chain condition ChainCondition 2002-07-30 21:47:50 2006-06-23 10:02:13 Definition chain condition countable chain condition $\kappa$-chain condition c.c.c. ccc % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} A partial order $P$ satisfies the \emph{$\kappa$-chain condition} if for any $S\subseteq P$ with $|S|=\kappa$ then there exist distinct $x,y\in S$ and some $p$ such that $p\leq x$ and $p\leq y$. If $\kappa=\aleph_1$ then $P$ is said to satisfy the \emph{countable chain condition} (c.c.c.)