A Euclidean space that is homeomorphic to a non-Euclidean space Is there a well-known example (preferably in dimensions 2 or higher) of two homeomorphic spaces: (1) a metric space with the Euclidean metric and (2) a metric space that is not Euclidean?
 A: Consider the metric spaces $\mathbb R^2$ with the Euclidean metric $d$ and $\mathbb R^2$ with the "max metric" $d'$ defined by $$d'(x,y) = \max\{|x_1-y_1|,|x_2-y_2|\}.$$ The open balls in the Euclidean metric space are open discs, whereas the open "balls" of the max metric space are open squares.
To show that these metric spaces are homeomorphic, we need to show that every open disc composed of open squares and vice versa. This is straightforward.
I am fairly certain that this can be generalized to $\mathbb R^n$ as well.
A: Even though the question has been already answered, let me point out two non-trivial generalisations to the infinite-dimensional case:

Theorem (M. Kadets). All infinite-dimensional separable Banach spaces are homeomorphic.

There is even a more surprising result:

Theorem (Cz. Bessaga). Every infinite-dimensional Hilbert space is diffeomorphic to its unit sphere.

References:

Cz. Bessaga, Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 14 (1966), 27–31.
M. I. Kadets, Proof of the topological equivalence of all separable infinite-dimensional Banach spaces, Functional Analysis and Its Applications, 1 (1967), 53–62.

A: To generalize on the fine examples given by Lee Mosher and Mike Miller, a finite dimensional vector space $\mathbb{R}^n$ has the same topology under any vector space norm.  The Euclidean metric is the Euclidean norm.  Lee's example is the 1-norm, and Mike's is the $p$-norm, for $1\le p \lt \infty$.  We could add to these the $\infty$-norm (max-norm) that is described in Benjamin's Answer.
A: Stereographic projection is a homeomorphism $S^n∖{p}→R^n$.Look at here.
