I was looking for an alternative proof for the finite intersection property:
$$ if \{ K_{\alpha} \}_{\alpha} \text{ $K_{\alpha}$ compact }, \cap^n_{i=1} K_{\alpha_i} \neq \emptyset, n < \infty \implies \cap_{\alpha} K_{\alpha} \neq \emptyset $$
I've studied the proof from Rudin 2.36 in detail and addressed the point I didn't understand in an earlier question I asked last week. I feel I have a good grasp of the proof mechanics and flow of it (and I've been able to reprove it myself). However, the concept on itself seems quite unintuitive still to me (finite intersection property).
Unlike the nested interval theorem or intersecting a bunch of closed sets nested intervals seems quite obvious even without a proof that there will be a point in the intersection. For example, if we shrink nested intervals on itself since they do include the edges its just not possible to have it be empty. However, the proof for finite intersection property depends on the definition of compactness that didn't seem exactly intuitive to me (perhaps thats the issue [the definition was that every open cover has a finite subcover]). However, the definition of limit point compactness is equivalent and for me a lot more natural so I became really curious to search for a proof of finite intersection property using that definition. Perhaps I just don't see what is so special about compact sets that allow this proof to really conceptually work (despite the proof in Rudin being 100% clear to me).
So this is what I've tried. First I recalled the definition:
Limit point compact means that every subset of $X$ has a limit point in $X$.
Which I sort of relate to as meaning that if you try to select an infinite subset $E$ if $X$ is limit point compact then any infinite subset like that will have a limit point inside $X$. This to me means that even if you tried to create an infinite set/sequence that went off to infinite it would necessarily halt inside $X$. Especially if it tries to escape $X$ it would halt at the "edges". This type of thinking has a similar quality to my intuition for nested interval theorem. Since the sets are limit point compact then you always converge to something within the set. So it nearly feels like your saying that the edges are always present. But I'm having a hard time using that fact to relate it to the intersection of all the sets $\cap_{\alpha} K_{\alpha}$. I think what makes it especially challenging me to use this definition is that in this theorem things are not nested so its not immediately clear to me how limit point compactness would come in however, if things were nested, then limit point compactness would mean the edges are included so collapsing everything to a point should mean we collapse to some point in someone's edge.
Does anyone know how to generalize this to any finite intersection using the definition of limit point compactness? It feels so close!!!
Obviously this question is not useful I think because I understand the proof using the standard definition of compactness i.e. that every open cover has a finite subcover.
