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 whose closures have empty interior. 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.
Definitions Throughout, X {\displaystyle X} will be a topological space. The definition of meagre set uses the notion of a nowhere dense subset of X , {\displaystyle X,} that is, a subset of X {\displaystyle X} whose closure has empty interior. See the corresponding article for more details. A subset of X {\displaystyle X} is called meagre in X , {\displaystyle X,} a meagre subset of X , {\displaystyle X,} or of the first category in X {\displaystyle X} if it is a countable union of nowhere dense subsets of X {\displaystyle X} . Otherwise, the subset is called nonmeagre in X , {\displaystyle X,} a nonmeagre subset of X , {\displaystyle X,} or of the second category in X . {\displaystyle X.} The qualifier "in X {\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 A {\displaystyle A} of X {\displaystyle X} is called comeagre in X , {\displaystyle X,} or residual in X , {\displaystyle X,} if its complement X ∖ A {\displaystyle X\setminus A} is meagre in X {\displaystyle X} . (This use of the prefix "co" is consistent with its use in other terms such as "cofinite".) A subset is comeagre in X {\displaystyle X} if and only if it is equal to a countable intersection of sets, each of whose interior is dense in X . {\displaystyle X.} Remarks on terminology The notions of nonmeagre and comeagre should not be confused. If the space X {\displaystyle X} is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets. If the space X {\displaystyle X} is nonmeagre, no set is at the same time meagre and comeagre, 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. As an additional point of terminology, if a subset A {\displaystyle A} of a topological space X {\displaystyle X} is given the subspace topology induced from X {\displaystyle X} , one can talk about it being a meagre space, namely being a meagre subset of itself (when considered as a topological space in its own right). In this case A {\displaystyle A} can also be called a meagre subspace of X {\displaystyle X} , meaning a meagre space when given the subspace topology. Importantly, this is not the same as being meagre in the whole space X {\displaystyle X} . (See the Properties and Examples sections below for the relationship between the two.) Similarly, a nonmeagre subspace will be a set that is nonmeagre in itself, which is not the same as being nonmeagre in the whole space. Be aware however that in the context of topological vector spaces some authors may use the phrase "meagre/nonmeagre subspace" to mean a vector subspace that is a meagre/nonmeagre set relative to the whole space. 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.
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