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Cardinal function

From Wikipedia, the free encyclopedia

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

Cardinal functions in set theory

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

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 nonempty opens 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
is a family of mutually disjoint non-empty open subsets of
  • 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

Cardinal functions in Boolean algebras

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
is a chain
  • Depth of a Boolean algebra is
is a well-ordered subset .
  • Incomparability of a Boolean algebra is
such that .
  • Pseudo-weight of a Boolean algebra is
such that

Cardinal functions in algebra

Examples of cardinal functions in algebra are:

External links

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

See also

References

  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.
  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. 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.
This page was last edited on 9 November 2021, at 01:40
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