00:01
Hi, in this video we have a set x such that there's at least two members, so there's at least two elements in the set, and we want to show that no metric generates the trivial topology.
00:35
So this is actually a very simple proof, and so if you remember what it means for a metric to generate a topology, it means that the open sets, or i'll say actually a basis for the topology, is the set of balls of radius r about points x.
01:07
So x is just any point, and r is any number, and br of x is just the set of points whose distance is within r.
01:23
So the key here is that if we have two points, x and y, and they're not the same point, then by definition of a metric, the distance from x to y has to be positive.
01:51
So then what we can do is we can choose r such that r is less than, or it's greater than zero, but less than the distance from x to y...