Meagre set

In the mathematical fields of general topology and descriptive set theory, a meagre set (also called a meager set or a set of first category) is a set that, considered as a subset of a (usually larger) topological space, is in a precise sense small or negligible. The meagre subsets of a fixed space form a -ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre.

General topologists use the term Baire space to refer to a broad class of topological spaces on which the notion of meagre set is not trivial (in particular, the entire space is not meagre). Descriptive set theorists mostly study meagre sets as subsets of the real numbers, or more generally any Polish space, and reserve the term Baire space for one particular Polish space.

The complement of a meagre set is a comeagre set or residual set.

Definition

Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X. Dually, a comeagre set is one whose complement is meagre, or equivalently, the intersection of countably many sets with dense interiors.