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 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.

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.

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.

The complement of a nowhere dense set is a dense set. More precisely, the complement of a nowhere dense set is a set with dense interior. Not every dense set has a nowhere dense complement. The complement of a dense set can have nowhere dense, and dense regions.

Relation to Borel hierarchy

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.

Terminology

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).

Properties

Any subset of a meagre set is meagre; any superset of a comeagre set is comeagre.
The union of countably many meagre sets is also meagre; the intersection of countably many comeagre sets is comeagre.

This follows from the fact that a countable union of countable sets is countable.

Banach Category Theorem: In any space X, the union of any family of open sets of the first category is of the first category.^[1]

Banach–Mazur game

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 Banach-Mazur game corresponding to $X, Y, W$ . In the Banach-Mazur 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 $W_1 \supset W_2 \supset W_3 \supset \dotsb$ . 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.

Examples

Subsets of the reals

The rational numbers are meagre as a subset of the reals and as a space – that is, they do not form a Baire space.
The Cantor set is meagre as a subset of the reals, but not as a space, since it is a complete metric space and is thus a Baire space, by the Baire category theorem.

Function spaces

The set of functions which have a derivative at some point is a meagre set in the space of all continuous functions.^[2]

Notes

↑ Measure and Category by John C. Oxtoby Chapter: "The Banach Category Theorem"
↑ Banach, S. (1931). "Über die Baire'sche Kategorie gewisser Funktionenmengen". Studia. Math. (3): 174–179.

External links

Is there a measure zero set which isn’t meagre?