We prove that arbitrary homomorphisms from one of the groups ${ \rm Homeo}(\ca)$, ${ \rm Homeo}(\ca)^\N$, ${ \rm Aut}(\Q,<)$, ${ \rm Homeo}(\R)$, or ${ \rm Homeo}(S^1)$ into a separable group are automatically continuous. This has consequences for the representations of these groups as discrete groups. For example, it follows, in combination with a result on V.G. Pestov, that any action of the discrete group ${ \rm Homeo}_+(\R)$ by homeomorphisms on a compact metric space has a fixed point.
Categories: math.LO math.GR