closed complex plane ClosedComplexPlane 2007-11-20 16:17:55 2007-11-20 22:42:48 Definition topology of the complex plane % 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 \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} The complex plane $\mathbb{C}$, i.e. the set of the complex numbers $z$ satisfying $$|z| < \infty,$$ is open but not closed, since it doesn't contain the accumulation points of all sets of complex numbers, for example of the set $\{1,\,2,\,3,\,\ldots\}$.\, One can \PMlinkescapetext{supplement} $\mathbb{C}$ to the {\em closed complex plane} $\mathbb{C}\cup\{\infty\}$ by adding to $\mathbb{C}$ the infinite point $\infty$ which \PMlinkescapetext{represents} the lacking accumulation points. One settles that\, $|\infty| = \infty$,\, where the latter $\infty$ means the real infinity. The resulting space is the one-point compactification of $\mathbb{C}$. The open sets are the open sets in $\mathbb{C}$ together with sets containing $\infty$ whose complement is compact in $\mathbb{C}$. Conceptually, one thinks of the additional open sets as those open sets ``around $\infty$''. The one-point compactification of $\mathbb{C}$ is also the complex projective line $\mathbb{CP}^1$, as well as the Riemann sphere.