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This paper extends the $T$-$\Omega$ formulation for eddy currents based onhigher order hierarchical basis functions so that it can deal with conductorsof arbitrary topology. To this aim we supplement the classical hierarchicalbasis functions with non-local basis functions spanning the first de Rhamcohomology group of the insulating region. Such non-local basis functions maybe efficiently found in negligible time with the recently introducedD{\l}otko--Specogna (DS) algorithm.

$T$-$\Omega$ formulation with higher order hierarchical basis functions for non simply connected conductors