Proof That The Empty Set, The Set Itself, and the Union and Intersection of Open Sets in a Metric Space Are Open Sets
Letbe a metric space. Then
1.and- whereis the empty set - are open sets.
2. the intersection of any two open sets is an open set.
3. The union of any family of open sets is an open set.
1. Ifandthenhenceis an open set.
For eachand henceis an open set.
2. Supposeandare open subsets ofandSinceandare open, open ballsandexist such thatand
Setto gethenceis an open set.
3. Letbe a union of a family of open sets.
An open setexists such that
and sinceis open,