\documentclass[11pt]{article}
%Options: draft shows overfull lines, reqno-leqno puts eq numbers on
%right/left
%\documentclass[11pt,draft, reqno,a4paper,psamsfonts]{amsart}
\usepackage{amsmath,amssymb} % preloaded by amsart above
\usepackage{graphicx} % replaces epsfig
\newcommand{\comments}[1]{} % to insert a comment
\usepackage{hyperref} % for a URL
% *** CHANGE DIMENSIONS ***
\voffset=-0.3truein % LaTeX has too much space at page top
\addtolength{\textheight}{0.3truein}
\addtolength{\textheight}{\topmargin}
\addtolength{\topmargin}{-\topmargin}
\textwidth 6.0in % LaTeX article default 360pt=4.98''
\oddsidemargin 0pt % \oddsidemargin .35in % default is 21.0 pt
\evensidemargin 0pt % \evensidemargin .35in % default is 59.0 pt
%\parindent=20pt
% *** MACROS ***
\newcommand{\Cal}{\mathcal} % Calligraphic - caps only
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\abs}[1]{\lvert #1 \rvert} % absolute value
\newcommand{\norm}[1]{\lVert #1 \rVert} % norm
\newcommand{\ip}[2]{\langle #1,\, #2\rangle} % ip = inner product
%==================== END PREAMBLE =========================================
\begin{document}
% Begin body of article here.
%\pagestyle{empty}
\vspace*{-.7in}
\huge
\centerline{\Huge \bf Basic Definitions}
\vskip20pt
In any metric space $S$:
\begin{itemize}
\item $S$ is \emph{bounded} if it is contained in some ball in $\R^n$.
\item $S$ is a \emph{neighborhood} of $p$ if $S$ contains some open
ball around $P$.
\item A point $p$ is a \emph{limit point} of $S$ if every neighborhood
of $p$ contains a point $q\in S$, where $q\ne p$.
\item If $p\in S$ is not a limit point of $S$, then it is called an
\emph{isolated point} of $S$.
\item $S$ is \emph{closed} if every limit point of $S$ is a point of
$S$.
\item A point $p\in S$ is an \emph{interior point of $S$} if $S$
contains a neighborhood of $p$.
\item $S$ is \emph{open} if every point of $S$ is an interior point of $S$.
\item Let $S'$ denote all of the limit points of $S$. Then the
\emph{closure $\bar{S}$ of $S$} is the set $S\cup S'$. It is the smallest
closed set containing $S$ and is thus the intersection of all the
closed sets containing $S$.
\item A subset $T\subset S$ is \emph{dense in $S$} if every point of
$S$ is either in $T$ or a limit point of $T$ (or both).
\item If $S$ is a metric space and $E\subset S$, let $E'$ be the limit
points of $E$. Then the {\it closure} of $E=E\cup E'$. It is the
smallest closed set that contains. It is also the intersection of
all the closed sets that contain $E$.
\item An \emph{open cover} of $S$ is a family of open sets
$T_\alpha\subset T$ with the property that every point of $S$ is in
at least one of these open sets.
\item A set $S \in \R$ with points $p$ has \emph{measure zero} if
given any
$\epsilon>0$ there is an open cover by open intervals $V_p$ so that
$$
\sum_p \text{length of the }V_p < \epsilon.
$$
The basic example is any countable set $S=\{x_1,\,x_2,\,\ldots \} \in
\R$. Let $V_1$ be an open interval of length less than $\epsilon/2$ containing
$x_1$, $V_2$ an open interval of length less than $\epsilon/2^2$ containing
$x_2$, \ldots $V_k$ an open interval of length less than $\epsilon/2^k$
containing
$x_k$, \ldots. Of course these intervals may overlap. However, since
we have a geometric series,
$$
\sum_k \text{length of the }V_k
< \sum_{k=1}^\infty \frac{\epsilon}{2^k} =\epsilon.
$$
\item A set $S$ is \emph{compact} if every open cover of $S$ has a sub-cover
consisting of a \emph{finite} number of these open sets.
\item $E$ has the \emph{Bolzano-Weierstrass property} if every
infinite subset $x_1$, $x_2$, \ldots of points in $E$ has at least
one limit point $p$ in $E$.
\bigskip
\item In a metric space $X$ (or any ``topological space'') a
\emph{separation} of $X$ is s pair $U$, $V$ ofnonempty disjoint open
subsets of $X$ whose union is $X$. The space $X$ is
\emph{connected} if a separation does not exist.
\smallskip
{\sc Example: } The subset $(0,\,2)\cup (2,\,3)$ in $\R$ is not
connected. The subset $(0,\,2)\cup [2,\,3)$ is connected.
\medskip
{\sc Remark: } For a subspace $Y$ of a larger topological space $X$
here is an alternate equivalent formulation (which the Rudin text
uses).
If $Y$ is a subset of $X$, a \emph{separation} of $Y$ is a pair of
nonempty sets $A$ and $B$ whose union is $Y$, neither of which
contains a limit point of the other. $Y$ is \emph{connected} if no
separation of $Y$ exists.
\medskip
{\sc Example } The following sets in the plane $\R^2$. The $x-$axis
and the graph of $y=1/x$ for $x>0$. It is not connected because
neither piece contains a limit point of the other.
\end{itemize}
\end{document}