There is this exercise: Show that countable compactness is equivalent to the following condition. If ${C_n}$ is a countable collection of closed sets in S satisfying the finite intersection hypothesis, then $\bigcap_{i=1}^\infty C_i$ is nonempty.
Definitions:
- A Space S is countably compact if every infinite subset of S has a limit point in S.
- A space S has the finite intersection property provided that if ${C_\alpha}$ is any collection of closed sets such that any finite number of them has a nonempty intersection, then the total intersection $\bigcap_\alpha C_\alpha$ is non-empty
- A family of closed sets, in any space, such that any finite number of them has a nonempty intersection, will be said to satisfy the finite intersection hypothesis.
Now there is also a related theorem in the book: Compactness is equivalent to the finite intersection property.
Sounds to me countable compactness and compactness are pretty much the same.
I am not asking for a solution to the exercise. My question is this:
What is the difference between Compactness and Countable Compactness in terms of closed Collections? Both things sound to me like this: Given a collection of closed sets, when a finite number of them has a nonempty intersection, all of them have a nonempty intersection.
BTW: the definitions, the theorems and the exercise are from Topology by Hocking/Young.