SphereMetricSpace 2004-11-04 16:52:09 2005-05-02 14:04:09 Definition metric space % 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 The set $\{ x \mid d(x,c) = r \}$ is called the \emph{sphere} of radius $r$ with centre $c$. This generalizes the notion of spheres to metric spaces. Note that the sphere in a metric space need not look like a sphere in Euclidean space. For instance, if we impose the metric $d(x,y) = max \{|x_1-y_1|, |x_2-y_2|, |x_3-y_3|\}$ on $\mathbb{R}^3$ instead of the Euclidean metric, spheres according to this metric are actually cubes! Even more bizarre situations can occur in general --- a sphere might be disconnected, or it may be discrete, or it may even be an empty set.