§ 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:
If a family of compact, countably infinite sets
S_a have all
finite intersections non-empty, then the intersection of the family
S = intersection of S_a. We know that
S must be compact since
S_a are compact, and the intersection of a countably infinite
number of compact sets is compact.
S be empty. Therefore, this means there must be a point
p ∈ P
p !∈ S_i for some arbitrary
§ 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 ...
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 thathas the highest index in the finite intersection.
- Each of the sets
Si are compact since they are closed and bounded subsets of
- Invoke theorem.