Definition.
A subset $E$ of a metric space $(X,\tau)$ is compact if every open cover of $E$ has a finite subcover.
Theorem (Compactness Theorem).
A set $\Gamma$ of formulas is satisfiable if and only if every finite subset $\hat \Gamma\subseteq \Gamma$ is satisfiable.
I've recently began studying metric spaces, and I remembered about the compactness theorem for propositional logic, so I wanted to know what's the connnection between it and the notion of compactness.
Could someone explain this in rather simple terms?