## Proof That Every Metric Space is Normal

Theorem

Every metric spaceis normal.

Proof

A topological space is normal if, given any two disjoint closed subsetsandthere are disjoint open setsandwithand

Letbe the topology induced onby the metricLetandbe disjoint closed sets, then there existssuch that for allfor some

Now takeand

andare disjoint,andsois normal.