  We study the Carath\'eodory metric on some generalized Teichm\"uller spaces.
Earle showed that the Carath\'eodory metric is complete on any Teichm\"uller
space. Miyachi extended this result for Asymptotic Teichm\"uller spaces. We
study the completeness of the Carath\'eodory metric on product Teichm\"uller
spaces and on the Teichm\"uller space of a closed set in the Riemann sphere.
