I have a set X={1,2,3...} of positive integers and tau (I'll denote T) that is the complement topology on X.
I have that the complement topology means that given a set O is open if and only if O= nulset or the C(O) is finite or empty. C(O) is the complement of O.
I need to show 2 things.
1) Prove X is connected.
X is connected if the only two subsets of X that are simultaneously both open and closed are X itself and the empty set. I can see this has something to do with the fact that the topology on X has two open sets, the nulset and the complement. I'm not sure how to apply them to each other. I need to show that the whole set X and the empty set are both open and closed and that they are the only two subsets with this quality.
2) Prove that {1,2} (a subset of X) is not connected.
I'm not sure how to begin this. I know that let A={1,2} then A is connected if the only two subsets of A that are simultaneously relatively open and relatively closed in A are the whole set A and the empty set.
I've listed the subsets of A as {1},{2},{nulset}, {1,2} which I guess means I need to show that {1} and {2} are relatively open and relatively closed which would imply that more than just the whole set and the nulset are closed and open? I'm not sure.
-I didn't realize that had this in urgent help. So I tried to figure out how to just move it here, but I couldn't. That is why I''m putting it twice. Sorry