Meagre set

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

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

A topological space is said to be meagre (respectively, nonmeagre) if it is a meagre (respectively, 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 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.[note 1]

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 called a residual subset of ${\displaystyle X}$ and is said to be comeagre in ${\displaystyle X}$ if its complement ${\displaystyle X\setminus A}$ is meagre in ${\displaystyle X}$ (this use of the prefix "co" is consistent with its use in other terms such as "cofinite"). A subset is comeagre in ${\displaystyle X}$ if and only if it is equal to an intersection of countably many sets, each of whose topological interior is a dense subset of ${\displaystyle X.}$

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).

Every Baire space is nonmeagre but there exist nonmeagre spaces that are not Baire spaces.[1] Since complete (pseudo)metric spaces as well as Hausdorff locally compact spaces are Baire spaces, they are also nonmeagre spaces.[1]

Any subset of a meagre set is a meagre set, as is the union of countably many meagre sets.[2] 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.[2]

Every nowhere dense subset is a meagre set.[2] Consequently, 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 ${\displaystyle X.}$[3]

Any topological space that contains an isolated point (such as any non-empty discrete space) is nonmeagre.[1]

Comeagre subset

Any superset of a comeagre set is comeagre, as is the intersection of countably many comeagre sets (because countable union of countable sets is countable).

Every non-empty discrete space is nonmeagre because this is true of any topological space that contains one or more isolated points.[1]

Meagre subsets and subspaces

The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space and it is also a meagre subspace.

A singleton subset ${\displaystyle \{x\}\subseteq X}$ is always a nonmeagre subspace of ${\displaystyle X}$ (that is, it is a nonmeagre topological space). If ${\displaystyle x}$ is an isolated point of ${\displaystyle X}$ (meaning that ${\displaystyle \{x\}}$ is an open subset) then ${\displaystyle \{x\}}$ is also a nonmeagre subset of ${\displaystyle X}$; the converse holds if ${\displaystyle X}$ is a T₁ space.

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 nonmeagre subspace (that is, ${\displaystyle \mathbb {R} }$ is not a meagre topological space).[1] A countable Hausdorff space without isolated points is meagre, whereas any topological space that contains an isolated point is nonmeagre.[1] 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.

The Smith–Volterra–Cantor set is a closed nowhere dense (and thus meagre) subset of the unit interval ${\displaystyle [0,1]}$ that has positive Lebesgue measure.

There is a subset ${\displaystyle H}$ of the real numbers such that for every nonempty open set ${\displaystyle U\subseteq \mathbb {R} }$, neither ${\displaystyle U\cap H}$ nor ${\displaystyle U\backslash H}$ is meager in ${\displaystyle U}$.

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]

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 ${\displaystyle B\subseteq S_{1}\cup S_{2}\cup \cdots }$ then at least one ${\displaystyle S_{n}}$ is of the second category in ${\displaystyle X.}$

A closed subset of ${\displaystyle X}$ that is of the second category in ${\displaystyle X}$ must have non-empty interior in ${\displaystyle X}$[3] (because otherwise it would be nowhere dense and thus of the first category).

A nonmeagre locally convex topological vector space is a barreled space.[1]

Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game. Let ${\displaystyle X}$ be a topological space, ${\displaystyle {\mathcal {W}}}$ be a family of subsets of ${\displaystyle X}$ 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 ${\displaystyle X.}$ 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.

• Baire category theorem – On topological spaces where the intersection of countably many dense open sets is dense
• Baire space – Concept in topology
• Generic property, for analogs to residual
• Negligible set, for analogs to meagre
• Nowhere dense set
• Property of Baire

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.

• Is there a measure zero set which isn't meagre?