LimitPointsOfSequences 2004-09-24 04:00:25 2005-03-02 23:24:32 Definition limit point of a sequence limit point of the sequence accumulation point of a sequence accumulation point of the sequence % 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 In a topological space $X$, a point $x$ is a \emph{limit point of the sequence} $x_0, x_1, \ldots$ if, for every open set containing $x$, there are finitely many indices such that the corresponding elements of the sequence do not belong to the open set. A point $x$ is an \emph{accumulation point of the sequence} $x_0, x_1, \ldots$ if, for every open set containing $x$, there are infinitely many indices such that the corresponding elements of the sequence belong to the open set. It is worth noting that the set of limit points of a sequence can differ from the set of limit points of the set of elements of the sequence. Likewise the set of accumulation points of a sequence can differ from the set of accumulation points of the set of elements of the sequence. Reference: L. A. Steen and J. A. Seebach, Jr. ``Counterxamples in Topology'' Dover Publishing 1970