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 a countable union of subsets that are not dense in any non-empty open set. Thus meager sets are, in a sense, "small", being small unions of small subsets.

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.

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

The definition of meagre set uses the notion of a nowhere dense subset of ${\displaystyle X,}$ that is, a subset of ${\displaystyle X}$ whose closure has empty interior. See the corresponding article for more details.

A subset of ${\displaystyle X}$ is called 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 ${\displaystyle X}$. Otherwise, the subset is called nonmeagre in ${\displaystyle X,}$ a nonmeagre subset of ${\displaystyle X,}$ or of the second category in ${\displaystyle X.}$ The qualifier "in ${\displaystyle X}$" can be omitted if the ambient space is fixed and understood from context.

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

A subset ${\displaystyle A}$ of ${\displaystyle X}$ is called comeagre in ${\displaystyle X,}$ or residual 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 a countable intersection of sets, each of whose interior is dense in ${\displaystyle X.}$