Whst's the relationship between these two theorems? When studying Computational Logic last year, we had to prove a theorem called "Compactness Theorem" ("Teorema de Compacidad in Spanish) which states that a set of formulae $\Gamma$ is satisfiable if and only if every finite subset of $\Gamma$ is satisfiable  I am now going over Rudin's Principles of Mathematical Analysis to "master the basics" and came upon a theorem that states that for a collection of compact sets $\lbrace K_\alpha \rbrace$ if the intersection of every finite subcollection is nonempty, then the intersection of the whole collection is nonempty.  These two theorems seem to state extremely similar things, so I was wondering how they're connected
Why is compactness in logic called compactness? In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set.   Most logic texts either don't explain the terminology, or allude to the topological property of compactness. I see an analogy as, given a topological space X and a subset of it S, S is compact iff for every open cover of S, there is a finite subcover of S. But, it doesn't seem strong enough to justify the terminology.   Is there more to the choice of the terminology in logic than this analogy?
Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$[NBHM_PhD Screening Test-2013, Topology] Which of the following sets are compact in $\mathbb{M}_n(\mathbb{R})$ (a) The set of all upper triangular matrices all of whose eigenvalues satisfy $|\lambda|≤2 $. (b) The set of all real symmetric matrices all of whose eigenvalues satisfy $|\lambda|≤2 $. (c) The set of all diagonalizable matrices all of whose eigenvalues satisfy $|\lambda|≤2 $.  (a) &amp; (c) are not true. But I am not sure about (b).can anybody help me please.