In joint work with Robin Deeley and Ian Putnam we constructed minimal homeomorphisms on spaces which look like a point from the perspective of K-theory and cohomology. Our motivation for such a construction stems from the classification programme for C*-algebras and the recent increased interest in understanding the connections between C*-algebras and topological dynamics. We set out to find a dynamical presentation of the Jiang-Su algebra, Z, an algebra of fundamental importance in C*-algebras. Though it is known the Z cannot arise as a crossed product from a minimal dynamical system, if we consider the topological equivalence relation that results from breaking the orbits of our minimal homeomorphism at a point, the resulting C*-algebra is isomorphic to Z whenever the system is uniquely ergodic.