I love compactness. I really do. It turns infinite things into (almost) finite things. I could gush about how great it is, but instead let me tell you about one problem where compact sets act as the delimiter.
Here is one way to characterize compactness:
A space X is compact if and only if any family of closed sets with the Finite Intersection Property (FIP) has a common point.
[Remember that a collection has the FIP if every finite subcollection has a common point (i.e. has non-empty intersection).]
This has a pretty clear connection to filters, as filters are collections of sets with the FIP (and the intersection is in the filter!) and closed under supersets. One example of a filter is the collection of all subsets of the real line that contain a closed interval around 0.
A closely related notion is that of being 2-linked. A collection A is 2-linked if any two sets in A have non-empty intersection. For example the collection of real intervals is 2-linked. Another example is the set of sides of a
polygon triangle. (Why not a square?)
Then of course we can talk about being 3-linked which means that any 3 sets have non-empty intersection (we will now say that this is called ‘meeting’). Obviously, is 2-linked, but not 3-linked. (edit: Yeah, so not only is this not obvious, but it is not true! I address this here.)
Then we could go on to define n-linked for an arbitrary natural number n.
Question 1: How is the FIP related to being n-linked?
Question 2: Can you find, for each n, an example of a collection that is n-linked but not n+1 linked?
Question 3: How is n-linkedness related to the dimension of the real line?
I’ll get to these later. But you should think about them. 1 and 2 are not hard. 3 takes some thought, but just try to come up with a conjecture.