Proof That The Empty Set, The Set Itself, and the Union and Intersection of Open Sets in a Metric Space Are Open Sets
Theorem
Let
be 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. If
and
then
henceis an open set.
For each
and 
hence
is an open set.
2. Suppose
and
are open subsets ofand
Sinceandare open, open balls
and
exist such that
and
Set
to get
hence
is an open set.
3. Let
be a union of a family of open sets.
Suppose
An open set
exists such that
and sinceis open,
