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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, geometri ...
, 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, norm, topology, etc.) and the linear functions defined o ...
.


Definitions

Throughout, X 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 X is called X, a of X, or of the in X 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 X (where a nowhere dense set is a set whose closure has empty interior). The qualifier "in X" can be omitted if the ambient space is fixed and understood from context. A subset that is not meagre in X is called X, a of X, or of the in X. A topological space is called (respectively, ) if it is a meagre (respectively, nonmeagre) subset of itself. A subset A of X is called in X, or in X, 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 ...
X \setminus A is meagre in X. (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 X if and only if it is equal to a countable intersection of sets, each of whose interior is dense in X. The notions of nonmeagre and comeagre should not be confused. If the space X is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets. If the space X 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 A of a topological space X 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 X, 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 A can also be called a ''meagre subspace'' of X, meaning a meagre space when given the subspace topology. Importantly, this is not the same as being meagre in the whole space X. (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 X is meagre. Consequently, any closed subset with empty interior is meagre. Thus a closed nonmeagre subset of X 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 A\subseteq Y\subseteq X, where Y 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 X. The set A may be meagre in X without being meagre in Y. However the following results hold: * If A is meagre in Y, then A is meagre in X. * If Y is open in X, then A is meagre in Y if and only if A is meagre in X. * If Y is dense in X, then A is meagre in Y if and only if A is meagre in X. And correspondingly for nonmeagre sets: * If A is nonmeagre in X, then A is nonmeagre in Y. * If Y is open in X, then A is nonmeagre in Y if and only if A is nonmeagre in X. * If Y is dense in X, then A is nonmeagre in Y if and only if A is nonmeagre in X. In particular, every subset of X that is meagre in itself is meagre in X. Every subset of X that is nonmeagre in X is nonmeagre in itself. And for an open set or a dense set in X, being meagre in X 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 X is nonmeagre if and only if every countable intersection of dense open sets in X 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 X, the union of an arbitrary family of meagre open sets is a meagre set.


Meagre subsets and Lebesgue measure

A meagre set in \R need not have Lebesgue measure zero, and can even have full measure. For example, in the interval ,1/math>
fat Cantor set In nutrition, biology, and chemistry, fat usually means any ester of fatty acids, or a mixture of such compounds, most commonly those that occur in living beings or in food. The term often refers specifically to triglycerides (triple e ...
s are closed nowhere dense and they can be constructed with a measure arbitrarily close to 1. The union of a countable number of such sets with measure approaching 1 gives a meagre subset of ,1/math> with measure 1. Dually, there can be nonmeagre sets with measure zero. The complement of any meagre set of measure 1 in ,1/math> (for example the one in the previous paragraph) has measure 0 and is comeagre in ,1 and hence nonmeagre in ,1/math> since ,1/math> is a Baire space. Here is another example of a nonmeagre set in \R with measure 0: :\bigcap_^\bigcup_^ \left(r_-\left(\tfrac\right)^, r_+\left(\tfrac\right)^\right) where \left(r_n\right)_^ 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_ set (countable union of closed sets), but is always contained in an F_ 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_ set (countable intersection of
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
sets), but contains a dense G_ set formed from dense open sets.


Examples

The empty set is a meagre subset of every topological space. In the nonmeagre space X= ,1cup( ,3cap\Q) the set ,3cap\Q is meagre. The set ,1/math> is nonmeagre and comeagre. In the nonmeagre space X= ,2/math> the set ,1/math> is nonmeagre. But it is not comeagre, as its complement (1,2] is also nonmeagre. A countable T1 space, T1 space without
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 meagre. So it is also meagre in any space that contains it as a subspace. For example, \Q is both a meagre subspace of \R (that is, meagre in itself with the subspace topology induced from \R) and a meagre subset of \R. The Cantor set is nowhere dense in \R and hence meagre in \R. But it is nonmeagre in itself, since it is a complete metric space. The set ( ,1cap\Q)\cup\ is not nowhere dense in \R, but it is meagre in \R. It is nonmeagre in itself (since as a subspace it contains an isolated point). The line \R\times\ is meagre in the plane \R^2. But it is a nonmeagre subspace, that is, it is nonmeagre in itself. The space (\Q \times \Q) \cup (\R\times\) (with the topology induced from \R^2) is meagre. Its meagre subset \R\times\ is nonmeagre in itself. There is a subset H of the real numbers \R that splits every nonempty open set into two nonmeagre sets. That is, for every nonempty open set U\subseteq \mathbb, the sets U\cap H and U \setminus H are both nonmeagre. In the space C( ,1 of continuous real-valued functions on ,1/math> with the topology of
uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
, the set A of continuous real-valued functions on ,1/math> that have a derivative at some point is meagre. Since C( ,1 is a complete metric space, it is nonmeagre. So the complement of A, which consists of the continuous real-valued
nowhere differentiable function In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstras ...
s on ,1 is comeagre and nonmeagre. In particular that set is not empty. This is one way to show the existence of continuous nowhere differentiable functions.


Banach–Mazur game

Meagre sets have a useful alternative characterization in terms of the
Banach–Mazur game In general topology, set theory and game theory, a Banach– Mazur game is a topological game played by two players, trying to pin down elements in a set (space). The concept of a Banach–Mazur game is closely related to the concept of Baire s ...
. Let Y be a topological space, \mathcal be a family of subsets of Y that have nonempty interiors such that every nonempty open set has a subset belonging to \mathcal, and X be any subset of Y. Then there is a Banach–Mazur game MZ(X, Y, \mathcal). In the Banach–Mazur game, two players, P and Q, alternately choose successively smaller elements of \mathcal to produce a sequence W_1 \supseteq W_2 \supseteq W_3 \supseteq \cdots. Player P wins if the intersection of this sequence contains a point in X; otherwise, player Q wins.


See also

* * , for analogs to residual * , for analogs to meagre *


Notes


Bibliography

* * * * * General topology Descriptive set theory