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Do you think you could help me with some of these? Thank you.

Suppose $A,B⊆X$ are disjoint and $a∈X\setminus B$. Prove the following:

  1. If $B$ is closed, then $d(a,B)>0$

  2. If $B$ is compact then there is some $b∈B$, such that $d(a,B)=d(a,b)$ (so $d(a,B)>0$)

  3. If $A$ is closed and $B$ is compact, then $d(A,B)>0$

  4. If $A$ and $B$ are compact, then there is some $a$ in $A$ and $b$ in $B$ such that $ d(A,B)=d(a,b)$

  5. Give an example to show that $d(A,B)=0$ is possible for disjoint $A,B⊆X$, with $A$ and $B$ closed.

  • 0
    1) Take a sequence $(b_1,b_2\ldots)$ in $B$ such that $d(a,b_n)$ goes to zero. Show that $a$ is then a limit point of $B$. 2) Take a sequence $(b_1,b_2,\ldots)$ in $B$ such that $d(a,b_n)$ goes to $d(a,B)$. Use compactness to get a convergent subsequence.2012-03-27
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    To prove 1) and 2) you use 3) and 4) because $\{a\} $ is compact set in any Hausdorff space, particularly in a metric space!2012-03-27
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    5) Consider the $x$-axis and the graph of the hyperbola $y=1/x$.2012-03-27

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