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.

DefinitionEdit

Throughout,   will be a topological space.

A subset   of a topological space   is called nowhere dense or 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   or of the first category in   if it is a countable union of nowhere dense subsets of   A subset is of the second category or nonmeagre in   if it is not of first category in  

A topological space is called meagre (resp. nonmeagre) if it is a meagre (resp. 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 applies to nonmeager subsets and subspaces.

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 comeagre in   if its complement   is meagre in   Equivalently, it is equal to an intersection of countably many sets, each of whose topological interior is a dense subset of   This use of the prefix "co" is consistent with its use in other terms such as "cofinite".

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).

Examples and sufficient conditionsEdit

Let   be a topological space.

Meagre subsets and subspaces

  • A singleton subset   is always a non-meagre subspace of   (that is, it is a non-meagre topological space). If   is an isolated point of   then   is also a non-meagre subset of  ; the converse holds if   is a T1 space.
  • Any subset of a meagre set is a meagre set.[1]
  • Every nowhere dense subset is a meagre set.[1]
  • The union of countably many meagre sets is also a meagre set.[1]
  • 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  [2]
  • A countable Hausdorff space without isolated points is meagre.[3]
  • Any topological space that contains an isolated point is non-meagre.[3]
  • Any discrete space is non-meagre.[3]
  • Every Baire space is non-meagre but there exist non-meagre spaces that are not Baire spaces.[3]
  • The set   is a meagre subset of   even though   is a non-meagre subspace (that is,   is not a meagre topological space).[3]
  • 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.
  • If   is a homeomorphism then a subset   is meagre if and only if   is meagre.[1]

Comeagre subset

  • Any superset of a comeagre set is comeagre.
  • the intersection of countably many comeagre sets is comeagre.
    • This follows from the fact that a countable union of countable sets is countable.

Function spacesEdit

The set of functions that have a derivative at some point is a meagre set in the space of all continuous functions.[4]

PropertiesEdit

  • Banach Category Theorem: In any space   the union of any countable family of open sets of the first category is of the first category.[5]
  • A non-meagre locally convex topological vector space is a barreled space.[3]
  • A closed subset of   that is of the second category in   must have non-empty interior in  [2]
  • If   is of the second category in   and if   are subsets of   such that   then at least one   is of the second category in  

Meagre subsets and Lebesgue measureEdit

A meagre set need not have measure zero. There exist nowhere dense subsets (which are thus meagre subsets) that have positive Lebesgue measure.[3]

Relation to Borel hierarchyEdit

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

Banach–Mazur gameEdit

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 alsoEdit

NotesEdit

  1. ^ a b c d Rudin 1991, p. 43.
  2. ^ a b Rudin 1991, pp. 42-43.
  3. ^ a b c d e f g Narici & Beckenstein 2011, pp. 371-423.
  4. ^ Banach, S. (1931). "Über die Baire'sche Kategorie gewisser Funktionenmengen". Studia Math. 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.

BibliographyEdit

External linksEdit