Isometry from higher to lower dimensions I was wondering if there was any isometry from $\mathbb{R^2}$ to $\mathbb{R}$, using the Euclidean metric?
no, in the real line, there are not three numbers at a common distance of $1$ apart for each pair. In the plane, there are equilateral triangles
An isometry would be both continuous and 1-1.  So, no way.