%%LaTeX 2e

\documentclass[11pt]{amsart}
\usepackage{graphicx}
\topmargin 0in \textheight 9in \textwidth 6in \oddsidemargin .25in
\evensidemargin .25in

\usepackage{amsmath}
\usepackage{amsfonts}
%\usepackage{showkeys}

\pagestyle{myheadings}

\begin{document}

\newcommand{\ds}{\displaystyle}
\newcommand{\ra}{\rightarrow}
\newcommand{\Ra}{\Rightarrow}
\newcommand{\Lra}{\Leftrightarrow}
\newcommand{\mc}{\mathcal}
\newcommand{\mrm}{\mathrm}
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
\newcommand{\cl}{\overline}
\newcommand{\hs}{\hspace}
\newcommand{\vs}{\vspace}
\newcommand{\interior}{\mathrm{Int}}
\newcommand{\complex}{\mathbb{C}}
\newcommand{\rsphere}{\mathbb{C}_\infty}
\newcommand{\nat}{\mathbb{N}}
\newcommand{\real}{\mathbb{R}}
\newcommand{\rat}{\mathbb{Q}}
\newcommand{\zed}{\mathbb{Z}}
\newcommand{\Disk}{\mathbb{D}}
\newcommand{\disk}{\Delta}
\newcommand{\wh}{\widehat}
\newcommand{\e}{\varepsilon}
\newcommand{\0}{\emptyset}
\newcommand{\var}{\mathrm{var}}
\newcommand{\ind}{\mathrm{ind}}
\newcommand{\dg}{\mathrm{degree}}
\newcommand{\win}{\mathrm{win}}
\newcommand{\bd}{\partial}
\newcommand{\im}{\mathrm{Im}}
\newcommand{\pr}{\mathrm{Pr}}
\newcommand{\sh}{\mathrm{Sh}}
\newcommand{\dm}{\mathrm{diam}}
\newcommand{\dist}{\mathrm{d}}
\newcommand{\sm}{\setminus}

\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{exr}[thm]{Exercise}
\newtheorem{exm}[thm]{Example}
\newtheorem{prob}[thm]{Problem}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{axm}[thm]{Axiom}
\newtheorem{rem}[thm]{Remark}
\newtheorem{ques}[thm]{Question}
\newtheorem{conj}[thm]{Conjecture}

\parindent=0in
\parskip=0.1in


\title
{Topology Notes \\ Summer, 2006}

\author[J.~C.~Mayer]{John C.~Mayer} \email[John C.~Mayer]{mayer@math.uab.edu}

\address[John C.~Mayer]
{Department of Mathematics\\ University of Alabama at Birmingham\\
Birmingham, AL 35294-1170}


\keywords{topology} \subjclass{Primary: 54F20}
\date{\today}

\maketitle

\setcounter{section}{2}

\section{Components}

We will define three  relations on a space, show that they are each
equivalence relations, and how they are related.  This will allow us
to define several useful subsets of a space that exploit its degree
of "connectedness."  Recall that an equivalence relation on a set
$X$ satisfies the three properties of {\em reflevivity}, {\em
symmetry}, and {\em transitivity}.  (See Munkries Sections 3 and
25.)

\begin{defn} \label{equiv} Let $X$ be a space.  Define the following
relations on $X$.
\begin{enumerate}
\item  $x\sim y$ iff there is no separation $X=A\cup B$ with $x\in
A$ and $y\in B$.
\item $x\simeq y$ iff there is a connected subset $C\subset X$ with
$x,y\in C$.
\item $x\approx y$ iff there is a path $f:[a,b]\to X$ with $f(a)=x$
and $f(b)=y$.
\end{enumerate}
\end{defn}

\begin{prop} Each of $\sim$, $\simeq$, and $\approx$ is an
equivalence relation on $X$. $\Box$ \end{prop}

\begin{prop} \label{impconn} $x\approx y \implies x\simeq y \implies x\sim y$.
$\Box$ \end{prop}

\begin{exm} No implication in Proposition~\ref{impconn} is
reversible. \end{exm}

\subsection*{Assignment 2, Part 2}

Hard: 8: unboxed Example 3.4 above.

Assignment 2, Parts 1 and 2, is due Friday, June 16.

\medskip


\begin{defn} For an equivalence relation $\sim$ on $X$, we use $[x]_\sim$
to denote the $\sim$-equivalence class of $x\in X$, defined by
$$[x]_\sim =\{y\in X\mid x\sim y\}.$$
(Similarly, define $[x]_\simeq$ and $[x]_\approx$ for the relations
$\simeq$ and $\approx$, respectively.)
\end{defn}

You should recall from previous work that equivalence classes are
pairwise disjoint and their union is the whole set $X$; that is,
they form a {\em partition} of $X$.  In particular, if two
equivalence classes meet, then they must be equal.

\begin{defn}  Let $x\in X$.  We define the following special
terminology for equivalence classes of the relations defined above
in Definition~\ref{equiv}.
\begin{enumerate}
\item $\sim$-equivalence classes are called {\em quasicomponents}.
\item $\simeq$-equivalence classes are called {\em components}.
\item $\approx$-equivalence classes are called {\em path components}.
\end{enumerate}
\end{defn}

\begin{prop} Let $X$ be a space. The following hold.
\begin{enumerate}
\item Each subset $Q\subset X$ which cannot be separated by a separation of $X$ is
contained in a quasicomponent of $X$.
\item Each connected subset $C\subset X$ is contained in a component
of $X$.
\item Each path connected subset $P\subset X$ is contained in a path
component of $X$.
\end{enumerate} \end{prop}

\begin{prop} Let $X$ be a space.  The following hold.
\begin{enumerate}
\item Each component of $X$ is connected, and is maximal with
respect to that property.
\item Each path component of $X$ is path connected, and is maximal with
respect to that property.
\end{enumerate} $\Box$ \end{prop}


\begin{prop} Let $X$ be a space. The following hold.
\begin{enumerate}
\item Each quasicomponent of $X$ is a union of components
of $X$.
\item Each component of $X$ is a union of path
components of $X$.
\end{enumerate} \end{prop}

\begin{prop} Let $X$ be a space. Quasicomponents and components are
closed in $X$.  Path components need not be closed in $X$. $\Box$
\end{prop}

\begin{defn} A space $X$ is {\em locally connected at} $x\in X$ iff
for every neighborhood $U$ of $x$, there is a connected neighborhood
$V$ of $x$ with $V\subset U$. If $X$ is locally connected at every
point, we say $X$ is {\em locally connected}. A space $X$ is {\em
locally path connected at} $x\in X$ iff for every neighborhood $U$
of $x$, there is a path connected neighborhood $V$ of $x$ with
$V\subset U$. If $X$ is locally path connected at every point, we
say $X$ is {\em locally path connected}. \end{defn}

\begin{thm} A space is locally connected iff for every open set $U$
in $X$, each component of $U$ is open in $X$. $\Box$ \end{thm}

\begin{thm} A space is locally path connected iff for every open set $U$
in $X$, each path component of $U$ is open in $X$. \end{thm}

\begin{thm} If a space $X$ is locally path connected, then the
components and the path components of $X$ are the same. $\Box$
\end{thm}

\begin{thm} If a space $X$ is locally  connected, then the
components and the quasicomponents of $X$ are the same.
\end{thm}


\subsection*{Assignment 3, Part 1}

Easy:  1--4: unboxed items above.

Hard: 5--7: page 162: 4, 8, 10c.

Assignment 3 may be added to on Thursday, June 15, and is due
Friday, June 23.

\end{document}
