We study locally compact group topologies on semisimple Lie groups. We show that the Lie group topology on such a group $S$ is very rigid: every 'abstract' isomorphism between $S$ and a locally compact and $\sigma$-compact group $\Gamma$ is automatically a homeomorphism, provided that $S$ is absolutely simple. If $S$ is complex, then non-continuous field automorphisms of the complex numbers have to be considered, but that is all.
To appear in Adv. Math.