Note: When
Note: When
The complement of
Suppose that
Now suppose that
Let's start by assuming that we have a set
Note that
Now let's assume that
Let's note that
Since
Suppose that
The other direction is easier, suppose that
Suppose that
Now suppose that
Set
On the other hand, we know that
We prove the contrapositive, that is
Suppose that
Now the reverse direction: assuming that we have some open set
Suppose that
Now suppose that every basis element containing