Metric space
Isometries

One interpretation of a "structure-preserving" map is one that fully preserves the distance function:

It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is called an isometry. One perhaps non-obvious example of an isometry between spaces described in this article is the map f: (R 2, d 1) → (R 2, d ∞) {\displaystyle f:(\mathbb {R} ^{2},d_{1})\to (\mathbb {R} ^{2},d_{\infty })} defined by f (x, y) = (x + y, x − y). {\displaystyle f(x,y)=(x+y,x-y).}

If there is an isometry between the spaces M and M, they are said to be isometric. Metric spaces that are isometric are essentially identical.
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