In mathematics, specifically in category theory, a generalized metric space is a

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...

but without the symmetry property and some other properties.namely, the property that distinct elements have nonzero distance between them and the property that the distance between two elements is always finite. Precisely, it is a category enriched over

, \infty /math>, the one-point compactification of \mathbb . 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 In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (vie ...

, a generalized metric space can be embedded into a much larger category in which, for instance, one can construct the

Cauchy completion In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...

of the space.

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