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%%% exercise set 4
%%% september 30, 2009
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\begin{document}

\begin{center}
Math 445,  Fall 2009 \hfill  Exercise Set \#4 \hfill Turn in: September 30 
\end{center}
 
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 {\bf 1.}   [\#1, page 78] ~  Let $M$ be a metric space with metric $D$. Prove that if $\{x_n \mid n =1,2, \ldots \} \subset M$ is a sequence which converges to points $x \in M$ and $y \in M$, then $x = y$.
  
  
  
  
  
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 {\bf 2.}   [\#4, page 78] ~  Given distinct points $x$ and $y$ in a metric space $M$, prove that there exist open sets $U$ and $V$ such that $x \in U$, $y \in V$, and their \underline{closures} $\overline{U} \cap \overline{V} = \emptyset$.
  
  
  
  
  
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 {\bf 3.}   [\#8, page 79] ~  Let $M$ be a metric space with metric $D$. Prove that the diameter of a set $A$ in $M$ equals the diameter of its closure, $\overline{A}$.
  
  
  
  
  
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 {\bf 4.}   [\#11*, page 79] ~  Prove that in a metric space,  the closure of a \underline{countable set} has cardinal number at most $c$. [Recall that $c$ is the cardinal of the continuum $\mR$, which equals the cardinal of the power set of the natural numbers, ${\mathcal P}(\mN)$.]
  
  
  
  
  
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 {\bf 5.}   [\#12*, page 79] ~  Prove that the following statements are equivalent for a metric space $M$:
 \begin{enumerate}
\item[(a)] Every subset of $M$ is either open or closed;
\item[(b)] At most one point of $M$ is not isolated.
\end{enumerate}
[Hint: Draw an example of a set with exactly one limit point.]
  
  
  
  
  
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 {\bf 6.}   [\#13*, page 79] ~  Let $M$ be a metric space in which the closure of every open set is open. Prove that $M$ is discrete. That is, show that every point of $M$ is an open set. 
  
  
  
  
  
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 {\bf 7.}   [\#14*, page 79] ~  Prove that in a metric space, every open set is the union of a countable number of closed sets. Deduce from this that every closed set is the intersection of a countable number of open sets. 
  
  
  
  
  
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 {\bf 8.}   [\#16, page 79] ~  Prove that a metric space is discrete if and only if every convergent sequence is ultimately constant. 
  
  
  
  
  
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 {\bf 9.}   [\#17, page 79] ~  Prove that a metric space is discrete if and only if it has no limit points. 
  
  
  
  
  
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 {\bf 10.}   [\#19*, page 79] ~  If a metric space $M$ has only countably many open sets, prove that $M$ is finite. 
  
  
  
  
  
   
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