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\begin{center} 
{\large {\bf Final Exam - Part 2 - Take home Exam}} \\
Topology, Math 441, Spring 2020 \\
Instructor: Abhijit Champanerkar \\
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\flushleft{\textbf{Date:} May 18th, 2020} \hfill \textbf{Points:} 100

\ \\
\hrule
\begin{itemize}
\item Please submit by {\bf emailing me} the solutions in pdf
  form either typed or written on plain or ruled paper (and scanned), by {\bf
    11:59 pm on Monday May 18th, 2020}.
  \item Justify your answers and write clear proofs for full credit.
  You can assume the continuity of single and multi-variable
    functions you have seen in Calculus and Advanced Calculus courses.
  \item {\bf Policy about plagiarsm and cheating:} Please note that
    you may be asked to explain and justify your solutions to me in an
    one-on-one online meeting.

  \item {\bf VERY IMPORTANT} 
    Please include the following honor code statement in your
    submission, right after your name: {\bf I, {\it student name},
      have submitted only those solutions that I fully understand
      myself.}

\item Submit any 5 problems. Please indicate at the begining of your 
answers which problems you are attempting. 

\end{itemize}
\hrule
\ \\
\noindent

\begin{enumerate}
%  \item Prove that the intersection of the plane $x+2y-5z=0$ with the 
% ellipsoid $x^2 +\frac{y^2}{4} + \frac{z^2}{9} =1$ is a closed subset of
%    $\R^3$.


\item {\it (20 points)}

  \begin{enumerate}
  \item Let $A$ denote the z-axis. Prove that $\R^3-A$ is connected and path-connected. 

  \item Prove that $\R^3-S^2$ is disconnected. Please justify all
    your steps.
  \end{enumerate}

\item {\it (20 points)}
  \begin{enumerate}
  \item Show that every set $A \subset \R$ is a compact subset in the finite complement topology of $\R$. 
  \item Give an example to show that arbitrary union of compact sets
    in a space $X$ is not necessarily compact.
  \end{enumerate}


\item {\it (20 points)} 
The sum of two subsets $A$ and $B$ of $\mathbb{R}^n$ is defined to be
$A+B=\{a+b \ |\  a\in A \text{ and }b\in B\}.$
\begin{enumerate}
\item Prove that the addition function is continuous i.e. 
$f: \R \times \R \to \R$ given by $f(a,b)=a+b$ is continuous. 
\item Prove that the sum of two compact sets in $\R$ is compact
\end{enumerate}




\item {\it (20 points)} Let $X$ be a topological space.  A family $\F$
  of subsets of $X$ is called {\em locally finite} if every point
  $x\in X$ has an open neighborhood $U$ such that only finitely many
  members of $\F$ have nonempty intersection with $U$.  Prove that if
  $\F$ is a locally finite family of closed sets then the union
  $\cup_{C\in\F}$ is closed.

\item {\it (20 points)} A topological space $X$ has the following
  property: Suppose that $\F =\{ F_j\ |\ j\in J\ \}$ is a collection
  of closed subsets of $X$ such that every finite intersection of
  subsets from $\F$ is non-empty, then $\cap_{j\in J} F_j \neq \phi$. 
Prove that $X$ is compact if and only if this propert holds. 


\item {\it (20 points)} Let
  $S^4 =\{ (x,y,z,u,v) \in \R^5 \ | x^2+y^2+z^2+u^2+v^2=1\ \} $ be the
  4-sphere i.e. the set of unit length vectors in $\R^5$. Prove that
  $S^4$ is a 4-manifold by finding explicit charts for it.

\item {\it (20 points)} 
  \begin{enumerate}
  \item Give an example to show that an identification of sides of
    polygon which has 3 or more sides identified cannot result in a
    surface.

  \item Compute the Euler characteristic of the surface obtained by
    identifying the sides of the polygon drawn below, and identify
    which surface it is.
\end{enumerate}

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\item {\it (20 points)}
  \begin{enumerate}
  \item Assuming that in the plane every simple closed curve separates
    the plane in two open sets, one of which is homeomorphic to an
    open disk, prove that any simple closed curve on $S^2$ separates
    it.

  \item Using cutsets prove that $S^2$ is not homeomorphic to $gT$ for any 
     $g \geq 1$. 

\end{enumerate}



  \end{enumerate}
\end{document}


