§ Using compactness to argue about covers
I've always seen compactness be used by starting with a possibly infinite
coverm and then filtering it into a finite subcover. This finite
subcover is then used for finiteness properties (like summing, min, max, etc.).
I recently ran across a use of compactness when one starts with the set
of all possible subcovers , and then argues about why a cover cannot be built
from these subcovers if the set is compact. I found it to be a very cool
use of compactness, which I'll record below:
§ Theorem:
If a family of compact, countably infinite sets S_a have all
finite intersections non-empty, then the intersection of the family S_a
is non-empty.
§ Proof:
Let S = intersection of S_a. We know that S must be compact since
all the S_a are compact, and the intersection of a countably infinite
number of compact sets is compact.
Now, let S be empty. Therefore, this means there must be a point p ∈ P
such that p !∈ S_i for some arbitrary i.
§ Cool use of theorem:
We can see that the cantor set is non-empty, since it contains a family
of closed and bounded sets S1, S2, S3, ... such that S1 ⊇ S2 ⊇ S3 ...
where each S_i is one step of the cantor-ification. We can now see
that the cantor set is non-empty, since:
- Each finite intersection is non-empty, and will be equal to the set that has the highest index in the finite intersection.
- Each of the sets
Si are compact since they are closed and bounded subsets of R - Invoke theorem.