Meagre set

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In the mathematical fields of general topology and descriptive set theory, a meagre set (also called a meager set or a set of first category) is a set that, considered as a subset of a (usually larger) topological space, is in a precise sense small or negligible. A topological space T is called meagre if it is a meager subset of itself; otherwise, it is called nonmeagre.

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. General topologists use the term Baire space to refer to a broad class of topological spaces on which the notion of meagre set is not trivial (in particular, the entire space is not meagre). Descriptive set theorists mostly study meagre sets as subsets of the real numbers, or more generally any Polish space, and reserve the term Baire space for one particular Polish space.

The complement of a meagre set is a comeagre set or residual set. A set that is not meagre is called nonmeagre and is said to be of the second category. Note that the notions of a comeagre set and a nonmeagre set are not equivalent.

Definition[edit]

Throughout, will be a topological space.

A subset of a topological space is called nowhere dense and rare in if its closure has empty interior. Equivalently, is nowhere dense in if for each open set the set is not dense in

A closed subset of is nowhere dense in if and only if its topological interior in is empty.

A subset of a topological space is said to be meagre in a meagre subset of and of the first category in if it is a countable union of nowhere dense subsets of A subset that is not of first category in is said to be nonmeagre in a nonmeagre subset of and of the second category in

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

Warning: If is a subset of then being a "meagre subspace" of means that when is endowed with the subspace topology (induced on it by ) then is a meagre topological space (that is, is a meagre subset of ). In contrast, being a "meagre subset" of means that is equal to a countable union of nowhere dense subsets of 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 is the set of all positive integers then is a meager subset of but not a meager subspace of If is not an isolated point of a T1 space (meaning that is not an open subset of ) then is a meager subspace of but not a meager subset of

A subset is called a residual subset of and is said to be comeagre 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 an intersection of countably many sets, each of whose topological interior is a dense subset of

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).

Sufficient conditions[edit]

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 is a homeomorphism then a subset is meagre if and only if is meagre.[2]

Every nowhere dense subset is a meagre set.[2] Consequently, any closed subset of whose interior in is empty is of the first category of (that is, it is a meager subset of ). Thus a closed subset of that is of the second category in must have non-empty interior in [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).

Examples[edit]

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 is always a nonmeagre subspace of (that is, it is a nonmeagre topological space). If is an isolated point of (meaning that is an open subset) then is also a nonmeagre subset of ; the converse holds if is a T1 space.

The set is a meagre subset of even though is a nonmeagre subspace (that is, 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 that has positive Lebesgue measure.

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]

Properties[edit]

Banach category theorem[5] — In any space the union of any countable family of open sets of the first category is of the first category.

If is of the second category in and if are subsets of such that then at least one is of the second category in

A closed subset of that is of the second category in must have non-empty interior in [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 subsets and Lebesgue measure[edit]

A meagre set need not have measure zero. There exist nowhere dense subsets (which are thus meagre subsets) that have positive Lebesgue measure.[1]

Relation to Borel hierarchy[edit]

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.

Banach–Mazur game[edit]

Meagre sets have a useful alternative characterization in terms of the Banach–Mazur game. Let be a topological space, be a family of subsets of that have nonempty interiors such that every nonempty open set has a subset belonging to and be any subset of Then there is a Banach–Mazur game corresponding to In the Banach–Mazur game, two players, and alternately choose successively smaller elements of to produce a sequence Player wins if the intersection of this sequence contains a point in ; otherwise, player wins.

Theorem — For any meeting the above criteria, player has a winning strategy if and only if is meagre.

See also[edit]

Notes[edit]

  1. ^ This distinction between "subspace" and "subset" is a consequence of the fact that in general topology, the word "space" means "topological space", which is a pair consisting of a set and topology, and (similarly) the word "subspace" means "topological subspace"; consequently, "subspace of " refers to the pair consisting of the subset together with the subspace topology that it inherits from whereas "subset of " refers only to the set. Consequently, if the subset lacks any topology then " is meagre of subset of " is not well-defined, leaving " is a meagre subset of " as the only possible meaning of " is meagre". But if is endowed with a topology then (by definition) " is meagre" means " is a meagre subset of " Saying " is a meagre subspace of " is just a combination of the following two statements: (1) " is a subspace of ", which by definition means that is endowed with a topology that is equal to the subspace topology induced by on it by (denote this topology by ), and (2) " is a meagre space", which by definition means " is a meagre subset of ". However, if happens to be endowed with a topology (say ) then the statement " is a meagre subset of " does not mean " is a meagre subset of " because in this statement, is being considered as a set (and not as a topological space). The same is true of a statement such as "let be a subspace of that is a meagre subset of " and its more succinct equivalent "let be a subspace that is meagre in " (note that the meaning is completely changed without the words "in ").
  1. ^ a b c d e f g h Narici & Beckenstein 2011, pp. 371–423.
  2. ^ a b c Rudin 1991, p. 43.
  3. ^ a b Rudin 1991, pp. 42–43.
  4. ^ Banach, S. (1931). "Über die Baire'sche Kategorie gewisser Funktionenmengen". Studia Math. 3 (1): 174–179. doi:10.4064/sm-3-1-174-179.
  5. ^ Oxtoby, John C. (1980). "The Banach Category Theorem". Measure and Category (Second ed.). New York: Springer. pp. 62–65. ISBN 0-387-90508-1.

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