order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...

, a

partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

''X'' is said to satisfy the countable chain condition, or to be ccc, if every

strong antichain In order theory, a subset ''A'' of a partially ordered set ''P'' is a strong downwards antichain if it is an antichain In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct eleme ...

in ''X'' is

countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...

Overview

There are really two conditions: the ''upwards'' and ''downwards'' countable chain conditions. These are not equivalent. The countable chain condition means the downwards countable chain condition, in other words no two elements have a common lower bound. This is called the "countable chain condition" rather than the more logical term "countable antichain condition" for historical reasons related to certain chains of open sets in topological spaces and chains in complete Boolean algebras, where chain conditions sometimes happen to be equivalent to antichain conditions. For example, if κ is a cardinal, then in a complete Boolean algebra every antichain has size less than κ if and only if there is no descending κ-sequence of elements, so chain conditions are equivalent to antichain conditions. Partial orders and spaces satisfying the ccc are used in the statement of

Martin's axiom In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consi ...

. In the theory of forcing, ccc partial orders are used because forcing with any generic set over such an order preserves cardinals and cofinalities. Furthermore, the ccc property is preserved by finite support iterations (see iterated forcing). For more information on ccc in the context of forcing, see . More generally, if κ is a cardinal then a poset is said to satisfy the κ-chain condition if every antichain has size less than κ. The countable chain condition is the ℵ₁-chain condition.

Examples and properties in topology

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...

is said to satisfy the countable chain condition, or Suslin's Condition, if the partially ordered set of non-empty

open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

s of ''X'' satisfies the countable chain condition, ''i.e.'' every

pairwise disjoint In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...

collection of non-empty open subsets of ''X'' is countable. The name originates from

Suslin's Problem In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC: showed that the statement can neither ...

. * Every separable topological space is ccc. Furthermore, the

product space In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seem ...

of at most

\mathfrak=2^

separable spaces is a separable space and, thus, ccc. * A

metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...

is ccc if and only if it's separable. * In general, a ccc topological space need not be separable. For example,

\^

with the

product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemi ...

is ccc, though ''not'' separable. * Paracompact ccc spaces are Lindelöf.

References

*{{Citation , last1=Jech , first1=Thomas , author1-link=Thomas Jech , title=Set Theory: Millennium Edition , publisher=

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 i ...

, location=Berlin, New York , series=Springer Monographs in Mathematics , isbn=978-3-540-44085-7 , year=2003 *Products of Separable Spaces, K. A. Ross, and A. H. Stone. The American Mathematical Monthly 71(4):pp. 398–403 (1964) Order theory Forcing (mathematics)