In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a separable Hilbert space. Within C${}^*$-algebra there are many examples of nice results that require separability. (Dixmier's problem: is every prime C${}^*$-algebra primitive? Yes for separable C*-algebras, no in general.)

I wondered whether there is a similar phenomenon in pure algebra. Are there good examples of results that hold for countable groups, countable dimensional vector spaces, etc., but fail in general?

One example I know about is Whitehead's problem, which has a positive solution for countable abelian groups, but is independent of ZFC in general.

any non-atomic Boolean algebra is free" is true for countable Boolean algebras but not in general (the countable case is a restatement [thru Stone duality] of the fact that any nonempty metrizable totally disconnected compact space is homeomorphic to the Cantor set). $\endgroup$ – YCor May 9 at 20:3512more comments