Generalized metric space

Not to be confused with Generalised metric.

In mathematics, specifically in category theory, a generalized metric space is a metric space but without the symmetry property and some other properties.^[1] Precisely, it is a category enriched over ${\displaystyle [0,\infty ]}$, the one-point compactification of ${\displaystyle \mathbb {R} }$. The notion was introduced in 1973 by Lawvere who noticed that a metric space can be viewed as a particular kind of a category.

The categorical point of view is useful since by Yoneda's lemma, a generalized metric space can be embedded into a much larger category in which, for instance, one can construct the Cauchy completion of the space.