ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions:...

mutually disjoint non-empty open intervals in R is countable (this is the countable chain condition for the order topology of R), is R necessarily order-isomorphic...

nonempty unbounded complete dense total order satisfying the countable chain condition is isomorphic to the realsPages displaying wikidata descriptions...

{P} } satisfies the countable chain condition (c.c.c.) if and only if every antichain in P {\displaystyle \mathbb {P} } is countable. (The name, which is...

line also satisfies the countable chain condition: every collection of mutually disjoint, nonempty open intervals in R is countable. In order theory, the...

size of a countable set, typically the cardinality of the set of natural numbers. countable chain condition The countable chain condition (ccc) for a...

Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich...

significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. A forcing or partially...

{\displaystyle (\aleph _{0})} , a countable form of Martin's axiom: in a preorder with the countable chain condition, every countable family of dense subsets has...

affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time...

Countable chain condition A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable....

Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. 2.  A Suslin cardinal is a cardinal λ such...

category, a concept in category theory CCC, Roman numeral for 300 Countable chain condition, in order theory CCCn, cube-connected cycles of order n in graph...

of c.c.c. forcings (see countable chain condition) are c.c.c. and thus preserve ω 1 {\displaystyle \omega _{1}} . Countable support iterations of proper...

following statement: MA(κ) For any partial order P satisfying the countable chain condition (hereafter ccc) and any set D = {Di}i∈I of dense subsets of P...

Polish group G, the σ-algebra of Haar null sets satisfies the countable chain condition if and only if G is locally compact. For any locally compact abelian...

all sets M is a countable transitive model of set theory G is a generic subset of P over M. P satisfies the countable chain condition if every antichain...

to the way an amoeba eats food. The amoeba order satisfies the countable chain condition. Kunen, Kenneth (2011), Set theory, Studies in Logic, vol. 34...

Knaster's condition downwards. The property is named after Polish mathematician Bronisław Knaster. Knaster's condition implies the countable chain condition (ccc)...

of cardinality strictly smaller than 2ℵ0 and the poset has the countable chain condition, we can instead use Martin's axiom. However, Martin's axiom is...

X_{n}=x_{n})>0.} The possible values of Xi form a countable set S called the state space of the chain. Markov chains are often described by a sequence of directed...

statement, denoted by MA(k): For any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P such that...

In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution...

requirements: First, the probability of a countable union of mutually exclusive events must be equal to the countable sum of the probabilities of each of these...

lattice Ascending chain condition Infinite descending chain Countable chain condition, often abbreviated as ccc Knaster's condition, sometimes denoted...

While loop Loop invariant Transfinite induction Descending chain condition Large countable ordinal Correctness (computer science) Weakest-preconditions...

measure spaces by observing that the countable union of countable unions is again a countable union, and that the countable unions of null sets are null. A...

I}X_{i}} is not empty. The union of any countable family of countable sets is countable (this requires countable choice but not the full axiom of choice)...

finite or countable (though it must be bounded and totally ordered). In that case, μ has to be a finite measure, and the lattice condition has to be defined...

discrete-time Markov chains. Let ( Ω , A , Pr ) {\displaystyle (\Omega ,{\cal {A}},\Pr )} be a probability space, let S {\displaystyle S} be a countable nonempty...