  Analogues of the classical Banach-Stone theorem for spaces of continuous
functions are studied in the context of the spaces of absolutely continuous
functions introduced by Ashton and Doust. We show that if $AC(\sigma_1)$ is
algebra isomorphic to $AC(\sigma_2)$ then $\sigma_1$ is homeomorphic to
$\sigma_2$. The converse however is false. In a positive direction we show that
the converse implication does hold if the sets $\sigma_1$ and $\sigma_2$ are
confined to a restricted collection of compact sets, such as the set of all
simple polygons.
