Definition:Pseudometric
From ProofWiki
Contents |
Definition
A pseudometric on a set $X$ is a real-valued function $d: X \times X \to \R$ which satisfies the following conditions for all $x, y, z \in X$:
- M1: $d \left({x, x}\right) = 0$
- M2: $d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$
- M3: $d \left({x, y}\right) = d \left({y, x}\right)$
Distance Function
The function $d: X \times X \to \R$ is called the distance function or simply distance.
Pseudometric Space
A pseudometric space $M = \left({A, d}\right)$ is an ordered pair consisting of a set $A \ne \varnothing$ followed by a pseudometric $d: A \times A \to \R$ which acts on that set.
The elements of $A$ are called the points of the space.
Comparison with Metric Space
- From Distance in Pseudometric Non-Negative, it can be seen that:
- $\forall x, y \in X: d \left({x, y}\right) \ge 0$
which is often taken as one of the axioms.
- Compare this definition with that for a metric.
The difference between a pseudometric and a metric is that a pseudometric does not insist that the distance function between distinct points is strictly positive.
Also see
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$
