# Meagre set

(Redirected from Meager set)

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. A topological space T is called meagre if it is a meager subset of itself; otherwise, it is called nonmeagre.

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. 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 numbers, 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. A set that is not meagre is called nonmeagre and is said to be of the second category. Note that the notions of a comeagre set and a nonmeagre set are not equivalent.

## Definition

Throughout, ${\displaystyle X}$  will be a topological space.

A subset ${\displaystyle B\subseteq X}$  of a topological space ${\displaystyle X}$  is called nowhere dense or rare in ${\displaystyle X}$  if its closure has empty interior. Equivalently, ${\displaystyle B}$  is nowhere dense in ${\displaystyle X}$  if for each open set ${\displaystyle U\subseteq X,}$  the set ${\displaystyle B\cap U}$  is not dense in ${\displaystyle U.}$

A closed subset of ${\displaystyle X}$  is nowhere dense in ${\displaystyle X}$  if and only if its topological interior in ${\displaystyle X}$  is empty.

A subset of a topological space ${\displaystyle X}$  is said to be meagre in ${\displaystyle X,}$  a meagre subset of ${\displaystyle X,}$  or of the first category in ${\displaystyle X}$  if it is a countable union of nowhere dense subsets of ${\displaystyle X.}$  A subset is of the second category or nonmeagre in ${\displaystyle X}$  if it is not of first category in ${\displaystyle X.}$

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

Warning: If ${\displaystyle S\subseteq X}$  is a subset of ${\displaystyle X}$  then ${\displaystyle S}$  being a "meagre subspace" of ${\displaystyle X}$  means that when ${\displaystyle S}$  is endowed with the subspace topology (induced on it by ${\displaystyle X}$ ) then ${\displaystyle S}$  is a meagre topological space (that is, ${\displaystyle S}$  is a meagre subset of ${\displaystyle S}$ ). In contrast, ${\displaystyle S}$  being a "meagre subset" of ${\displaystyle X}$  means that ${\displaystyle S}$  is equal to a countable union of nowhere dense subsets of ${\displaystyle X.}$  The same applies to nonmeager subsets and subspaces.

For example, if ${\displaystyle S:=\mathbb {N} }$  is the set of all positive integers then ${\displaystyle S}$  is a meager subset of ${\displaystyle \mathbb {R} }$  but not a meager subspace of ${\displaystyle \mathbb {R} .}$  If ${\displaystyle x\in X}$  is not an isolated point of a T1 space ${\displaystyle X}$  (meaning that ${\displaystyle \{x\}}$  is not an open subset of ${\displaystyle X}$ ) then ${\displaystyle \{x\}}$  is a meager subspace of ${\displaystyle X}$  but not a meager subset of ${\displaystyle X.}$

A subset ${\displaystyle A\subseteq X}$  is comeagre in ${\displaystyle X}$  if its complement ${\displaystyle X\setminus A}$  is meagre in ${\displaystyle X.}$  Equivalently, it is equal to an intersection of countably many sets, each of whose topological interior is a dense subset of ${\displaystyle X.}$  This use of the prefix "co" is consistent with its use in other terms such as "cofinite".

Importantly, being of the second category is not the same as being comeagre — a set may be neither meagre nor comeagre (in this case it will be of second category).

## Examples and sufficient conditions

Let ${\displaystyle X}$  be a topological space.

Meagre subsets and subspaces

• A singleton subset ${\displaystyle \{x\}\subseteq X}$  is always a non-meagre subspace of ${\displaystyle X}$  (that is, it is a non-meagre topological space). If ${\displaystyle x}$  is an isolated point of ${\displaystyle X}$  then ${\displaystyle \{x\}}$  is also a non-meagre subset of ${\displaystyle X}$ ; the converse holds if ${\displaystyle X}$  is a T1 space.
• Any subset of a meagre set is a meagre set.[1]
• Every nowhere dense subset is a meagre set.[1]
• The union of countably many meagre sets is also a meagre set.[1]
• Any closed subset of ${\displaystyle X}$  whose interior in ${\displaystyle X}$  is empty is of the first category of ${\displaystyle X}$  (that is, it is a meager subset of ${\displaystyle X}$ ). Thus a closed subset of ${\displaystyle X}$  that is of the second category in ${\displaystyle X}$  must have non-empty interior in ${\displaystyle X.}$ [2]
• A countable Hausdorff space without isolated points is meagre.[3]
• Any topological space that contains an isolated point is non-meagre.[3]
• Any discrete space is non-meagre.[3]
• Every Baire space is non-meagre but there exist non-meagre spaces that are not Baire spaces.[3]
• The set ${\displaystyle S=(\mathbb {Q} \times \mathbb {Q} )\cup \mathbb {R} }$  is a meagre subset of ${\displaystyle \mathbb {R} ^{2}}$  even though ${\displaystyle \mathbb {R} }$  is a non-meagre subspace (that is, ${\displaystyle \mathbb {R} }$  is not a meagre topological space).[3]
• 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.
• The Cantor set is meagre as a subset of the reals, but not as a subset of itself, since it is a complete metric space and is thus a Baire space, by the Baire category theorem.
• If ${\displaystyle h:X\to X}$  is a homeomorphism then a subset ${\displaystyle S\subseteq X}$  is meagre if and only if ${\displaystyle h(S)}$  is meagre.[1]

Comeagre subset

• Any superset of a comeagre set is comeagre.
• the intersection of countably many comeagre sets is comeagre.
• This follows from the fact that a countable union of countable sets is countable.

### Function spaces

The set of functions that have a derivative at some point is a meagre set in the space of all continuous functions.[4]

## Properties

• Banach Category Theorem: In any space ${\displaystyle X,}$  the union of any countable family of open sets of the first category is of the first category.[5]
• A non-meagre locally convex topological vector space is a barreled space.[3]
• A closed subset of ${\displaystyle X}$  that is of the second category in ${\displaystyle X}$  must have non-empty interior in ${\displaystyle X.}$ [2]
• If ${\displaystyle B\subseteq X}$  is of the second category in ${\displaystyle X}$  and if ${\displaystyle S_{1},S_{2},\ldots }$  are subsets of ${\displaystyle X}$  such that ${\displaystyle B\subseteq \bigcup _{n\in \mathbb {N} }S_{n},}$  then at least one ${\displaystyle S_{n}}$  is of the second category in ${\displaystyle X.}$

### Meagre subsets and Lebesgue measure

A meagre set need not have measure zero. There exist nowhere dense subsets (which are thus meagre subsets) that have positive Lebesgue measure.[3]

### 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 (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.

## Banach–Mazur game

Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game. Let ${\displaystyle Z}$  be a topological space, ${\displaystyle {\mathcal {W}}}$  be a family of subsets of ${\displaystyle Z}$  that have nonempty interiors such that every nonempty open set has a subset belonging to ${\displaystyle {\mathcal {W}},}$  and ${\displaystyle Z}$  be any subset of ${\displaystyle Z.}$  Then there is a Banach–Mazur game corresponding to ${\displaystyle X,{\mathcal {W}},Z.}$  In the Banach–Mazur game, two players, ${\displaystyle P}$  and ${\displaystyle Q,}$  alternately choose successively smaller elements of ${\displaystyle {\mathcal {W}}}$  to produce a sequence ${\displaystyle W_{1}\supseteq W_{2}\supseteq W_{3}\supseteq \cdots .}$  Player ${\displaystyle P}$  wins if the intersection of this sequence contains a point in ${\displaystyle X}$ ; otherwise, player ${\displaystyle Q}$  wins.

Theorem: For any ${\displaystyle {\mathcal {W}}}$  meeting the above criteria, player ${\displaystyle Q}$  has a winning strategy if and only if ${\displaystyle X}$  is meagre.