In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
general topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, a meagre set (also called a meager set or a set of first category) is a
subset of a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
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. See below for definitions of other related terms.
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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
of
countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
many meagre sets is meagre.
Meagre sets play an important role in the formulation of the notion of
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
and of the
Baire category theorem, which is used in the proof of several fundamental results of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
.
Definitions
Throughout,
will be a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
.
A subset of
is called
a of
or of the in
if it is a countable union of
nowhere dense
In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. ...
subsets of
(where a nowhere dense set is a set whose closure has empty interior). The qualifier "in
" can be omitted if the ambient space is fixed and understood from context.
A subset that is not meagre in
is called
a of
or of the in
A topological space is called (respectively, ) if it is a meagre (respectively, nonmeagre) subset of itself.
A subset
of
is called in
or in
if its
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-clas ...
is meagre in
. (This use of the prefix "co" is consistent with its use in other terms such as "
cofinite
In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is coco ...
".)
A subset is comeagre in
if and only if it is equal to a countable
intersection of sets, each of whose interior is dense in
The notions of nonmeagre and comeagre should not be confused. If the space
is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets. If the space
is nonmeager, no set is at the same time meagre and comeager, 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.
As an additional point of terminology, if a subset
of a topological space
is given the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced from
, one can talk about it being a meagre space, namely being a meagre subset of itself (when considered as a topological space in its own right). In this case
can also be called a ''meagre subspace'' of
, meaning a meagre space when given the subspace topology. Importantly, this is not the same as being meagre in the whole space
. (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
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
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
René ('' born again'' or ''reborn'' in French) is a common first name in French-speaking, Spanish-speaking, and German-speaking countries. It derives from the Latin name Renatus.
René is the masculine form of the name ( Renée being the femin ...
in his thesis of 1899. The ''meagre'' terminology was introduced by
Bourbaki in 1948.
Properties
Every nowhere dense subset of
is meagre. Consequently, any closed subset with empty interior is meagre. Thus a closed nonmeagre subset of
must have nonempty interior.
(1) Any subset of a meagre set is meagre; (2) any countable union of meagre sets is meagre. Thus the meagre subsets of a fixed space form a
σ-ideal of subsets, a suitable notion of
negligible set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.
As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integr ...
. And, equivalently to (1), any superset of a nonmeagre set is nonmeagre.
Dually, (1) any superset of a comeagre set is comeagre; (2) any countable intersection of comeagre sets is comeagre.
Suppose
where
has the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
induced from
The set
may be meagre in
without being meagre in
However the following results hold:
* If
is meagre in
then
is meagre in
* If
is open in
then
is meagre in
if and only if
is meagre in
* If
is dense in
then
is meagre in
if and only if
is meagre in
And correspondingly for nonmeagre sets:
* If
is nonmeagre in
then
is nonmeagre in
* If
is open in
then
is nonmeagre in
if and only if
is nonmeagre in
* If
is dense in
then
is nonmeagre in
if and only if
is nonmeagre in
In particular, every subset of
that is meagre in itself is meagre in
Every subset of
that is nonmeagre in
is nonmeagre in itself. And for an open set or a dense set in
being meagre in
is equivalent to being meagre in itself, and similarly for the nonmeagre property.
Any topological space that contains an
isolated point
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
is nonmeagre (because no set containing the isolated point can be nowhere dense). In particular, every nonempty
discrete space is nonmeagre.
A topological space
is nonmeagre if and only if every countable intersection of dense open sets in
is nonempty.
Every nonempty
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
is nonmeagre. In particular, by the
Baire category theorem every nonempty
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
and every nonempty
locally compact Hausdorff In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which e ...
space is nonmeagre.
In any topological space
the union of an arbitrary family of meagre open sets is a meagre set.
Meagre subsets and Lebesgue measure
A meagre set in
need not have
Lebesgue measure zero, and can even have full measure. For example, in the interval