Can the intersection over a finite set be written as a sum?

[Raziel2701]

I know the union can be, but how about the intersection? I am trying to prove that:

Suppose (X,T) is a finite topological space, n is a positive integer and $$U_i\in T$$ for 1<= i <= n. Use mathematical induction to prove $$\bigcap U_i \in T$$, where the intersection goes from i=1 to n.

 

[lanedance]

can you show the intersection of 2 open sets is open?

 

[Mark44]

I don't see that open or closed enters into this problem, unless I'm missing something. For the base case, show that if two sets U₁ and U₂ are in T, then their intersection is also in T.

 

[lanedance]

fair point, depending where you start from you can do it stright from the definition of the sets in T, but those sets are the open sets