1. ## separation axioms

prove that :
every metric space is T3

2. Originally Posted by flower3
prove that :
every metric space is T3
What we need to show is that given a closed set C and a point $x \not \in C$ that we can find disjoint open sets around each point.

define $r=\inf \{d(x,y) | y \in C \}$

Now lets make a disjoint open set that covers C

$O = \cup_{y \in C}B_{r/2}(y)$ since the arbitarty untion of open sets is open O is an open set such that $C \subset O$

now consider the ball $B_{r/2}(x)$ is an open set containing x.

These two sets are disjoint(why?) and fulfill the requirements

3. These two sets are disjoint(why?)
basic open sets??!!!