Detectable Properties of Planar Figures

A description is given of those properties of planar figures that can be detected by viewing the figure through a retina-like device. The answer given involves certain topologies on the set of figures in the plane. A predicate is detectable through an acceptable sequence of finer and finer partitions if and only if the set of all figures satisfying this predicate can be expressed as a Boolean combination of a suitable collection of sets open in the topology induced by the sequence of partitions. A predicate is detectable through all acceptable sequences of partitions if and only if it can be expressed as a Boolean combination of sets open in the Hausdorff topology on the set of all planar figures.