Injective Metric Space - Injectivity

Injectivity

A retraction of a metric space X is a function ƒ mapping X to a subspace of itself, such that

1. for all x, ƒ(ƒ(x)) = ƒ(x); that is, ƒ is the identity function on its image, and
2. for all x and y, d(ƒ(x), ƒ(y)) ≤ d(x, y); that is, ƒ is nonexpansive.

A retract of a space X is a subspace of X that is an image of a retraction. A metric space X is said to be injective if, whenever X is isometric to a subspace Z of a space Y, that subspace Z is a retract of Y.