The Compactness Theorem is equivalent to the compactness of the Stone space of the Lindenbaum–Tarski algebra of the first-order language L. (This is also the space of 0-types over the empty theory.)
A point in the Stone space SL is a complete theory T in the language L. That is, T is a set of sentences of L which is closed under logical deduction and contains exactly one of σ or ¬σ for every sentence σ of the language. The topology on the set of types has for basis the open sets U(σ) = {T: σ ∈ T} for every sentence σ of L. Note that these are all clopen sets since U(¬σ) is complementary to U(σ).
To see how the Compactness Theorem implies the compactness of SL, suppose the basic open sets U(σi), i ∈ I, form a cover of SL. This means that every complete theory T contains at least one of the sentences σi. I claim that this cover has a finite subcover. If not, then the set {¬σi: i ∈ I} is finitely consistent. By the Compactness Theorem, the set consistent and hence (by Zorn's Lemma) is contained in a maximally consistent set T. This theory T is a point of the Stone space which is not contained in any U(σi), which contradicts our hypothesis that the U(σi), i ∈ I, form a cover of the space.
To see how the compactness of SL implies the Compactness Theorem, suppose that {σi: i ∈ I} is an inconsistent set of sentences in L. Then U(¬σi), i ∈ I, forms a cover of SL. This cover has a finite subcover, which corresponds to a finite inconsistent subset of {σi: i ∈ I}. Therefore, every inconsistent set has a finite inconsistent subset, which is the contrapositive of the Compactness Theorem.