Meagre set
Encyclopedia
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
. The meagre subsets of a fixed space form a sigma-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 number
s, 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.
Dually
, a comeagre set is one whose complement
is meagre, or equivalently, the intersection
of countably many sets with dense interiors.
A subset B of X is nowhere dense if there is no neighbourhood on which B is dense
: for any nonempty open set U in X, there is a nonempty open set V contained in U such that V and B are disjoint.
The complement of a nowhere dense set is a dense set, but not every dense set is of this form. More precisely, the complement of a nowhere dense set is a set with dense interior
.
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 Gδ set (countable intersection of open
sets), but contains a dense Gδ set formed from dense open sets.
. If Y is a topological space, W is a family of subsets of Y which have nonempty interior such that every nonempty open set has a subset in W, and X is any subset of Y, then there is a Banach-Mazur game corresponding to X, Y, W. In the Banach-Mazur game, two players, P1 and P2, alternate choosing successively smaller (in terms of the subset relation) elements of W to produce a descending sequence
If the intersection of this sequence contains a point in X, P1 wins; otherwise, P2 wins. If W is any family of sets meeting the above criteria, then P2 has a winning strategy if and only if X is meagre.
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
and descriptive set theory
Descriptive set theory
In mathematical logic, descriptive set theory is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces...
, a meagre set (also called a meager set or a set of first category) is a set that, considered as a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of a (usually larger) topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, is in a precise sense small or negligible
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 integral of a measurable function.Negligible sets define...
. The meagre subsets of a fixed space form a sigma-ideal
Sigma-ideal
In mathematics, particularly measure theory, a σ-ideal of a sigma-algebra is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is perhaps in probability theory.Let be a measurable space...
of subsets; that is, any subset of a meagre set is meagre, and the union of countably
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
many meagre sets is meagre.
General topologists use the term Baire space
Baire space
In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
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 number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, or more generally any Polish space
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
, and reserve the term Baire space
Baire space (set theory)
In set theory, the Baire space is the set of all infinite sequences of natural numbers with a certain topology. This space is commonly used in descriptive set theory, to the extent that its elements are often called “reals.” It is often denoted B, N'N, or ωω...
for one particular Polish space.
The complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
of a meagre set is a comeagre set or residual set.
Definition
Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.Dually
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
, a comeagre set is one whose complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
is meagre, or equivalently, the intersection
Intersection (set theory)
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....
of countably many sets with dense interiors.
A subset B of X is nowhere dense if there is no neighbourhood on which B is dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
: for any nonempty open set U in X, there is a nonempty open set V contained in U such that V and B are disjoint.
The complement of a nowhere dense set is a dense set, but not every dense set is of this form. More precisely, the complement of a nowhere dense set is a set with dense interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
.
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 Fσ set (countable union of closed sets), but is always contained in an Fσ 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
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
(contains a dense open set), a comeagre set need not be a Gδ set (countable intersection of open
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
sets), but contains a dense Gδ set formed from dense open sets.
Terminology
A meagre set is also called a set of first category; a nonmeagre set (that is, a set that is not meagre) is also called a set of second category. Second category does not mean comeagre – a set may be neither meagre nor comeagre (in this case it will be of second category).Properties
- Any subset of a meagre set is meagre; any superset of a comeagre sets is comeagre.
- The union of countable many meagre sets is also meagre; the intersection of countably many comeagre sets is comeagre.
-
- This follows from the fact that a countable union of countable sets is countable.
Banach–Mazur game
Meagre sets have a useful alternative characterization in terms of the Banach–Mazur gameBanach–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 . The concept of a Banach–Mazur game is closely related to the concept of Baire spaces...
. If Y is a topological space, W is a family of subsets of Y which have nonempty interior such that every nonempty open set has a subset in W, and X is any subset of Y, then there is a Banach-Mazur game corresponding to X, Y, W. In the Banach-Mazur game, two players, P1 and P2, alternate choosing successively smaller (in terms of the subset relation) elements of W to produce a descending sequence
If the intersection of this sequence contains a point in X, P1 wins; otherwise, P2 wins. If W is any family of sets meeting the above criteria, then P2 has a winning strategy if and only if X is meagre.Subsets of the reals
- The rational numbers are meagre as a subset of the reals and as a space – they are not a Baire spaceBaire spaceIn mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
. - The Cantor setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
is meagre as a subset of the reals, but not as a space, since it is a complete metric space – it is thus a Baire spaceBaire spaceIn mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
, by the Baire category theoremBaire category theoremThe Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
.
Function spaces
- The set of functions which have a derivative at some point is a meagre set in the space of all continuous functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s.
See also
- Baire category theoremBaire category theoremThe Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
- Generic propertyGeneric propertyIn mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic...
, for analogs to residual - Negligible setNegligible setIn 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 integral of a measurable function.Negligible sets define...
, for analogs to meagre