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%%% exercise set 7
%%% November 20, 2009
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\begin{document}

\begin{center}
Math 445,  Fall 2009 \hfill  Exercise Set \#7 \hfill Turn in: November 20  
\end{center}
 
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 {\bf 1.} ~ Let $X$ be a topological space, and suppose that $A, B \subset X$ are compact subsets.  Show that $A \cup B$ is again compact.
 
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 {\bf 2.} ~ Let $X, Y$ be   topological spaces, $f \colon X \to Y$ a continuous map, and assume that $X$ is \emph{compact} and $Y$ is \emph{Hausdorff}. Show that $f$ is a closed map. That is, if $F \subset X$ is closed, then $f(F) \subset Y$ is closed. 
 
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 {\bf 3.} [\#3, page 103] ~   Let $f \colon X \to Y$ be a continuous one-to-one mapping of a compact metric space $X$ onto a metric space $Y$. Prove that $f^{-1} \colon Y \to X$ is continuous (and thus $f$ is a homeomorphism.)
 
 
 
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 {\bf 4.} ~ Show that a topological space $X$ is Hausdorff if and only if the diagonal 
$$\Delta(X) = \{(x,x) \mid x \in X\} \subset X \times X$$ is closed for the product topology.

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       The graph of a function $f \colon X \to Y$ is the set
  $\ds  \cG_f ~ = ~ \{ (x,f(x)) \mid x \in X\}$.



 {\bf 5.} ~ Let $X, Y$ be   topological spaces, and suppose that $Y$ is compact Hausdorff. 

 
a)  Show that if 
 $f \colon X \to Y$ is continuous, then the graph $\cG_f $ of $f$  is closed in $X \times Y$. 
    
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  b)  Show that if the graph  $\cG_f $ of $f$  is closed, then 
 $f \colon X \to Y$ is continuous.  


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 {\bf 6.} [\#5, page 104] ~ Let $A, B \subset X$ be   disjoint subsets of a metric space $X$.
 Suppose that $A$ is closed, and $B$ is compact. Prove that the distance between $A$ and $B$ is positive. That is,
show that 
 $$\inf ~ \{D(a,b) \mid a \in A , ~ b \in B\} > 0$$
 
   
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 {\bf 7.} ~  Let $\{X,d\}$  be a compact metric space.  Suppose that $f \colon X \to X$ is a function which satisfies
$$d(f(x), f(y)) ~  < ~  d(x, y) ~~ {\rm for\;\; all} ~~ x \ne y$$  
Prove that $f$ has a   {unique fixed point}.
 
  
 \vfill
 
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