AnotherDefinitionOfCofinality 2003-08-20 03:54:56 2003-08-20 18:57:37 Definition cofinality supremum approximation cofinality % 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{cofinality} \PMlinkescapeword{parent} Let $\kappa$ be a limit ordinal (e.g. a cardinal). The {\em cofinality of $\kappa$} $\operatorname{cf}(\kappa)$ could also be defined as: $$\operatorname{cf}(\kappa)=\inf \{ |U| : U \subseteq \kappa \text{s.t. } \sup \; U = \kappa \} $$ ($\sup \; U$ is calculated using the natural order of the ordinals). The cofinality of a cardinal is always a regular cardinal and hence $\operatorname{cf}(\kappa) = \operatorname{cf}(\operatorname{cf}(\kappa))$. This definition is equivalent to the parent definition.