# Infinite subset is dense

1. Sep 3, 2005

### Oxymoron

My problem is:

Consider an infinite set $X$ with the finite complement topology. I want to show that any infinite subset $A$ of $X$ is dense in $X$.

Now, I can show that every point of $X$ is a limit point of $A$.

Can this help me in any way to show that $A$ is dense in $X$. Or could someone provide me with an alternative method.

By the way, my understanding of dense is that a subset of a set is dense if its closure coincides with the set.

2. Sep 3, 2005

### Hurkyl

Staff Emeritus

If every point of X is a limit point of A, then is A dense in X?

3. Sep 3, 2005

### Oxymoron

Yes, that is my question.

4. Sep 3, 2005

### Hurkyl

Staff Emeritus
So what do the definitions say?

5. Sep 3, 2005

### Oxymoron

Wait a sec, if every point of X is a limit point of A, then A is dense in X!

6. Sep 3, 2005

### matt grime

SO, A is any infinite subset and you want to find the smallest closed set containing A. Since there are three kinds of closed set:

the empty set

a set containing a finite number of points

all of X

isn't the answer obvious? I mean, whichof those can contain A, given A is infinite?

7. Sep 3, 2005

### Oxymoron

is it all of X?

8. Sep 3, 2005

### matt grime

well, can you name a finite (or empty set) that contains an infinite subset?

9. Sep 3, 2005

### Oxymoron

nope, I can't.

So am I correct in thinking that by proving that every point of $X$ is the limit point of $A$, then $A$ is dense?

10. Sep 3, 2005

### Hurkyl

Staff Emeritus
That is correct -- the point of my first post was to eliminate the unnecessary stuff, hoping you could see this when that's all that's left.

However, the approach matt has mentioned is a much easier way to do this problem... and is a fairly important theme to understand in general.

11. Sep 3, 2005

### Oxymoron

I agree, Matt's method was MUCH easier.

12. Sep 3, 2005

### HallsofIvy

Staff Emeritus
Once again- look at the DEFINITION of "dense"!