TypesOfLimitPoints 2004-09-24 01:09:20 2005-10-23 21:31:03 Definition $\omega$-accumulation points condensation points cluster points % 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} \usepackage{amsthm} % 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 \newcommand{\mc}{\mathcal} \newcommand{\mb}{\mathbb} \newcommand{\mf}{\mathfrak} \newcommand{\ol}{\overline} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcommand{\La}{\Leftarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\nor}{\vartriangleleft} \newcommand{\Gal}{\text{Gal}} \newcommand{\GL}{\text{GL}} \newcommand{\Z}{\mb{Z}} \newcommand{\R}{\mb{R}} \newcommand{\Q}{\mb{Q}} \newcommand{\C}{\mb{C}} \newcommand{\<}{\langle} \renewcommand{\>}{\rangle} Let $X$ be a topological space and $A\subset X$ be a subset. A point $x\in X$ is an \emph{$\omega$-accumulation point of $A$} if every open set in $X$ that contains $x$ also contains infinitely many points of $A$. A point $x\in X$ is a \emph{condensation point of $A$} if every open set in $X$ that contains $x$ also contains uncountably many points of $A$. If $X$ is in addition a metric space, then a \emph{cluster point} of a sequence $\{x_n\}$ is a point $x\in X$ such that every $\epsilon>0$, there are infinitely many point $x_n$ such that $d(x,x_n)<\epsilon$. These are all clearly examples of limit points.