# how to show dense in Finite complement topology

• Apr 15th 2009, 08:09 PM
monkey.brains
how to show dense in Finite complement topology
Let U be any open set in the finite complement topology on http://www.mathhelpforum.com/math-he...a3150bfd-1.gif.

Show that U is dense in http://www.mathhelpforum.com/math-he...a3150bfd-1.gif
• Apr 16th 2009, 11:35 AM
Moo
Hello,
Quote:

Originally Posted by monkey.brains
Let U be any open set in the finite complement topology on http://www.mathhelpforum.com/math-he...a3150bfd-1.gif.

Show that U is dense in http://www.mathhelpforum.com/math-he...a3150bfd-1.gif

U is an open set in this topology. Let's assume that U is not the empty set.
Hence $\mathbb{R}\backslash U$ is finite and is a closed set of this topology.
Similarly, it is obvious that any closed set of this topology is finite, except $\mathbb{R}$, which is closed and open.

We have $U \cup \{\mathbb{R}\backslash U\}=\mathbb{R}$
But $\mathbb{R}$ is an infinite set, and $\mathbb{R}\backslash U$ is a finite set.
Thus U is an infinite set.

A definition of U being dense in $\mathbb{R}$, is that the unique closed set containing U is $\mathbb{R}$ 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 $\mathbb{R}$
This proves that U is dense in $\mathbb{R}$

Now you have to study the situation where U= $\emptyset$. It looks simple, but I'm quite confused... can you try it ? :p