In $1D$ case, $\mathbb{R}^1$ as a typical case of $1D$ Euclidean space is totally ordered. I was wondering if any $1D$ Euclidean space $E^1$ is ordered as well? What is it
For higher dimensional case, are both $\mathbb{R}^n$ and $E^n$ ordered? What are the orders?
Thanks and regards!
Update:
I meant for an order for an Euclidean space:
that can induce or be compatible with the topology of the Euclidean space;
or that is defined in their definition, for example, the order of $\mathbb{R}^1$ is the one defined in one of $\mathbb{R}^1$'s definition as an ordered field with supreme property;
or that is used by default or most commonly.