Hello,
U is an open set in this topology. Let's assume that U is not the empty set.
Hence is finite and is a closed set of this topology.
Similarly, it is obvious that any closed set of this topology is finite, except , which is closed and open.
We have
But is an infinite set, and is a finite set.
Thus U is an infinite set.
A definition of U being dense in , is that the unique closed set containing U is itself.
But a closed set in the finite complement topology is, as said before, finite.
Thus there is no possible closed set containing U, except
This proves that U is dense in
Now you have to study the situation where U= . It looks simple, but I'm quite confused... can you try it ?