TITLE: Infinite intersection of infinite sets
QUESTION [5 upvotes]: 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 :)

REPLY [3 votes]: This may be helpful for you:

Let $X=\mathbb N$, and let $\mathcal U=\{\{1,3, 5\}, \{1,3, 5, 0\}, \{1,3, 5, 0, 2 \}, \{1,3, 5, 0, 2 , 4\},....\}$.
  Clearly, the infinite intersection of $\mathcal U$ is always $\{1,3,5\}$, hence finite.


I may introduce a definition of $calibre-{\aleph_1}$ which between Countable chain condition and separable. 

A topological space $X$ is called $calibre-{\aleph_1}$ if for any uncountable nonempty open sets of $X$, there is an uncountable subcollection such that $\bigcap\{U_\xi: \xi < \aleph_1\}\not=\emptyset$.