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Category of metric spaces
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==Objects== The [[Empty set|empty]] metric space is the [[initial object]] of '''Met'''; any [[singleton (mathematics)|singleton]] metric space is a [[terminal object]]. Because the initial object and the terminal objects differ, there are no [[zero object]]s in '''Met'''. The [[injective object]]s in '''Met''' are called [[injective metric space]]s. Injective metric spaces were introduced and studied first by {{harvtxt|Aronszajn|Panitchpakdi|1956}}, prior to the study of '''Met''' as a category; they may also be defined intrinsically in terms of a [[Helly family|Helly property]] of their metric balls, and because of this alternative definition Aronszajn and Panitchpakdi named these spaces ''hyperconvex spaces''. Any metric space has a smallest injective metric space into which it can be isometrically [[Embedding|embedded]], called its metric envelope or [[tight span]].
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