Meagre set

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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 some precise sense small or negligible. The meagre subsets of a fixed space form a sigma-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.

[edit] 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, where a subset B of X is nowhere dense if, for any nonempty open set U in X, there is a nonempty open set V contained in U such that V and B are disjoint. That is, there is no neighbourhood on which B is dense.

Equivalently, a comeagre (or comeager) set is one that includes the intersection of countably many open dense sets. Then a meagre set is the complement of a comeagre set.

[edit] Terminology

Meagre sets are also called sets of first category; nonmeagre sets (that is, sets that are not meagre) are also called sets of second category.