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| category of metric spaces ： ウィキペディア英語版 |
category of metric spaces
In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by .
==Arrows==
The monomorphisms in Met are the injective metric maps, maps that do not map two points into a single point. The epimorphisms are the metric maps in which the domain of the map has a dense image in the range. The isomorphisms are the isometries, metric maps that are one-to-one, onto, and distance-preserving.
As an example, the inclusion of the rational numbers into the real numbers is a monomorphism and an epimorphism, but it is clearly not an isomorphism; this example shows that Met is not a balanced category.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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