Cardinal function

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In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.

Cardinal functions in set theory

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The "additivity" of I is the smallest number of sets from I whose union is not in I any more. As any ideal is closed under finite unions, this number is always at least  ; if I is a σ-ideal, then  
 
The "covering number" of I is the smallest number of sets from I whose union is all of X. As X itself is not in I, we must have add(I ) ≤ cov(I ).
 
The "uniformity number" of I (sometimes also written  ) is the size of the smallest set not in I. Assuming I contains all singletons, add(I ) ≤ non(I ).
 
The "cofinality" of I is the cofinality of the partial order (I, ⊆). It is easy to see that we must have non(I ) ≤ cof(I ) and cov(I ) ≤ cof(I ).
In the case that   is an ideal closely related to the structure of the reals, such as the ideal of Lebesgue null sets or the ideal of meagre sets, these cardinal invariants are referred to as cardinal characteristics of the continuum.
  • For a preordered set   the bounding number   and dominating number   are defined as
 
 
  • In PCF theory the cardinal function   is used.[1]

Cardinal functions in topology

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Cardinal functions are widely used in topology as a tool for describing various topological properties.[2][3] Below are some examples. (Note: some authors, arguing that "there are no finite cardinal numbers in general topology",[4] prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the definitions given below, for example by adding " " to the right-hand side of the definitions, etc.)

  • Perhaps the simplest cardinal invariants of a topological space   are its cardinality and the cardinality of its topology, denoted respectively by   and  
  • The weight   of a topological space   is the cardinality of the smallest base for   When   the space   is said to be second countable.
    • The  -weight of a space   is the cardinality of the smallest  -base for   (A  -base is a set of non-empty open sets whose supersets includes all opens.)
    • The network weight   of   is the smallest cardinality of a network for   A network is a family   of sets, for which, for all points   and open neighbourhoods   containing   there exists   in   for which  
  • The character of a topological space   at a point   is the cardinality of the smallest local base for   The character of space   is   When   the space   is said to be first countable.
  • The density   of a space   is the cardinality of the smallest dense subset of   When   the space   is said to be separable.
  • The Lindelöf number   of a space   is the smallest infinite cardinality such that every open cover has a subcover of cardinality no more than   When   the space   is said to be a Lindelöf space.
  • The cellularity or Suslin number of a space   is
 
  • The hereditary cellularity (sometimes called spread) is the least upper bound of cellularities of its subsets:   or   where "discrete" means that it is a discrete topological space.
  • The extent of a space   is   So   has countable extent exactly when it has no uncountable closed discrete subset.
  • The tightness   of a topological space   at a point   is the smallest cardinal number   such that, whenever   for some subset   of   there exists a subset   of   with   such that   Symbolically,   The tightness of a space   is   When   the space   is said to be countably generated or countably tight.
    • The augmented tightness of a space     is the smallest regular cardinal   such that for any     there is a subset   of   with cardinality less than   such that  

Basic inequalities

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Cardinal functions in Boolean algebras

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Cardinal functions are often used in the study of Boolean algebras.[5][6] We can mention, for example, the following functions:

  • Cellularity   of a Boolean algebra   is the supremum of the cardinalities of antichains in  .
  • Length   of a Boolean algebra   is
 
  • Depth   of a Boolean algebra   is
 .
  • Incomparability   of a Boolean algebra   is
 .
  • Pseudo-weight   of a Boolean algebra   is
 

Cardinal functions in algebra

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Examples of cardinal functions in algebra are:

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  • A Glossary of Definitions from General Topology [1] [2]

See also

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References

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  1. ^ Holz, Michael; Steffens, Karsten; Weitz, Edmund (1999). Introduction to Cardinal Arithmetic. Birkhäuser. ISBN 3764361247.
  2. ^ Juhász, István (1979). Cardinal functions in topology (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-062-2. Archived from the original (PDF) on 2014-03-18. Retrieved 2012-06-30.
  3. ^ Juhász, István (1980). Cardinal functions in topology - ten years later (PDF). Math. Centre Tracts, Amsterdam. ISBN 90-6196-196-3. Archived from the original (PDF) on 2014-03-17. Retrieved 2012-06-30.
  4. ^ Engelking, Ryszard (1989). General Topology. Sigma Series in Pure Mathematics. Vol. 6 (Revised ed.). Heldermann Verlag, Berlin. ISBN 3885380064.
  5. ^ Monk, J. Donald: Cardinal functions on Boolean algebras. "Lectures in Mathematics ETH Zürich". Birkhäuser Verlag, Basel, 1990. ISBN 3-7643-2495-3.
  6. ^ Monk, J. Donald: Cardinal invariants on Boolean algebras. "Progress in Mathematics", 142. Birkhäuser Verlag, Basel, ISBN 3-7643-5402-X.