I was trying to prove that a collection of sets is a topological space over a set and I came over this question which I can't figure out now:
Can the infinite intersection of infinite sets (all subsets of a given set) be finite but not empty?
If it is possible can anyone construct any example?
EDIT:
So the actual question was:
Let $X$ be a an infinite set. Does the following collection of subsets of $X$ given by $T=\{U \in X : X \backslash U$ is infinite or empty $\}$ defines a topology?
Answer : No.
Which now by your help I can prove. Thnx everyone 🙂
Best Answer
Take the sets $S_n = \{n^k \colon k \ge 0\}$, $\bigcap_{n \ge 1} S_n = \{1\}$.