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 small or negligible in a precise sense detailed below.
A set that is not meagre is called nonmeagre, or of the second category.
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.
The definition of meagre set uses the notion of a nowhere dense subset of
" can be omitted if the ambient space is fixed and understood from context.
if and only if it is equal to a countable intersection of sets, each of whose interior is dense in
Remarks on terminology The notions of nonmeagre and comeagre should not be confused.
, meaning a meagre space when given the subspace topology.
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.
[4][5] The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space.
A countable T1 space without isolated point is meagre.
It is nonmeagre in itself (since as a subspace it contains an isolated point).
[6] A countable Hausdorff space without isolated points is meagre, whereas any topological space that contains an isolated point is nonmeagre.
[6] 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[6] (because no set containing the isolated point can be nowhere dense).
, which consists of the continuous real-valued nowhere differentiable functions on
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.
[9] 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[10]).
In particular, by the Baire category theorem every nonempty complete metric space and every nonempty locally compact Hausdorff space is nonmeagre.
The Banach category theorem[12] states that in any space
However the following results hold:[5] And correspondingly for nonmeagre sets: In particular, every subset of
is equivalent to being meagre in itself, and similarly for the nonmeagre property.
is nonmeagre if and only if every countable intersection of dense open sets in
Consequently, any closed subset with empty interior is meagre.
The union of a countable number of such sets with measure approaching
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
Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game.
alternately choose successively smaller elements of
wins if the intersection of this sequence contains a point in