This is not quite the original definition. Carlsson et al. choose Z to be \mathbb{R} or \mathbb{R}^2, and cover it with open sets such that at most two intersect. This restriction means that the output is in the form of a complex network. Because the topology of a finite point cloud is trivial, clustering methods (such as single linkage) are used to produce the analogue of connected sets in the preimage f^{-1}(U) when MAPPER is applied to actual data.
