# 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 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 (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 for each neighbourhood U of X, the set $B\cap U$ is not dense in U. Equivalently, B is nowhere dense if its closure contains no nonempty open set.

The complement of a nowhere dense set is a dense set; in fact, it has dense interior. The complement of a dense set need not be nowhere dense; indeed, the complement can have both nowhere dense and dense regions.

A meagre set need not have measure zero.

### 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 Fσ set (countable union of closed sets), but is always contained in an Fσ 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 Gδ set (countable intersection of open sets), but contains a dense Gδ 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.

## Banach–Mazur game

Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game. Let $Z$ be a topological space, ${\mathcal {W}}$ be a family of subsets of $Z$ that have nonempty interiors such that every nonempty open set has a subset belonging to ${\mathcal {W}}$ , and $X$ be any subset of $Z$ . Then there is a Banach–Mazur game corresponding to $X,{\mathcal {W}},Z$ . In the Banach–Mazur game, two players, $P$ and $Q$ , alternately choose successively smaller elements of ${\mathcal {W}}$ to produce a sequence $W_{1}\supseteq W_{2}\supseteq W_{3}\supseteq \dotsb$ . Player $P$ wins if the intersection of this sequence contains a point in $X$ ; otherwise, player $Q$ wins. Theorem: For any ${\mathcal {W}}$ meeting the above criteria, player $Q$ has a winning strategy if and only if $X$ is meagre.

## Examples

### Function spaces

• The set of functions that have a derivative at some point is a meagre set in the space of all continuous functions.

## See also

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