How do you show that the closure of an open ball is a closed ball?
Let A be the the closure of the open ball B
Let x be in
We want to show is open (because a set is closed if its complement is open)
Suppose is not open, then for all r > 0, the open ball intersects A. But then x is in the closure of A which equals A. Contradiction
southprkfan1, does that show that the closure is a closed ball?
Oh, is does show that the complement of a closed set is open. BUT?
Here is a hint: In a metric space the closure of a set is the set of points a distance of zero from the set.