We define the Polish space 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 R, showing that it is F-sigma as a subset of R x R and bi-reducible to E-0. We also explicitly describe when a non-degenerate rank-1 system is topologically isomorphic to its inverse.
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