What does *final* intersection mean?
Hello, thank you very much for reading.
I'm totally stuck on the following statement, which I need to prove:
Let {Ki} be the group of compact sets, which also satisfy the fact that every *final* intersection of them is NOT empty.
I need to prove that any intersection of the Ki's is not empty (non-final intersections as well).
I'm trying to play with the fact that if a set is compact, it is partial to a final unification of open sets.... But I can't see how to prove it :-\
Replies are most welcomed and wanted!!!
thanks,
Tomer.
I'm sorry, Enlish isn't my native tounge.
By final I mean... a finite number of sets in the intersection (as opposed to infinite).
Actually, I have already submitted my assignment, with this question partially solved, and afterwords asked my proffesor to explain the answer for me, so it's not relavent anymore.
Thank you very much, regardless!
This is called the finite intersection property for compact sets.
The FIP theorem works in any topological space (compact or not). It says that if a collection of compact subsets has the property that the intersection of any finite subcollection is nonempty then the intersection of the entire collection is nonempty.
To prove it, you can replace the ambient space by one of the sets in the collection. In that way, you can assume that you are working in a compact space.