  By means of relativistic density functional theory (DFT) calculations we
study electron band structure of the topological insulator (TI) Bi$_2$Se$_3$
thin films deposited on the ferromagnetic insulator (FMI) EuS substrate. In the
Bi$_2$Se$_3$/EuS heterostructure, the gap opened in the spectrum of the
topological state has a hybridization character and is shown to be controlled
by the Bi$_2$Se$_3$ film thickness, while magnetic contribution to the gap is
negligibly small. We also analyzed the effect of Eu doping on the magnetization
of the Bi$_2$Se$_3$ film and demonstrated that the Eu impurity induces magnetic
moments on neighboring Se and Bi atoms an order of magnitude larger than the
substrate-induced moments. Recent magnetic and magneto-transport measurements
in EuS/Bi$_2$Se$_3$ heterostructure are discussed.
