eli5 What relationship does the Cartesian product R² and R³ have with the Cartesian plane and three-dimensional space?

Question

eli5 What relationship does the Cartesian product R² and R³ have with the Cartesian plane and three-dimensional space?

In: 2

Answer 1

Abstractly, a Cartesian plane/space describes a class of objects with particular characteristics (Euclidean plane/space with a Cartesian coordinate system). R^2 /R^3 is a particular realization, a particular instance of that class.

The distinction don’t really matter though, since any Cartesian plane/space can be identified with R^2 /R^3 in an obvious manner.

There are some small benefits in distinguishing Euclidean plane/space (in the abstract sense) from R^2 /R^3 , but there are no benefits to distinguish Cartesian plane/space from R^2 /R^3 , because every Cartesian plane/space already come automatically with an identification.