In the mathematical fields of general topology and descriptive set theory, a meagre set is a set that, considered as a subset of a topological space, is in a precise sense small or negligible. A topological space is called meagre if it is a meager subset of itself; otherwise, it is called nonmeagre. The meagre subsets of a fixed space form a σ-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. 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. A set that is not meagre is called nonmeagre and is said to be of the second category. Note that the notions of a comeagre set and a nonmeagre set are not equivalent.
Definition
Let be a topological space. Note that a closed subset of is nowhere dense if and only if its interior in is empty. Note that second category does not mean comeagre — a set may be neither meagre nor comeagre.
Examples and sufficient conditions
Let be a topological space. ;Meagre subsets and subspaces
A singleton set is always a non-meagre subspace ; but it is a non-meagre subset if and only if it is an isolated point.
Just as a nowhere dense subset need not be closed, but is always contained in a closed nowhere dense subset, a meagre set need not be an set, but is always contained in an set made from nowhere dense sets. Dually, just as the complement of a nowhere dense set need not be open, but has a dense interior, a comeagre set need not be a set, but contains a dense set formed from dense open sets.
Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game. Let be a topological space, be a family of subsets of that have nonempty interiors such that every nonempty open set has a subset belonging to, and be any subset of. Then there is a Banach–Mazur game corresponding to. In the Banach–Mazur game, two players, and, alternately choose successively smaller elements of to produce a sequence. Player wins if the intersection of this sequence contains a point in ; otherwise, player wins.
