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%%% exercise set 3
%%% september 21, 2009
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Math 445,  Fall 2009 \hfill  Exercise Set \#3 \hfill Turn in: September 21 
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 {\bf 1.}   [\#11, page 70] ~     Let $D_1$ and $D_2$ be metrics on a single space $M$. Which of the following are metrics on $M$: $D_1 + D_2$, $\max \{D_1 , D_2\}$, $\min \{D_1, D_2\}$?
 

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   {\bf 2.}   [\#14, page 71] ~     Let $M$ be a metric space in which the distance function assumes only the values $0,1,3$. Define $x \sim y$ to means $D(x,y) \leq 1$. Prove that $\sim$ is an equivalence relation on $M$. Show also that $\sim$ determines the metric $D$.
 

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  {\bf 3.}   [\#1, page 74] ~     Let $M, D$ be a metric space.   Prove that:
  
  a) For every $x \in M$, the complement $V_x = M- \{x\}$ is open. [Points are closed.]
  
  b) For any set $X \subset M$, then $X$ is the intersection of open sets. [The problem is to find enough open sets. A finite number will not suffice, unless $X$ is itself open.]

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  {\bf 2.}   [\#2, page 74] ~     Let $x, y \in M$ be distinct points in a metric space $M, D$. Prove that there exists  disjoint open sets $U, V \subset M$ with $x \in U$ and $y \in V$.  

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  {\bf 5.}   [\#5, page 74] ~       Let $M = \mR$ be the real line, with the metric $D(x,y) = |x-y|$. Prove that there are no isolated points in $\mR$. [A point $x \in M$ is \emph{isolated} if there exists an open set $U$ such that $U \cap M = \{x\}$.]

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  {\bf 6.}   [\#8, page 74] ~       Let $x$ be a point of as metric space $M$. Prove that the following two statements are equivalent:
  
  a) $x$ is not isolated.
  
  b) Every neighborhood of $x$ contains an infinite number of points of $M$.

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  {\bf 7.}   [\#9, page 74] ~       Let $M$ be an infinite metric space. Prove that $M$ contains an open set $U$ such that both $U$ and its complement $M - U$ are infinite.

 
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