Discontinuous linear representations of $GL(n,\mathbb{C})$ can be obtained from the so-called "wild" (field) automorphisms of $\mathbb{C}$; but these wild automorphisms in turn require some choice to construct. Is it consistent with ZF that all linear representations of $GL(n,\mathbb{C})$ are continuous?



Discontinuous representations of GL(n,C) in ZF
It is consistent with $\sf ZF+DC$ that every homomorphism between Polish groups is continuous.


This holds in Solovay's model, whose existence requires an inaccessible cardinal, but also in Shelah's model where every set of real numbers have the Baire property -- something which does not require any additional consistency strength.


