prove that :
every metric space is T3
What we need to show is that given a closed set C and a point $\displaystyle x \not \in C$ that we can find disjoint open sets around each point.
define $\displaystyle r=\inf \{d(x,y) | y \in C \}$
Now lets make a disjoint open set that covers C
$\displaystyle O = \cup_{y \in C}B_{r/2}(y)$ since the arbitarty untion of open sets is open O is an open set such that $\displaystyle C \subset O$
now consider the ball $\displaystyle B_{r/2}(x)$ is an open set containing x.
These two sets are disjoint(why?) and fulfill the requirements