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\begin{center} 
{\large {\bf Homework 5 }} \\
Topology, Math 441, Spring 2020 \\
Topic: Interior, closures etc, and Hausdorff spaces \\
{\bf Due:} Monday March 23rd, 2020 \\
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{\bf Reading:} Pages 23-37 from Chapter 3 of the text book. Read subspace topology from Chapter 4.  \\

{\bf Problems:}
\begin{enumerate}


\item Find the interior, closure and boundary of the following sets. \\
(a) $\Q \subset \R$, (b) $\R \subset \R^2$, (c) $A=\{ (x,y)\ | \ y \geq 0 \} \subset \R^2$.  

\item Let $X$ be a topological space and let $A \subset X$. Let 
 $\partial A$ denote the boundary of the set $A$. Prove the
  following statements. 
  \begin{enumerate}
  \item   $\partial A$ is a closed set.
\item  $\partial A = \phi$ if and only if $A$ is both open and closed. 
\item  $\partial A \cap A = \phi$ if and only if $A$ is open. 
\item $\partial A = {\rm Cl}(A) \cap {\rm Cl}(X-A)$.
  \end{enumerate}

\item 
  \begin{enumerate}
  \item Prove that $x\in {\rm Cl}(A)$ if and only if every neighbourhood of 
$x$ interesects $A$. 

\item A subset $A\subset X$ is {\bf dense} if ${\rm Cl}(A)=X$. Prove 
$A\subset X$ is dense if and only if for any $x \in X$ and 
any neighbourhood $U$ of $x$, $U\cap A \neq \phi$. 

  \end{enumerate}

\item A space $X$ is {\bf separable} if $X$ has a countable dense set. 
Prove that a separable metric space is second countable.


\item Prove that $\R_{{\rm fc}}$ is not Hausdorff (Hint: Prove that any two
  (non-empty) open sets in $\R_{{\rm fc}}$ intersect).

\item Let $X$ be a Hausdorff space. Prove that any finite set is closed in $X$.

\item  Let $(n^2)$ be a sequence in
  $\R_{{\rm fc}}$. Prove that for any $x \in \R$, $n^2 \to x$.







\end{enumerate}



{\bf Handin:} 1a, 2ad, 3a, 6, 7  



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