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Concept# Meagre set

Summary

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 , which is used in the proof of several fundamental results of functional analysis.
Throughout, will be a topological space.
The definition of meagre set uses the notion of a nowhere dense subset of that is, a subset of whose closure has empty interior. See the corresponding article for more details.
A subset of is called a of or of the in if it is a countable union of nowhere dense subsets of . Otherwise, the subset is called a of or of the in The qualifier "in " can be omitted if the ambient space is fixed and understood from context.
A topological space is called (respectively, ) if it is a meagre (respectively, nonmeagre) subset of itself.
A subset of is called in or in if its complement is meagre in . (This use of the prefix "co" is consistent with its use in other terms such as "cofinite".)
A subset is comeagre in if and only if it is equal to a countable intersection of sets, each of whose interior is dense in
Remarks on terminology
The notions of nonmeagre and comeagre should not be confused. If the space is meagre, every subset is both meagre and comeagre, and there are no nonmeagre sets. If the space 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.

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In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function. Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere.

In topology and related areas of mathematics, a subset A of a topological space X is said to be dense in X if every point of X either belongs to A or else is arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, is dense in if the smallest closed subset of containing is itself.

In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. For example, the integers are nowhere dense among the reals, whereas the interval (0, 1) is not nowhere dense. A countable union of nowhere dense sets is called a meagre set. Meagre sets play an important role in the formulation of the , which is used in the proof of several fundamental results of functional analysis.

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