How about the compactness theorem which appears almost everywhere in Model Theory? It says a set of the first-order sentences has a model if and only if every finite subset of it has a model. There is a similarity between it and the finite intersection property for compact topological spaces too. This latter says a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The interesting thing that exists about that is one can find a topological proof for model theoretical compactness theorem. Another interesting thing is that among the three most important topological features, I mean completeness, connectedness, and compactness, the compactness appears in many places. Something like a deep property.