In mathematics , residual is a subset in a Baire space that can be represented as the intersection of a countable number of dense sets open everywhere. Equivalently, a residual set is a complement to a set of the first category. In a certain sense, we can assume that the residual sets are “large” from a topological point of view.
The concept of residualness is often used to characterize typicality in infinite-dimensional spaces that are not equipped with any natural measure. In particular, many statements in the theory of dynamical systems are formulated for mappings belonging to the residual (in the corresponding topology) set: this is precisely the result that satisfies the countable number of consecutive small perturbations.
Examples
The set of Liouville numbers is residual, and, therefore, its elements are “typical” from a topological point of view (although they are atypical from the point of view of measure theory — Liouville numbers have measure zero).
Links
Finch, Barnaby. Residual set on the Wolfram MathWorld website.