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 sigmaideal 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.
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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.
Recall that a subset B of X is nowhere dense if there is no neighbourhood on which B is dense: 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.
Note that the complement of a nowhere dense set is a dense set, but not every dense set is of this form. More precisely, the complement of a nowhere dense set is a set with dense interior.
Just as a nowhere dense subset need not be closed, but is always contained in a closed nowhere dense subset (viz, its closure), a meagre set need not be an F_{σ} set (countable union of closed sets), but is always contained in an F_{σ} set made from nowhere dense sets (by taking the closure of each set).
Dually, just as the complement of a nowhere dense set need not be open, but has a dense interior (contains a dense open set), a comeagre set need not be a G_{δ} set (countable intersection of open sets), but contains a dense G_{δ} set formed from dense open sets.
A meagre set is also called a set of first category; a nonmeagre set (that is, a set that is not meagre) is also called a set of second category. Second category does not mean comeagre – a set may be neither meagre nor comeagre (in this case it will be of second category).
Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game. If Y is a topological space, W is a family of subsets of Y which have nonempty interior such that every nonempty open set has a subset in W, and X is any subset of Y, then there is a BanachMazur game corresponding to X, Y, W. In the BanachMazur game, two players, P_{1} and P_{2}, alternate choosing successively smaller (in terms of the subset relation) elements of W to produce a descending sequence If the intersection of this sequence contains a point in X, P_{1} wins; otherwise, P_{2} wins. If W is any family of sets meeting the above criteria, then P_{2} has a winning strategy if and only if X is meagre.
