John Carlos Baez on Nostr: As an abstract group, ignoring its topology, the unit circle is isomorphic to ℚ/ℤ ...

As an abstract group, ignoring its topology, the unit circle is isomorphic to ℚ/ℤ ⊕ ℝ. The first summand contains the roots of unity, and the second contains everything else.

I consider this as evidence that you should not ignore the topology.

Also, this result relies on the axiom of choice - or at least a special case of that axiom, like choosing a basis for the reals as a rational vector space, and choosing a same-sized basis for the elements of the circle that aren't roots of unity, and concluding that they're isomorphic.