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\begin{center} 
{\large {\bf Homework 4 }} \\
Topology, Math 441, Spring 2020 \\
Topic: Closed sets and limit points \\
{\bf Extended Due Date:} Friday March 11th, 2020 \\
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{\bf Reading:} Pages 29-33 from Chapter 3 of the text book.  \\
\ \\


{\bf Problems:}
\begin{enumerate}
\item For a topological space $X$ with a basis for its topology, we
  gave a characterization of open sets defined in terms of basis open
  sets.  For any $x \in X$ we define a {\bf neighbourhood of $x$} to
  be any open set containing $x$. Show that a set $U \subset X$ is
  open if and only if every point $x \in U$ has a neighbourhood
  contained in $U$.

\item Prove that the following subsets are closed. 
  \begin{enumerate}
\item  $\Z \subset \R$.
\item $A=\{1/n \; |\; n \in \N \; \} \cup \{\; 0\; \} \subset \R$. 
  \item A closed rectangle $[a,b] \times [c,d] \subset \R^2$ (Hint: use 
the fact that open rectangles are basis open sets for topology on 
$\R^2$).
\item The $(n-1)$-dimensional sphere
  $S^{n-1}=\{\; \overline{x} \; |\; d(\overline{x},\overline{0})=1 \; \}
  \subset \R^n$.
\item $[a,b) \subset \R_{\ell}$ i.e. $\R$ with lower limit topology. 


  \end{enumerate}
\item Prove that $\Q \subset \R$ is neither closed nor open.


\item Find the closures of the following sets. 
  \begin{enumerate}
\item $(0,1] \subset \R_{\ell}$ i.e. $\R$ with lower limit topology.
\item $\Q \subset \R$.
  \end{enumerate}


\item Let $X$ be a topological space. 
  \begin{enumerate}
  \item Prove that if $A \subset B$ then $\cl{A}  \subset \cl{B} $.
\item $A \subset X$. Prove that $\cl{X-A}= X - \intr{A}$. 
  \end{enumerate}



\end{enumerate}



{\bf Handin:} 1, 2bce, 3, 4b, 5b



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