  We probe the local magnetic properties of interfaces between the insulating
ferromagnet EuS and the topological insulator Bi$_2$Se$_3$ using low energy
muon spin rotation (LE-$\mu$SR). We compare these to the interface between EuS
and the topologically trivial metal, titanium. Below the magnetic transition of
EuS, we detect strong local magnetic fields which extend several nm into the
adjacent layer and cause a complete depolarization of the muons. However, in
both Bi$_2$Se$_3$ and titanium we measure similar local magnetic fields,
implying that their origin is mostly independent of the topological properties
of the interface electronic states. In addition, we use resonant soft X-ray
angle resolved photoemission spectroscopy (SX-ARPES) to probe the electronic
band structure at the interface between EuS and Bi$_2$Se$_3$. By tuning the
photon energy to the Eu anti-resonance at the Eu $M_5$ pre-edge we are able to
detect the Bi$_2$Se$_3$ conduction band, through a protective Al$_2$O$_3$
capping layer and the EuS layer. Moreover, we observe a signature of an
interface-induced modification of the buried Bi$_2$Se$_3$ wave functions and/or
the presence of interface states.
