  The interplay between various symmetries and electronic bands topology is one
of the core issues for topological quantum materials. Spontaneous magnetism,
which leads to the breaking of time-reversal symmetry, has been proven to be a
powerful approach to trigger various exotic topological phases. In this work,
utilizing the combination of angle-resolved photoemission spectroscopy,
magneto-optical Kerr effect microscopy and first-principles calculations, we
present the direct evidence on the realization of the long-sought spontaneous
time-reversal symmetry breaking induced topological phase transition in soft
ferromagnetic EuB$_6$. We successfully disentangle the bulk band structure from
complicated surface states, and reveal the hallmark band inversion occurring
between two opposite-parity bulk bands below the phase transition temperature.
Besides, our magneto-optical Kerr effect microscopy result confirms the
simultaneous formation of magnetic domains in EuB$_6$, implying the intimate
link between the topological phase transition and broken time-reversal symmetry
therein. Our results demonstrate that EuB$_6$ provides a potential platform to
study the interplay between the topological phases and tunable magnetic orders.
