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Quasimetrics

Occasionally, a quasimetric is defined as a function that satisfies all axioms for a metric with the possible exception of symmetry. The name of this generalisation is not entirely standardized.

Quasimetrics are common in real life. For example, given a set X of mountain villages, the typical walking times between elements of X form a quasimetric because travel uphill takes longer than travel downhill. Another example is the length of car rides in a city with one-way streets: here, a shortest path from point A to point B goes along a different set of streets than a shortest path from B to A and may have a different length.

A quasimetric on the reals can be defined by setting d (x, y) = { x − y if x ≥ y, 1 otherwise. {\displaystyle d(x,y)={\begin{cases}x-y&{\text{if }}x\geq y,\\1&{\text{otherwise.}}\end{cases}}} The 1 may be replaced, for example, by infinity or by 1 + y − x {\displaystyle 1+{\sqrt {y-x}}} or any other subadditive function of y - x. This quasimetric describes the cost of modifying a metal stick: it is easy to reduce its size by filing it down, but it is difficult or impossible to grow it.

Given a quasimetric on X, one can define an R -ball around x to be the set { y ∈ X | d (x, y) ≤ R } {\displaystyle \{y\in X|d(x,y)\leq R\}}. As in the case of a metric, such balls form a basis for a topology on X, but this topology need not be metrizable. For example, the topology induced by the quasimetric on the reals described above is the (reversed) Sorgenfrey line.