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 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.

1 See also
2 Definition
3 Terminology
4 Properties
5 Banach–Mazur game
6 Examples
7 Notes

[edit] See also

Baire category theorem

[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

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.

[edit] Properties

The intersection of meagre sets is meagre.
The union of countable many meagre sets is also meagre.

[edit] Banach–Mazur game

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Further information might be found on the talk page or at requests for expansion.

Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game.

[edit] Examples

The set of the integers ( $\mathbb{Z}$ ) and consequently the set of the naturals ( $\mathbb{N}$ ).
The Cantor set.
The set of functions which have a derivative at some point is a meager set in the space of all continuous functions.^[1]