Theorem
Letbe a metric space. Then
1.and
- where
is 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. Ifand
then
hence
is an open set.
For eachand
hence
is an open set.
2. Supposeand
are open subsets of
and
Since
and
are open, open balls
and
exist such that
and
Setto get
hence
is an open set.
3. Letbe a union of a family of open sets.
Suppose
An open setexists such that
and sinceis open,