- The property of being a linear ordering has finite character, i.e. a relation linearly orders a set if it linearly orders all of its finite subsets. This is a trivial corollary of the definition.
- The property of being linearly independent has finite character, i.e. a set of vectors is linearly independent if every one of its finite subsets is linearly independent. This is immediate from the definition.
- $n$-colorability of graphs has finite character, i.e. a graph is $n$-colorable if all of its finite subsets are $n$-colorable. This does not follow at once from the definition, but requires proof.
- Satisfiability of a set of first-order formulas has finite character, i.e. the set is satisfiable if every one of its finite subsets is satisfiable. This is the compactness theorem of first-order logic.
So my question is: Which examples of this concept are most worth knowing about, because of either utility or beauty or begin consequential in mathematics or for other reasons? Maybe the top two dozen or so examples?