Mappings that preserve neighbourhood relationships are relevant in both practical and biological contexts. It is important to be clear about precisely what preserving neighbourhoods could mean. We give a definition of a “perfectly neighbourhood preserving” map, which we call a topographic homeomorphism, and prove that this has certain desirable properties. When a topographic homeomorphism does not exist (the usual case), many choices are available for quantifying the quality of a map. We introduce a particular measure, C, which has the form of a quadratic assignment problem. We also discuss other measures that have been proposed, some of which are related to C. A comparison of seven measures applied to the same simple mapping problem reveals interesting similarities and differences between the measures, and challenges common intuitions as to what constitutes a “good” map.