Meagre set

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. 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 of countably many meagre sets is meagre.

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.

Definitions

Throughout, $X$ will be a topological space.

A subset of $X$ is called meagre in $X,$ a meagre subset of $X,$ or of the first category in $X$ if it is a countable union of nowhere dense 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 nonmeagre in $X,$ a nonmeagre subset of $X,$ or of the second category in $X.$

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

A subset $A$ of $X$ is called comeagre in $X,$ or residual in $X,$ if its complement $X\setminus A$ is meagre in $X$. (This use of the prefix "co" is consistent with its use in other terms such as "cofinite".) 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.

The terms first category and second category were the original ones used by René Baire in his thesis of 1899. The meagre terminology was introduced by Bourbaki in 1948.

Meager subsets versus meager subspaces

It may be important to distinguish nonmeagre subspaces from nonmeagre subsets. If $S\subseteq X$ is a subset of $X$ then $S$ being a "meagre subspace" of $X$ means that when $S$ is endowed with the subspace topology (induced on it by $X$) then $S$ is a meagre topological space (that is, $S$ is a meagre subset of $S$). In contrast, $S$ being a "meagre subset" of $X$ means that $S$ is equal to a countable union of nowhere dense subsets of $X.$ The same warning applies to nonmeagre subsets versus nonmeagre subspaces. More details on how to tell these notions apart (and why the slight difference in these terms is reasonable) are given in this footnote.

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

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. 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 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 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 is nonmeagre. In particular, by the Baire category theorem every nonempty complete metric space and every nonempty locally compact Hausdorff space is nonmeagre.

Banach category theorem: 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 $\mathbb {R}$ need not have Lebesgue measure zero, and can even have full measure. For example, in the interval $[0,1]$ fat Cantor sets 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 $[0,1]$ with measure $1.$

Dually, there can be nonmeagre sets with measure zero. The complement of any meagre set of measure $1$ in $[0,1]$ (for example the one in the previous paragraph) has measure $0$ and is comeagre in $[0,1],$ and hence nonmeagre in $[0,1]$ since $[0,1]$ is a Baire space.

Here is another example of a nonmeagre set in $\mathbb {R}$ with measure $0$:

$\bigcap _{m=1}^{\infty }\bigcup _{n=1}^{\infty }\left(r_{n}-\left({\tfrac {1}{2}}\right)^{n+m},r_{n}+\left({\tfrac {1}{2}}\right)^{n+m}\right)$

where $\left(r_{n}\right)_{n=1}^{\infty }$ 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 $F_{\sigma }$ set (countable union of closed sets), but is always contained in an $F_{\sigma }$ 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 $G_{\delta }$ set (countable intersection of open sets), but contains a dense $G_{\delta }$ set formed from dense open sets.

Examples

The empty set is a meagre subset of every topological space.

In the nonmeagre space $X=[0,1]\cup ([2,3]\cap \mathbb {Q} )$ the set $[2,3]\cap \mathbb {Q}$ is meagre. The set $[0,1]$ is nonmeagre and comeagre.

In the nonmeagre space $X=[0,2]$ the set $[0,1]$ is nonmeagre. But it is not comeagre, as its complement $(1,2]$ is also nonmeagre.

A countable T₁ space without isolated point is meagre. So it is also meagre in any space that contains it as a subspace. For example, $\mathbb {Q}$ is both a meagre subspace of $\mathbb {R}$ (that is, meagre in itself with the subspace topology induced from $\mathbb {R}$) and a meagre subset of $\mathbb {R} .$

The Cantor set is nowhere dense in $\mathbb {R}$ and hence meagre in $\mathbb {R} .$ But it is nonmeagre in itself, since it is a complete metric space.

The line $\mathbb {R} \times \{0\}$ is meagre in the plane $\mathbb {R} ^{2}.$ But it is a nonmeagre subspace, that is, it is nonmeagre in itself.

The space $(\mathbb {Q} \times \mathbb {Q} )\cup (\mathbb {R} \times \{0\})$ (with the topology induced from $\mathbb {R} ^{2}$) is meagre. Its meagre subset $\mathbb {R} \times \{0\}$ is nonmeagre in itself.

There is a subset $H$ of the real numbers $\mathbb {R}$ that splits every nonempty open set into two nonmeagre sets. That is, for every nonempty open set $U\subseteq \mathbb {R}$, the sets $U\cap H$ and $U\setminus H$ are both nonmeagre.

In the space $C([0,1])$ of continuous real-valued functions on $[0,1]$ with the topology of uniform convergence, the set $A$ of continuous real-valued functions on $[0,1]$ that have a derivative at some point is meagre. Since $C([0,1])$ is a complete metric space, it is nonmeagre. So the complement of $A$, which consists of the continuous real-valued nowhere differentiable functions on $[0,1],$ 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. Let $Y$ be a topological space, ${\mathcal {W}}$ be a family of subsets of $Y$ that have nonempty interiors such that every nonempty open set has a subset belonging to ${\mathcal {W}},$ and $X$ be any subset of $Y.$ Then there is a Banach–Mazur game $MZ(X,Y,{\mathcal {W}}).$ In the Banach–Mazur game, two players, $P$ and $Q,$ alternately choose successively smaller elements of ${\mathcal {W}}$ 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.

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

See also

• Barrelled space – Type of topological vector space
• Generic property – Property holding for typical examples, for analogs to residual
• Negligible set – Mathematical set regarded as insignificant, for analogs to meagre
• Property of Baire – Difference of an open set by a meager set

Bibliography

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• Oxtoby, John C. (1980). Measure and Category. Springer Verlag.
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