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%%% exercise set 8
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\begin{center}
Math 445,  Fall 2009 \hfill  Exercise Set \#8 \hfill Turn in: November 30  
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 {\bf 1.}    ~    Consider a Hausdorff topological space  $T = \{A , \cT\}$. 
Consider the collection of open subsets
$$\cT' = \{U \subset A \mid  A- U ~ \text{is ~ compact ~ in} ~ T\} \cup \{\emptyset\}$$
a) Show that $T' = \{A, \cT'\}$ is a topological space. (i.e., that $\cT'$ satisfies the axioms of a topology.) 
  
 b) Show that $T'$ is a compact topological space.
 
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 {\bf 2.}    ~        Let $X = [0,2]$ be the closed interval in $\mR$, but with  a topology $\cT$ on $X$ as follows: 
$$U \in \cT  \quad \Longleftrightarrow \quad  \text{either} ~ 1 \not\in U,  ~ \text{or}  ~  (0,2) \subset U$$
Find the closure of the subset $A =  \{\frac{1}{2}\}$  of $X$.

 
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 {\bf 3.}    ~  Prove that there is no continuous bijective map $f \colon    \mS^1 \to [0,1]$.
[Hint: think connected.]


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{\bf 4.}   [\#1, page 97] ~      Prove that any subspace of a separable metric space is separable.     
 
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 {\bf 5.}   ~     Let $X$ be a locally path connected space. Show that every \emph{open} connected subset of $X$ is path connected.   

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 {\bf 6.}     [\#3a, page 97] ~     Let $X$ be a separable metric space, and let $h \colon X \to Y$ be a continuous onto map. Prove that $Y$ is separable.
 
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  {\bf 7.}   [\#3b, page 97] ~     Let $X$ be a complete metric space, and let $h \colon X \to Y$ be a continuous onto map. Either prove that $Y$ is complete, or give a counter-example.
 
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 {\bf 8.}   [\#4*, page 97] ~     Let $M$ be a metric space. Prove that $M$ is separable if and only if every collection of disjoint open sets of $M$ is countable.     
 
  

 
  
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