first countable FirstCountable 2002-02-19 10:41:38 2002-02-19 10:41:38 Definition % 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 Let $X$ be a topological space and let $x\in X$. $X$ is said to be \emph{\PMlinkescapetext{first countable} at $x$} if there is a sequence $(B_n)_{n\in\mathbb{N}}$ of open sets such that whenever $U$ is an open set containing $x$, there is $n\in\mathbb{N}$ such that $x\in B_n\subseteq U$. The space $X$ is said to be \emph{\PMlinkescapetext{first countable}} if for every $x\in X$, $X$ is first countable at $x$.