}$ and so ${\displaystyle d(x',y)\leq d(x,x')+d(x,y)=d(x,y)}$ and vice versa. Then ${\displaystyle d^{*}}$ is a metric on ${\displaystyle X^{*}}$ and ${\displaystyle (X^{*},d^{*})}$ is a well-defined metric space, called the metric space induced by the pseudometric space ${\displaystyle (X,d)}$.^[6][7]

The metric identification preserves the induced topologies. That is, a subset ${\displaystyle A\subseteq X}$ is open (or closed) in ${\displaystyle (X,d)}$ if and only if ${\displaystyle \pi (A)=[A]}$ is open (or closed) in ${\displaystyle \left(X^{*},d^{*}\right)}$ and ${\displaystyle A}$ is

.

An example of this construction is the

.

See also

• Generalised metric – Metric geometry
• Metric signature – Number of positive, negative and zero eigenvalues of a metric tensor
• Metric space – Mathematical space with a notion of distance
• Metrizable topological vector space – A topological vector space whose topology can be defined by a metric

Notes

References

• ISBN 3-540-18178-4
  .
• ISBN 0-486-68735-X
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• Willard, Stephen (2004) [1970], General Topology (Dover reprint of 1970 ed.), Addison-Wesley
• This article incorporates material from Pseudometric space on
  Creative Commons Attribution/Share-Alike License
  .
• "Example of pseudometric space". PlanetMath.