  We define the Polish space $\mathcal{R}$ of non-degenerate rank-1 systems.
Each non-degenerate rank-1 system can be viewed as a measure-preserving
transformation of an atomless, $\sigma$-finite measure space and as a
homeomorphism of a Cantor space. We completely characterize when two
non-degenerate rank-1 systems are topologically isomorphic. We also analyze the
complexity of the topological isomorphism relation on $\mathcal{R}$, showing
that it is $F_{\sigma}$ as a subset of $\mathcal{R} \times \mathcal{R}$ and
bi-reducible to $E_0$. We also explicitly describe when a non-degenerate rank-1
system is topologically isomorphic to its inverse.
