What is the precise definition of the ambient space

Question

For instance, what is the ambient space of a singleton $\{x\}$, where $x \in \mathbb{R}$?

Can it be the singleton itself? $\mathbb{R}$? $\mathbb{R}^n$? or some arbitary set that happens to contain $\{x\}$?

Answer 1

Usually when this comes up, you are looking at some sort of space isomorphic in some way to a space of lower dimension/"size", but the space is embedded in a larger space. In this case the "ambient space" is the higher dimensional space where your manifold or polyhedron or whatever it is is actually originally defined, although you can often work in a lower dimensional representation of the space where your set lives to solve problems, e.g. polyhedra living in an affine space which is a higher dimensional space than the dimension of the polyhedron can be projected down to lower dimensional space to do computations.

Answer 2

It depends. If you study $\{x\}$ by itself, then the ambient space is also $\{x\}$. If you study $\{x\}$ as a subset of $\mathbb R$, the ambient space is $\mathbb R$. And so on.