In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is a countable union of subsets whose closures have empty interior. Thus meager sets are, in a sense, "small", being small unions of small subsets.

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.

Meagre sets play an important role in the formulation of the notion of Baire space and of the Baire category theorem, which is used in the proof of several fundamental results of functional analysis.

Definitions

will be a topological space.

The definition of meagre set uses the notion of a nowhere dense whose closure has empty interior. See the corresponding article for more details.

is called a or of the if it is a countable union of nowhere dense. Otherwise, the subset is called a or of the " can be omitted if the ambient space is fixed and understood from context.

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

is called or if its complement. (This use of the prefix "co" is consistent with its use in other terms such as "cofinite if and only if it is equal to a countable intersection

Remarks on terminology

is nonmeagre, no set is at the same time meagre and comeagre, every comeagre set is nonmeagre, and there can be nonmeagre sets that are not comeagre, that is, with nonmeagre complement. See the Examples section below.

is given the subspace topology can also be called a meagre subspace. (See the Properties and Examples sections below for the relationship between the two.) Similarly, a nonmeagre subspace will be a set that is nonmeagre in itself, which is not the same as being nonmeagre in the whole space. Be aware however that in the context of topological vector spaces some authors may use the phrase "meagre/nonmeagre subspace" to mean a vector subspace that is a meagre/nonmeagre set relative to the whole space.

The terms first category and second category were the original ones used by René Baire in his thesis of 1899. The meagre terminology was introduced by Bourbaki in 1948.

Examples

The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space.

is nonmeagre and comeagre.

is also nonmeagre.

A countable T1 space without isolated point

The Cantor set But it is nonmeagre in itself, since it is a complete metric space.

. It is nonmeagre in itself (since as a subspace it contains an isolated point).

But it is a nonmeagre subspace, that is, it is nonmeagre in itself.

is a meagre sub is a nonmeagre sub is not a meagre topological space). A countable Hausdorff space without isolated points is meagre, whereas any topological space that contains an isolated point is nonmeagre. 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.

Any topological space that contains an isolated point is nonmeagre (because no set containing the isolated point can be nowhere dense). In particular, every nonempty discrete space is nonmeagre.

are both nonmeagre.

with the topology of uniform convergence that have a derivative at some point is meagre., which consists of the continuous real-valued nowhere differentiable functions is comeagre and nonmeagre. In particular that set is not empty. This is one way to show the existence of continuous nowhere differentiable functions.

On an infinite-dimensional Banach space, there exists a discontinuous linear functional whose kernel is nonmeagre. Also, under Martin's axiom, on each separable Banach space, there exists a discontinuous linear functional whose kernel is meagre (this statement disproves the Wilansky–Klee conjecture).

Characterizations and sufficient conditions

Every nonempty Baire space is nonmeagre. By the Baire category theorem, this applies to nonempty complete (pseudo)metric spaces as well as locally compact Hausdorff spaces. However, there exist nonmeagre spaces that are not Baire spaces.

Every nowhere dense subset is a meagre set.).

All subsets and all countable unions of meagre sets are meagre. Thus the meagre subsets of a fixed space form a σ-ideal of subsets, a suitable notion of negligible set. Dually, all supersets and all countable intersections of comeagre sets are comeagre. Every superset of a nonmeagre set is nonmeagre.

The the union of any family of open sets of the first category is of the first category.

has the subspace topology However the following results hold:

And correspondingly for nonmeagre sets:

is equivalent to being meagre in itself, and similarly for the nonmeagre property.

is nonempty.

Properties

(because otherwise it would be nowhere dense and thus of the first category).

Meagre subsets and Lebesgue measure

There exist nowhere dense subsets (which are thus meagre subsets) that have positive Lebesgue measure.

need not have Lebesgue measure fat Cantor sets, like the Smith–Volterra–Cantor set

is a Baire space.

is a sequence that enumerates the rational numbers.

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_{\sigma} set 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_{\delta} set (countable intersection of open set formed from dense open sets.

Banach–Mazur game

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

Erdos–Sierpinski duality

Many arguments about meagre sets also apply to null sets, i.e. sets of Lebesgue measure 0. The Erdos–Sierpinski duality theorem states that if the continuum hypothesis holds, there is an involution from reals to reals where the image of a null set of reals is a meagre set, and vice versa. In fact, the image of a set of reals under the map is null if and only if the original set was meagre, and vice versa.

See also

• , for analogs to residual

• , for analogs to meagre

Notes

Bibliography