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\begin{document}

\title{Topology problems}
\author{David Glickenstein}
\maketitle

\section{Lecture 1: Topological spaces}

\begin{enumerate}
\item Give all possible topologies (up to homeomorphism) on a set of three
points. Give two collections of open subsets of three points which contain
both the set of all points and $\varnothing$ but are not topologies.

\item Show that all open intervals (including unbounded ones) in $\mathbb{R}$
are homeomorphic.

\item If $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ are continuous, show that
the composition $g\circ f:X\rightarrow Z$ is continuous.

\item If $Y\subset X$ is given the subspace topology, the inclusion map
$\iota:Y\rightarrow X$ is continuous. Conclude that if $f:Z\rightarrow Y$ is
continuous, then if we expand the range and consider $\bar{f}:Z\rightarrow X$
(the function is defined the same) then $\bar{f}$ is continuous.

\item Show that the quotient topology on $\left[  0,1\right]  /0\sim1$ is
homeomorphic to $\left\{  \left(  x,y\right)  \in\mathbb{R}^{2}:x^{2}%
+y^{2}=1\right\}  $ with the subspace topology.

\item Show that the two given definitions of topology generated by a basis are
equivalent and are, in fact, topologies.

\item Given two topologies $\mathcal{T}$ and $\mathcal{T}^{\prime}$ on a space
$X,$ if $\mathcal{T\subset T}^{\prime}$ then $\mathcal{T}$ is said to be
coarser than $\mathcal{T}^{\prime}$ (because it contains fewer open sets) and
$\mathcal{T}^{\prime}$ is said to be finer than $\mathcal{T}$ (because it
contains more open sets). Suppose $\mathcal{B}$ is a basis for $\mathcal{T}$
and $\mathcal{B}^{\prime}$ is a basis for $\mathcal{T}^{\prime}.$ Show that
$\mathcal{T}$ is coarser than $\mathcal{T}^{\prime}$ if and only if for each
$x\in X$ and each basis element $B\in\mathcal{B}$ containing $x$ there is a
basis element $B^{\prime}\in\mathcal{B}^{\prime}$ such that $x\in B^{\prime
}\subset B.$

\item Show that the quotient topology is the coarsest topology such that the
quotient map is continuous.

\item One can generalize the product topology to an arbitrary Cartesian
product $%
%TCIMACRO{\dprod \limits_{j\in J}}%
%BeginExpansion
{\displaystyle\prod\limits_{j\in J}}
%EndExpansion
X_{j}$ in two ways:

\begin{enumerate}
\item The topology generated by basis sets of the form $%
%TCIMACRO{\dprod \limits_{j\in J}}%
%BeginExpansion
{\displaystyle\prod\limits_{j\in J}}
%EndExpansion
U_{j}$ where $U_{j}$ is open in $X_{j}.$ This is called the \emph{box
topology}.

\item The topology generated by the subbasis $%
%TCIMACRO{\dbigcup \limits_{j\in J}}%
%BeginExpansion
{\displaystyle\bigcup\limits_{j\in J}}
%EndExpansion
\left\{  \pi_{j}^{-1}\left(  U_{j}\right)  :U_{j}\text{ open in }%
X_{j}\right\}  ,$ where $\pi_{k}:%
%TCIMACRO{\dprod \limits_{j\in J}}%
%BeginExpansion
{\displaystyle\prod\limits_{j\in J}}
%EndExpansion
X_{j}\rightarrow X_{k}$ is the projection map assigning $\pi_{k}\left(
\left(  x_{j}\right)  _{j\in J}\right)  =x_{k}.$ This is called the
\emph{product topology}.
\end{enumerate}

\noindent Show that the product topology is generated by the basis $%
%TCIMACRO{\dprod \limits_{j\in J}}%
%BeginExpansion
{\displaystyle\prod\limits_{j\in J}}
%EndExpansion
U_{j}$ where $U_{j}$ is open in $X_{j}$ and all but finitely many $U_{j}$ are
equal to the entire space $X_{j}.\emph{\ }$Conclude that the box topology is
finer than the product topology. Show that the product topology is the
coarsest topology such that the projection maps are all continuous.

\item A function $f:X\rightarrow Y$ is \emph{continuous at a point} $x\in X$
if for every neighborhood $V$ of $f\left(  x\right)  $ there is a neighborhood
$U$ of $x$ such that $f\left(  U\right)  \subset V.$ Explain why for a
function $f:\mathbb{R\rightarrow\mathbb{R}}$, this is the $\varepsilon
$-$\delta$ definition of continuity. Show that a function is continuous if and
only it is continuous at every point.

\item Show that every metric space is Hausdorff.

\item Given a set $X,$ the \emph{finite complement topology} on $X$ is the
topology such that the complement of every finite set is open. Show that this
is a topology. Show that if $X$ is finite, then this is the discrete topology.
Show that the only closed sets are $X,$ $\varnothing,$ and finite sets.

\item Consider the following topology on $\mathbb{R}^{n}$ called the Zariski
topology, used in algebraic geometry. We take as the closed sets the sets
\[
F\left(  S\right)  =\left\{  x\in\mathbb{R}^{n}:f\left(  x\right)  =0\forall
f\in S\right\}
\]
where $S$ is a set of polynomials in $n$ variables. Show that this is a
topology on $\mathbb{R}^{n}.$ Show that any two open sets must intersect, and
hence the topology cannot be Hausdorff.
\end{enumerate}

\section{Lecture 2: Compactness}

\begin{enumerate}
\item Prove that the sphere is compact.

\item Show that bounded open intervals in $\mathbb{R}$ are not homeomorphic to
bounded closed intervals.

\item Show that the only compact subsets of a discrete space are a finite set
of points.

\item Recall that the finite complement topology on $X$ is the topology such
that the complement of every finite set is open (see exercises from lecture
1). Show that $X$ and every subspace of $X$ is compact, although only $X,$
$\varnothing,$ and finite sets are closed. Why aren't all compact subsets closed?

\item Consider the following space. Let $X^{n}$ be the set of all lines
through the origin in $\mathbb{R}^{n+1}.$ One can give this a topology as a
quotient by letting $X^{n}=(\mathbb{R}^{n+1}\setminus\left\{  0\right\}
)/\sim$ where $x\sim\lambda x$ for all $\lambda\in\mathbb{R\setminus}\left\{
0\right\}  .$ Show that $X$ is compact and Hausdorff. $X^{n}$ is called real
projective space, or $\mathbb{RP}^{n}$. One may do the same thing taking
instead $\mathbb{C}^{n+1}$ and using the equivalence $x\sim\lambda x$ for all
$\lambda\in\mathbb{C\setminus}\left\{  0\right\}  .$ This space is called
$\mathbb{CP}^{n}.$ Show that $\mathbb{CP}^{n}$ is compact and Hausdorff.

\item Show that if $X$ is compact and $Y$ is Hausdorff and $f:X\rightarrow Y$
is a continuous bijection, then $f$ is a homeomorphism.

\item A metric space is said to be \emph{sequentially compact} if every
sequence has a convergent subsequence whose limit is in the space. Show that a
metric space is compact if and only if it is sequentially compact.

\item A metric space is said to be \emph{complete} if every Cauchy sequence
converges. It is said to be \emph{totally bounded} if for every $\varepsilon
>0$ it can be covered by finitely many balls of radius $\varepsilon.$ Show
that a metric space is compact if and only if it is complete and totally bounded.

\item A space is \emph{locally compact} if every point has a compact
neighborhood. There is a canonical way to add one point to a locally compact
Hausdorff space to get a compact space. If X is locally compact Hausdorff,
define $\bar{X}=X\cup\left\{  \infty\right\}  .$ The open sets of $\bar{X}$
are the open sets of $X$ together with sets $\left(  X\setminus K\right)
\cup\left\{  \infty\right\}  $ where $K$ is a compact subset of $X.$ Prove
that $\bar{X}$ is a compact Hausdorff space. Prove that the natural inclusion
$X\rightarrow\bar{X}$ is continuous and that the image of $X$ is open in
$\bar{X}.$ $\bar{X}$ is called the \emph{one-point compactification} of $X.$
Discuss the one-point compactification of $\mathbb{R}^{n}.$

\item The \emph{Cantor set} is defined as follows. Let $E_{1}=\left[
0,1\right]  .$ Let $E_{2}=\left[  0,1\right]  \setminus\left(  1/3,2/3\right)
,$ $E_{3}=E_{2}\setminus\left(  1/9,2/9\right)  \cup\left(  7/9,8/9\right)  ,$
and successive $E_{k}$ defined by deleting the middle third of the open
intervals in $E_{k-1}.$ The Cantor set is defined as
\[
E=%
%TCIMACRO{\dbigcap \limits_{k=1}^{\infty}}%
%BeginExpansion
{\displaystyle\bigcap\limits_{k=1}^{\infty}}
%EndExpansion
E_{k}.
\]
Show that the Cantor set $E$ is compact. Show that the Cantor set is uncountable.

\item One can define a metric structure on the closed subsets of a metric
space $\left(  X,d\right)  $ as follows. If $F,G$ are closed subsets of $X,$
then
\[
d_{H}\left(  F,G\right)  =\inf\left\{  \varepsilon:F\subset B_{\varepsilon
}\left(  G\right)  \text{ and }G\subset B_{\varepsilon}\left(  F\right)
\right\}
\]
where $B_{\varepsilon}\left(  F\right)  =\left\{  x\in X:d\left(  x,F\right)
<\varepsilon\right\}  $ and $d\left(  x,F\right)  =\inf\left\{  d\left(
x,y\right)  :y\in F\right\}  .$ Show that $d_{H}$ defines a metric. The metric
$d_{H}$ is called the Hausdorff metric. Show that if $X$ is compact, then so
is the set of closed subsets of $X$ in the metric topology generated by
$d_{H}.$
\end{enumerate}

\section{Lecture 3: Connectedness}

\begin{enumerate}
\item Find all of the topologies (up to homeomorphism) on four points which
make it connected.

\item Prove that the closure of a connected set is connected.

\item Prove that a path connected space is connected.

\item Show that any infinite set is connected in the finite complement
topology. Is $\left[  0,1\right]  $ path connected in the finite complement topology?

\item Let $K=\left\{  1/n:n=1,2,3,\ldots\right\}  .$ Define comb space to be
$C=\left(  [0,1]\times0\right)  \cup\left(  K\times\left[  0,1\right]
\right)  \cup\left(  0\times\left[  0,1\right]  \right)  $ with the subspace
topology ($C$ is a subset of $\left[  0,1\right]  \times\left[  0,1\right]
$). Define the deleted comb space to be $C^{\prime}=C\setminus0\times\left(
0,1\right)  .$ Show that the deleted comb space is connected but not path connected.

\item A space is \emph{totally disconnected} if the only connected subsets are
one-point sets. Show that the set of rational numbers between $0$ and $1$
(i.e. $\mathbb{Q\cap}\left[  0,1\right]  $) with the subspace topology is
totally disconnected. Show that the Cantor set is totally disconnected.

\item Show that none of $\left(  0,1\right)  ,$ $[0,1),$ and $\left[
0,1\right]  $ are homeomorphic (use the notion of a connected set).

\item Show that $\mathbb{R}$ and $\mathbb{R}^{2}$ are not homeomorphic (use
the notion of a connected set).

\item A space $X$ is said to be \emph{locally connected at} $x$ if for every
neighborhood $U$ of $x$ there is a connected neighborhood $V$ of $x$ contained
in $U.$ If $X$ is locally connected at each of its points $X$ is said to be
\emph{locally connected}. Show that the subset $\left\{  0\right\}  \cup%
%TCIMACRO{\dbigcup \limits_{n=1}^{\infty}}%
%BeginExpansion
{\displaystyle\bigcup\limits_{n=1}^{\infty}}
%EndExpansion
\left\{  1/n\right\}  $ is not locally connected. Show that $[-1,0)\cup(0,1]$
is locally connected but not connected. Show that the deleted comb space is
connected but not locally connected.

\item A space $X$ is said to be \emph{locally path connected at} $x$ if for
every neighborhood $U$ of $x$ there is a path connected neighborhood $V$ of
$x$ contained in $U.$ If $X$ is locally path connected at each of its points
$X$ is said to be \emph{locally path connected}. Show that if $X$ is locally
path connected and connected, then it is path connected.

\item Prove that an arbitrary product of connected sets with the product
topology is connected. Does your proof hold if the box topology is used?
\end{enumerate}


\end{document}