The conncetion between logic and topology.

[MathematicalPhysicist]

i think that i read that the compactness theorem in logic has a similar theorem in topology.
i wanted to inquire, are there any other theorems in logic which have similar, dual theorems in topology or other branches in maths?

 

[matt grime]

You must be unique in thinking that compactness is a logical statement with an analogue in topology...

Compactness says, in whatever sense, that something is true for an infinite collection if and only if it is true for something to do with a finite subcollection (possibly all finite subcollections). The analogy is formal, mostly, though D(X) the derived category of sheaves on X is compactly generated in the triangulated category theory sense if and only if X is a compact manifold

 

[MathematicalPhysicist]

should i be flattered by your first sentence?

i just learned the theorem in a logic context, i haven't yet delved in it in topology.

 

[HallsofIvy]

There is a definition of "compactness" in logic and a "compactness" in topology but they are nothing alike: they just have the name in common.
(When I google on "compact" and "logic" to check on this I also find that there is a "Logic compact" camcorder on the market which swamps everything else!)

In topology, a set is said to be "compact" if and only if every cover of the set by open sets contains a finite sub-cover. That means the set will have many of the nice properties of finite sets.

In logic, if I remember correctly, the "compactness" property says that if every finite subset of a collection of axioms has a model, then the entire collection has a model.

I don't see a lot of connection there- except for the "finite" part which may be why the term "compactness".