# Infinite subset is dense

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.

Staff Emeritus
Gold Member

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

Oxymoron
Yes, that is my question.

Staff Emeritus
Gold Member
So what do the definitions say?

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

Homework Helper
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?

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

is it all of X?

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

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?

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

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.

Oxymoron
I agree, Matt's method was MUCH easier.