Topology: Finite Complement & Defining Limit Points

[doodlepin]

Question is: X is a set and tau is a collection of subsets O of X such that X - O is either finite or all of X. Show this is a topology and completely described when a point x in X is a limit of a subset A in X.

I already proved that this satisfies the conditions for defining a topology (called the finite complement topology) But I am having a lot of trouble with defining when a point is a limit. I know that any open subset (defined by tau) containing such a limit point must have a non-empty intersection with A for it to be a limit point.

I have tried considering multiple cases: I know if X is finite then it is the discrete topology and therefore no limit points exist. So the non trivial case if when X is infinite.
Now, if A is finite then for any open subset containing x, O, the complement with A would obviously be finite so therefore non empty?
I'm sort of stuck and could use a nudge in some helpful direction.

 

[micromass]

You are correct in saying that the only case you need to find out is when X is infinite.
Now there are two more cases to consider:
1) A is finite: I claim that A has no limit points in this case. Hint: consider X/A
2) A is infinite: I claim that every point is a limit point in this case. Take U open, then X/U is finite. So it can not happen that $$A\subseteq X\setminus U$$. I'll let you complete the proof...

 

[doodlepin]

wow you are totally right. Thank you so much. Sometimes you start thinking too hard about something and you start to over analyze it. :)