Show that: $\left\{{p : |p-p_o|>r}\right\}$ is open for any $p_o$ and any $r\geq0$.

How will I start this proof? What should I need to show i.e should a delta neighborhood needed? If so, how should I use it?

  • 1
    $\begingroup$ Who is the space $X$? Which topology are you consider on $X$? $\endgroup$
    – InsideOut
    Aug 12 '15 at 12:38

Fix $r$ and $p_0$

Show that every point in your set is an inner point by finding a neighborhood that is contained in your point.

To find such a neighborhood remember that a metric satisfies the triangle inequality. Rearrange it to get a sufficient condition such that points in the neighborhood remain in your set.

I hope I haven't given away too much.

  • $\begingroup$ Since this is your first answer here, I thought I'd point you to a useful link on how to format math. $\endgroup$
    – wltrup
    Aug 12 '15 at 14:08
  • $\begingroup$ Thank you! The link is very useful. Does it make a difference that I am on the mobile version? The subscript does not render. $\endgroup$ Aug 12 '15 at 14:33
  • $\begingroup$ I don't think it should make a difference but I'm not sure. Are you sure you typed \$p_0\$ ? $\endgroup$
    – wltrup
    Aug 12 '15 at 14:35
  • $\begingroup$ You got it working now. $\endgroup$
    – wltrup
    Aug 12 '15 at 14:41
  • $\begingroup$ @wltrup That fixed it. So its basically the same as Latex. Thanks! $\endgroup$ Aug 12 '15 at 14:42

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