Generalized metric space - Misplaced Pages

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. Precisely, it is a category enriched over $[0, \infty]$ , the one-point compactification of $\mathbb {R}$ . 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.

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namely, the property that distinct elements have nonzero distance between them and the property that the distance between two elements is always finite.

References

Lawvere, F. William (1973). "Metric spaces, generalized logic, and closed categories". Rendiconti del Seminario Matematico e Fisico di Milano. 43: 135–166. doi:10.1007/BF02924844.
- Lawvere, F. W. (2002). "Metric spaces, generalized logic and closed categories" (PDF). Reprints in Theory and Applications of Categories (1): 1–37.
Borceux, Francis; Dejean, Dominique (1986). "Cauchy completion in category theory". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 27 (2): 133–146.
Bonsangue, M.M.; Van Breugel, F.; Rutten, J.J.M.M. (1998). "Generalized metric spaces: Completion, topology, and powerdomains via the Yoneda embedding". Theoretical Computer Science. 193 (1–2): 1–51. doi:10.1016/S0304-3975(97)00042-X. hdl:1887/4083537.

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