In
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 intr ...
, a
partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
''X'' is said to satisfy the countable chain condition, or to be ccc, if every
strong antichain in ''X'' is
countable
In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
.
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 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, also written as κ-c.c., if every antichain has size less than κ. The countable chain condition is the ℵ
1-chain condition.
Examples and properties in topology
A
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is said to satisfy the countable chain condition, or
Suslin's Condition, if the partially ordered set of non-empty
open subset
In mathematics, an open set is a generalization of an open interval in the real line.
In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...
s of ''X'' satisfies the countable chain condition, ''i.e.'' every
pairwise disjoint
In set theory in mathematics and Logic#Formal logic, formal logic, two Set (mathematics), sets are said to be disjoint sets if they have no element (mathematics), element in common. Equivalently, two disjoint sets are sets whose intersection (se ...
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 neith ...
.
* Every
separable topological space has ccc. Furthermore, a
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-seemi ...
of arbitrary amount of separable spaces has ccc.
* A
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
has ccc if and only if it's separable.
* In general, a topological space with ccc need not be separable. For example, a
Cantor cube 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-seemin ...
has ccc for any cardinal
, though ''not'' separable for
.
* Paracompact ccc spaces are
Lindelöf.
* An example of a topological space with ccc is the real line.
References
*
*Products of Separable Spaces, K. A. Ross, and A. H. Stone. The American Mathematical Monthly 71(4):pp. 398–403 (1964)
*Kunen, Kenneth. ''Set Theory: An Introduction to Independence Proofs.''
{{refend
Order theory
Forcing (mathematics)