Proof That The Empty Set, The Set Itself, and the Union and Intersection of Open Sets in a Metric Space Are Open Sets

Theorem

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.

Proof

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.

Suppose

An open setexists such that

and sinceis open,