Generalization of a theorem in Real Analysis

[Silviu]

Homework Statement

If $\{K_\alpha\}$ is a collection of compact subsets of a metric space X, such that the intersection of every finite subcollection of {$K_\alpha$} is nonempty, then $\cap K_\alpha$ is nonempty. Generalize this theorem and proof the generalization. Why doesn't it make sense to state the theorem in its general form?

Homework Equations

The Attempt at a Solution

I know how to prove the actual theorem, but I don't really know what is its generalization in order to attempt to prove it. I don't want yet a proof of the generalization, just its statement. Thank you.

 

[Mark44]

Homework Statement

If $\{K_\alpha\}$ is a collection of compact subsets of a metric space X, such that the intersection of every finite subcollection of {$K_\alpha$} is nonempty, then $\cap K_\alpha$ is nonempty. Generalize this theorem and proof the generalization. Why doesn't it make sense to state the theorem in its general form?

Homework Equations

The Attempt at a Solution

I know how to prove the actual theorem, but I don't really know what is its generalization in order to attempt to prove it. I don't want yet a proof of the generalization, just its statement. Thank you.

Seems to me that the generalization would change "finite subcolledtion" to "infinite subcollection."

 

[Silviu]

Seems to me that the generalization would change "finite subcolledtion" to "infinite subcollection."

Thank you for your reply. However, doesn't that make the statement redundant? If any infinite intersection is non-empty, the intersection of all K is by assumption non-empty, so there is nothing left to prove (and it seems actually weaker than the other one).

 

[Ray Vickson]

Homework Statement

If $\{K_\alpha\}$ is a collection of compact subsets of a metric space X, such that the intersection of every finite subcollection of {$K_\alpha$} is nonempty, then $\cap K_\alpha$ is nonempty. Generalize this theorem and proof the generalization. Why doesn't it make sense to state the theorem in its general form?

Homework Equations

The Attempt at a Solution

I know how to prove the actual theorem, but I don't really know what is its generalization in order to attempt to prove it. I don't want yet a proof of the generalization, just its statement. Thank you.

There are generalizations of metric spaces. See, eg., https://en.wikipedia.org/wiki/Normal_space . In some of those spaces you might be able to generalize the concept of "compactness"; see, eg., https://en.wikipedia.org/wiki/Compact_space

Perhaps a version of your cited theorem remains true for some of those generalizations.