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.
A topological space T 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 (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**.
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

Throughout, will be a topological space.

A subset of a topological space is called **nowhere dense** or **rare** in if its closure has empty interior. Equivalently, is nowhere dense in if for each open set the set is not dense in

A closed subset of is nowhere dense in if and only if its topological interior in is empty.

A subset of a topological space is said to be **meagre in** a **meagre sub set of** or of the

**first category**in if it is a countable union of nowhere dense subsets of A subset is of the

**second category**or

**nonmeagre**in if it is not of first category in

A topological space is called **meagre** (resp. **nonmeagre**) if it is a meagre (resp. nonmeagre) subset of itself.

**Warning**: If is a subset of then being a "meagre sub" of means that when is endowed with the subspace topology (induced on it by ) then is a meagre topological space (that is, is a meagre subset of ). In contrast, being a "meagre sub*space*" of means that is equal to a countable union of nowhere dense subsets of The same applies to nonmeager subsets and subspaces.*set*

For example, if is the set of all positive integers then is a meager sub*set* of but not a meager sub*space* of If is not an isolated point of a T_{1} space (meaning that is not an open subset of ) then is a meager sub*space* of but not a meager sub*set* of

A subset is **comeagre** in if its **co**mplement is meagre in Equivalently, it is equal to an intersection of countably many sets, each of whose topological interior is a dense subset of This use of the prefix "**co**" is consistent with its use in other terms such as "**co**finite".

Importantly, being of the second category is *not* the same as being comeagre — a set may be neither meagre nor comeagre (in this case it will be of second category).

## Examples and sufficient conditions

Let be a topological space.

**Meagre sub sets and subspaces**

- A singleton subset is always a non-meagre sub
**space**of (that is, it is a non-meagre topological space). If is an isolated point of then is also a non-meagre sub**set**of ; the converse holds if is a T_{1}space. - Any subset of a meagre set is a meagre set.
^{[1]} - Every nowhere dense subset is a meagre set.
^{[1]} - The union of countably many meagre sets is also a meagre set.
^{[1]} - Any closed subset of whose interior in is empty is of the first category of (that is, it is a meager subset of ). Thus a closed subset of that is of the second category in must have non-empty interior in
^{[2]} - A countable Hausdorff space without isolated points is meagre.
^{[3]} - Any topological space that contains an isolated point is non-meagre.
^{[3]} - Any discrete space is non-meagre.
^{[3]} - Every Baire space is non-meagre but there exist non-meagre spaces that are not Baire spaces.
^{[3]}- Since complete metric spaces as well as Hausdorff locally compact spaces are Baire spaces, they are also non-meagre spaces.

- The set is a meagre sub
*set*of even though is a non-meagre sub*space*(that is, is not a meagre topological space).^{[3]} - Because the rational numbers are countable, they 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 subset of itself, since it is a complete metric space and is thus a Baire space, by the Baire category theorem. - If is a homeomorphism then a subset is meagre if and only if is meagre.
^{[1]}

**Comeagre subset**

- Any superset of a comeagre set is comeagre.
- the intersection of countably many comeagre sets is comeagre.
- This follows from the fact that a countable union of countable sets is countable.

### Function spaces

The set of functions that have a derivative at some point is a meagre set in the space of all continuous functions.^{[4]}

## Properties

- Banach Category Theorem: In any space the union of any countable family of open sets of the first category is of the first category.
^{[5]} - A non-meagre locally convex topological vector space is a barreled space.
^{[3]} - A closed subset of that is of the second category in must have non-empty interior in
^{[2]} - If is of the second category in and if are subsets of such that then at least one is of the second category in

### Meagre subsets and Lebesgue measure

A meagre set need not have measure zero.
There exist nowhere dense subsets (which are thus meagre subsets) that have positive Lebesgue measure.^{[3]}

### 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.

## Banach–Mazur game

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.

**Theorem**: For any meeting the above criteria, player has a winning strategy if and only if is meagre.

## See also

- Baire category theorem – On topological spaces where the intersection of countably many dense open sets is dense
- Baire space
- Generic property, for analogs to residual
- Negligible set, for analogs to meagre
- Nowhere dense set
- Property of Baire

## Notes

- ^
^{a}^{b}^{c}^{d}Rudin 1991, p. 43. - ^
^{a}^{b}Rudin 1991, pp. 42-43. - ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}Narici & Beckenstein 2011, pp. 371-423. **^**Banach, S. (1931). "Über die Baire'sche Kategorie gewisser Funktionenmengen".*Studia Math*.**3**(1): 174–179.**^**Oxtoby, John C. (1980). "The Banach Category Theorem".*Measure and Category*(Second ed.). New York: Springer. pp. 62–65. ISBN 0-387-90508-1.

## Bibliography

- Narici, Lawrence; Beckenstein, Edward (2011).
*Topological Vector Spaces*. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. - Rudin, Walter (1991).
*Functional Analysis*. International Series in Pure and Applied Mathematics.**8**(Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.