Club 2002-07-29 20:12:59 2008-02-15 20:13:05 Definition club closed unbounded closed unbounded closed set unbounded set closed unbounded set club set % 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} If $\kappa$ is a cardinal then a set $C\subseteq\kappa$ is \emph{closed} iff for any $S\subseteq C$ and $\alpha<\kappa$, $\sup(S\cap \alpha)=\alpha$ then $\alpha\in C$. (That is, if the limit of some sequence in $C$ is less than $\kappa$ then the limit is also in $C$.) If $\kappa$ is a cardinal and $C\subseteq\kappa$ then $C$ is \emph{unbounded} if, for any $\alpha<\kappa$, there is some $\beta\in C$ such that $\alpha<\beta$. If a set is both closed and unbounded then it is a \emph{club} set.