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%%% exercise set 5
%%% October 9, 2009
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\begin{center}
Math 445,  Fall 2009 \hfill  Exercise Set \#5 \hfill Turn in: October 9  
\end{center}
 
 \bigskip
 
 The following five problems are all related. Each  one builds on the previous ones.
 \bigskip
 
 {\bf 1.}   [\#10, page 78] ~     Let $A, B \subset M$ be subsets of a metric space $M$. Definethe distance between the sets to be 
 $$D(A,B) = \inf\{ D(a,b) \mid a \in A ~ , ~ b \in B\}$$
  
  a) Suppose that  $B = \{x\}$ consists of a single point.  Prove    that $D(A,B) = 0$ if and only if   $x \in \overline{A}$.
    \medskip
  
  b) Give an example in the Euclidean plane of two closed subsets, $A, B \subset \mR^2$, such that $A \cap B = \emptyset$ and yet $D(A,B) = 0$. [Hint: the sets $A$ and $B$ cannot be bounded.]
  
 \bigskip
 
 {\bf 2.}   [\#2, page 82] ~     Let $u \in M$ be a fixed point in a metric space $M$. The function $f(x) = D(u,x)$ maps $M$ into the real numbers, $f \colon M \to \mR$. Prove that $f$ is continuous.
  
  
 \bigskip
 
 {\bf 3.}   [\#3, page 82] ~     Let $A \subset M$  be a fixed subset of a metric space $M$. The function $f(x) = D(A,x)$ maps $M$ into the real numbers, $f \colon M \to \mR$. Prove that $f$ is continuous.
  
 \bigskip
 
 {\bf 4.}   [\#4, page 82] ~      Let  $A \subset M$ be a \emph{closed} subset and $y$ a point in a metric space $M$, with $y \not\in A$.
 Prove that there exists a continuous real-valued function on $M$ which vanishes on $A$ but not at $y$.
 
 \bigskip
 
   {\bf 5.}    ~      Let $A \subset M$ be a   subset and $y$ a point in a metric space $M$. 
   Suppose that $y \not\in \overline{A}$. 
 Prove that there exists a continuous real-valued function $f \colon M \to [0,\infty)$ which vanishes on $A$ but not at $y$.
 

  
     
      
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