Cantors intersection theorem states :
If $\cdots\subset C_3 \subset C_2 \subset C_1$ is a decreasing sequence of non-empty sequentially compact subsets of $\Bbb R^n$, then the intersection of these sets is non-empty.
My question :
How does one use this theorem to prove properties of metric spaces. specifically does it help at all in proofs regarding showing a set is open/closed , complete or compact ?
Edit: I merely wish to know is it possible to use it tro prove any of these properties, as it is a proof I learned in my metric spaces class but I can not find an immediate use for it nor have I seen it used in any proofs I have come across regarding properties of metric spaces. So I wanted to know if this was perhaps just thrown in to prepare for point set topology where it may become more useful or is it useful now and I just haven't realised yet in what way (Edit after 2 votes to close because the question seemed too broad)
One usage that I like is to prove a theorem from Sierpinski. Namely The Euclidean plane $\mathbb R^2$ is not the union of disjoint circles.
The idea of the proof is to proceed by contradiction and to find a sequence of decreasing closed disks such that the next one has a radius less than half of the previous one. The intersection of those disks is not empty according to Cantor’s intersection theorem, and is reduced to a single point. A circle can’t pass through this intersection point.
Cantor’s intersection theorem can also be used to prove that Cantor set is not empty.
Not sure this is what you are looking for though...