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\begin{center} 
{\large {\bf Homework 7 }} \\
Topology, Math 441, Spring 2020 \\
Topic: Quotient Topology, Connectedness \\
{\bf Due:} Friday April 17th, 2020 \\
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{\bf Reading:} Its important to read the following sections to understand 
quotient topology.  
\begin{enumerate}
\item pg 56-61 from the text book. 
\item  Section 3.3 from book \emph{Introduction to Topology} by Adams-Franzosa 
(skip Example 3.18).
\item Section 3.4 from book \emph{Introduction to Topology} by Adams-Franzosa.

\end{enumerate}

{\bf Problems:}

\begin{enumerate}

\item Give an example to show that the quotient space of a Hausdorff
  space need not be Hausdorff.

\item 
  \begin{enumerate}
  \item Prove that $f:X \to Y$ is a quotient map if subset $K\subset Y$
  is closed if and only if $f^{-1}(K)$ is closed.

\item A map $f: X \to Y$ is closed if it takes closed sets to closed
  sets. Using (a) prove that a surjective, continuous, closed map is a
  quotient map.
  \end{enumerate}

\item Consider the equivalence relation on $\R^2$ defined by $(x_1,x_2) 
\sim (y_1,y_2)$ if $x_1^2+x_2^2=y_1^2+y_2^2$. Describe the quotient space 
$\R^2/\sim$. 

\item Describe or draw a picture of the resulting quotient space. Assume 
that points are identified only with themselves unless they ar eexplicitly said to be identified with other points. 

\begin{enumerate}
\item The interval $[0,4] \subset \R$, with integer points identified. 
\item The sphere with the equator collapsed to a point. 
\item The real line $\R$ with $[-1,1]$ collapsed to a point. 
\end{enumerate}

%\item Prove that $X$ is connected iff every non-empty proper subset has a 
%non-empty boundary. 

\item Prove that $\R$ with the following topology is connected: 
  \begin{enumerate}
  \item $\R_{PP=0}$, the particular point topology with $p=0$. 
\item $\R_{EP=0}$, the excluded point topology with $p=0$.
\item $\R_{co-fin}$, the co-finite topology on $\R$. 
  \end{enumerate}

\item Show that if $A \subset X$ is connected then $A$ is contained in
  a component of $X$.

\item Prove that $S^2 \subset \R^3$ is connected. 

\item Use cutsets to prove the following: 
  \begin{enumerate}
  \item $[a,b]$ is not homeomorphic to $(a,b)$. 
  \item $[a,b)$ is not homeomorphic to $(a,b)$. 
  \item $(a,b]$ is not homeomorphic to $(a,b)$. 
  \item $(a,b]$ is not homeomorphic to $[a,b]$. 
\end{enumerate}

\item Prove that no two-point set separates $\R^2$.


\item[*10.] Let $X$ be a topological space and $Y$ be a set. Let $p:X \to Y$
  be a surjective map. Prove that the quotient topology on $Y$ induced by $p$ 
(i.e. using the equivalence relation on $X$ given by $x \sim y \iff p(x)=p(y)$ 
is the finest  topology under which $p$ is continuous. 
\end{enumerate}



{\bf Handin:} 3, 4ab, 5bc, 7, 8ab, 9



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