  Rule-based explanations provide simple reasons explaining the behavior of
machine learning classifiers at given points in the feature space. Several
recent methods (Anchors, LORE, etc.) purport to generate rule-based
explanations for arbitrary or black-box classifiers. But what makes these
methods work in general? We introduce a topological framework for rule-based
explanation methods and provide a characterization of explainability in terms
of the definability of a classifier relative to an explanation scheme. We
employ this framework to consider various explanation schemes and argue that
the preferred scheme depends on how much the user knows about the domain and
the probability measure over the feature space.
