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We study the problem of discretization in a Hausdorff space followed in [WTR 98]. We recall the definitions and properties of the Hausdorff discretization of a compact set. We also study the relationship between the covering discretizations and the Hausdorff discretization. For a cellular metric every covering discretization minimizes the Hausdorff distance, and conversely, if the supercover discretization minimizes the Hausdorff distance then the metric is cellular. The supercover discretization is the Hausdorff discretization iff the metric is proportional to d ∞. We compare also the Hausdorff discretization and the Bresenham discretization [Bres 65]. Actually, the Bresenham discretization of a segment of IR 2 is not always a good discretization relatively to a Hausdorff metric.