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 isobviousthat 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 ?