prove that :
every metric space is T3
What we need to show is that given a closed set C and a point that we can find disjoint open sets around each point.
define
Now lets make a disjoint open set that covers C
since the arbitarty untion of open sets is open O is an open set such that
now consider the ball is an open set containing x.
These two sets are disjoint(why?) and fulfill the requirements