  Differential calculus on Euclidean spaces has many generalisations. In
particular, on a set $X$, a diffeological structure is given by maps from open
subsets of Euclidean spaces to $X$, a differential structure is given by maps
from $X$ to $\mathbb{R}$, and a Fr\"{o}licher structure is given by maps from
$\mathbb{R}$ to $X$ as well as maps from $X$ to $\mathbb{R}$. We illustrate the
relations between these structures through examples.
