  This paper extends the $T$-$\Omega$ formulation for eddy currents based on
higher order hierarchical basis functions so that it can deal with conductors
of arbitrary topology. To this aim we supplement the classical hierarchical
basis functions with non-local basis functions spanning the first de Rham
cohomology group of the insulating region. Such non-local basis functions may
be efficiently found in negligible time with the recently introduced
D{\l}otko--Specogna (DS) algorithm.
