Affine space with a single element

Question

I was thinking that if an affine space is consisted of a single element, this has to be the zero element. Is that correct ?

Answer 1

Well, technically no: any singleton is canonically an affine space trhough the only possible action of $k^0$ $$\{p\}\times\{0\}\to \{p\}\\ (p,0)\mapsto p$$

In fact, if you think about it, every point of $p\in\Bbb R^2$ is considered a zero-dimensional affine subspace of $\left(\Bbb R^2,\overrightarrow{\Bbb R^2}\right)$, not only the origin.

It is true, though, that its space of translations must be $\{0\}$.