1
$\begingroup$

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.

  • 1
    5) Consider the $x$-axis and the graph of the hyperbola $y=1/x$.2012-03-27

1 Answers 1

1
  1. What can you say about the complement of a closed set? $x$ is in it.

  2. $x\mapsto d(a,x)$ is continuous.

I'll stop there, just noting that similar ideas will deal with 3 and 4.