Топологиялық кеңістікте "кішкентай" жиын

"Small" subset of a topological space
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.

The empty set is always a closed nowhere dense (and thus meagre) subset of every topological space. In the nonmeagre space the set is meagre. The set is nonmeagre and comeagre. In the nonmeagre space the set is nonmeagre. But it is not comeagre, as its complement 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, is both a meagre subspace of (that is, meagre in itself with the subspace topology induced from ) and a meagre subset of

The Cantor set is nowhere dense in and hence meagre in But it is nonmeagre in itself, since it is a complete metric space. The set is not nowhere dense in , but it is meagre in It is nonmeagre in itself (since as a subspace it contains an isolated point). The line is meagre in the plane But it is a nonmeagre subspace, that is, it is nonmeagre in itself. The set is a meagre subset of even though its meagre subset is a nonmeagre subspace (that is, is not a meagre topological space). A countable Hausdorff space without isolated points is meagre, whereas any topological space that contains an isolated point is nonmeagre. 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. 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. There is a subset of the real numbers that splits every nonempty open set into two nonmeagre sets. That is, for every nonempty open set , the sets and are both nonmeagre. In the space of continuous real valued functions on with the topology of uniform convergence, the set of continuous real valued functions on that have a derivative at some point is meagre. Since is a complete metric space, it is nonmeagre. So the complement of , which consists of the continuous real valued nowhere differentiable functions on is comeagre and nonmeagre. In particular that set is not empty. This is one way to show the existence of continuous nowhere differentiable functions.

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 set (countable union of closed sets), but is always contained in an 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 set (countable intersection of open sets), but contains a dense set formed from dense open sets.

Many arguments about meagre sets also apply to null sets, i. e. sets of Lebesgue measure 0. The Erdos–Sierpinski duality theorem states that if the continuum hypothesis holds, there is an involution from reals to reals where the image of a null set of reals is a meagre set, and vice versa. In fact, the image of a set of reals under the map is null if and only if the original set was meagre, and vice versa.