Proof that Closed Disjoint Subsets of a Metric Space are Contained in Open Disjoint Subsets
Letbe a metric. Ifandare closed subsets ofwith then open setsandexist such thatand
For eachdefineand for each
Both setsandare open because each is the union of a family of open sets. Since includes eachSimilarly B subset B_1 .
Supposethen for someand for some
This is a contradiction hence