  Abnormal electronic properties of complex systems require new ideas
concerning explanation of their behavior and possibility of realization. In
this acticle we show that a colossal magnetoresistance is realized in the state
of the topological Kondo insulator, that is similar to the Kondo insulator
state in the Kondo lattice. The mechanism of the phenomenon is the following:
in the spin gapless phase an external magnetic field induces the gap in the
spectrum of spin excitations, the gap in the spectrum of fermions is opened due
to a hybridization between spin and fermion subsystems at half filling, as the
result the magnetic field leads to metal-insulator (or bad metal - insulator)
phase transition. A model of the topological Kondo lattice defined on a
honeycomb lattice is studied for the case when spinless fermion bands are half
filled. It is shown that the hybridization between local moments and itinerant
fermions should be understood as the hybridization between corresponding
Majorana fermions of the spin and charge sectors. The system is a topological
insulator, single fermion and spin excitations at low energies are massive. We
will show that a spin gap induces a gap in the charge channel, it leads to an
appearance of a topological insulator state with chiral gapless edge modes and
the Chern number one or two depending on the exchange integrals' values. The
relevance of this to the traditional Kondo insulator state is discussed.
