  We study the Balmer spectrum of the category of finite G-spectra for a
compact Lie group G, extending the work for finite G by Strickland,
Balmer-Sanders, Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton and others. We
give a description of the underlying set of the spectrum and show that the
Balmer topology is completely determined by the inclusions between the prime
ideals and the topology on the space of closed subgroups of G. Using this, we
obtain a complete description of this topology for all abelian compact Lie
groups and consequently a complete classification of thick tensor-ideals. For
general compact Lie groups we obtain such a classification away from a finite
set of primes p.
