Definition:Pseudometric
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Definition
A pseudometric on a set $A$ is a real-valued function $d: A \times A \to \R$ which satisfies the following conditions:
| \((\text M 1)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds \map d {x, x} = 0 \) | ||||||
| \((\text M 2)\) | $:$ | \(\ds \forall x, y, z \in A:\) | \(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \) | ||||||
| \((\text M 3)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds \map d {x, y} = \map d {y, x} \) |
The difference between a pseudometric and a metric is that a pseudometric does not insist that the distance function between distinct elements is strictly positive.
Pseudometric Space
A pseudometric space $M = \struct {A, d}$ is an ordered pair consisting of a set $A \ne \O$ followed by a pseudometric $d: A \times A \to \R$ which acts on that set.
Also known as
A pseudometric on a pseudometric space can be referred to as a distance function in the same way as a metric on a metric space.
Also see
- Distance in Pseudometric is Non-Negative, where it is shown that $\forall x, y \in A: \map d {x, y} \ge 0$ which is often taken as one of the axioms.
- Results about pseudometric spaces can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $5$: Metric Spaces