Dense Sets
A subset
of a set
is dense in X if the closure ofis equal toie
With this definition the following statements are equivalent.
1.is dense in
2. If
is a closed set,is dense inand
then
3. Each nonempty open set incontains an element of
The complement
ofhas empty interior.
Proof
1. Ifis dense inthenSince
but
sinceis closed hence
therefore
2. Let
be an open set with
Then
contradicting 2 because
is closed.
3. Suppose
is open therefore there is a nonempty set
but
hence
and
contains no points of
Hencecontains some element of
4.
hence
