Definition:Quasimetric
From ProofWiki
(Redirected from Definition:Quasimetric Space)
Contents |
Definition
A quasimetric 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: $d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right)$
- M4: $x \ne y \implies d \left({x, y}\right) > 0$
Note the numbering system of these conditions. They are numbered this way so as to retain consistency with the metric space axioms, of which these are a subset.
Quasimetric Space
A quasimetric space $M = \left({X, d}\right)$ is an ordered pair consisting of a set $X \ne \varnothing$ followed by a quasimetric $d: X \times X \to \R$ which acts on that set.
Also see
- Compare this definition with that for a metric.
The difference between a quasimetric and a metric is that a quasimetric does not insist that the distance function between distinct points is commutative, that is, that $d \left({x, y}\right) = d \left({y, x}\right)$.
- Results about quasimetric spaces can be found here.
Sources
- Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 5$
