  In this paper we investigate the relationship between the properties of a
compact set $\sigma \subseteq \mathbb{C}$ and the structure of the space
$BV(\sigma)$ of functions of bounded variation (in the sense of Ashton and
Doust) defined on $\sigma$. For the subalgebras of absolutely continuous
functions on $\sigma$, it is known that for certain classes of compact sets one
obtains a Gelfand--Kolmogorov type result: the function spaces $AC(\sigma_1)$
and $AC(\sigma_2)$ are isomorphic if and only if the domain sets $\sigma_1$ and
$\sigma_2$ are homeomorphic. Our main theorem is that in this case the
isomorphism must extend to an isomorphism of the $BV(\sigma)$ spaces. An
application is given to the spectral theory of $AC(\sigma)$ operators.
