  We re-examine the band structure of the stripe charge ordered state of
$\alpha$-(BEDT-TTF)$_2$I$_3$ under pressure by using an extended Hubbard model
within the Hartree mean-field theory. By increasing pressure, we find a
topological transition from a conventional insulator with a single-minimum in
the dispersion relation at the M-point in the Brillouin zone, towards a new
phase which exhibits a double-minimum. This transition is characterized by the
appearance of a pair of Dirac electrons with a finite mass. Using the
Luttinger-Kohn representation at the M-point, it is shown that such a variation
of the band structure can be described by an effective $2 \times 2$ low energy
Hamiltonian with a single driving parameter. The topological nature of this
transition is confirmed by the calculation of the Berry curvature which
vanishes in the conventional phase and has a double peak structure with
opposite signs in the new phase. We compare the structure of this transition
with a simpler situation which occurs in two-component systems, like
boron-nitride.
