  In a certain class of scalar-Gauss-Bonnet gravity, the black holes and the
neutron stars can undergo spontaneous scalarization - a strong gravity phase
transition triggered by a tachyonic instability due to the non-minimal coupling
between the scalar field and the spacetime curvature. Studies of this
phenomenon have so far been restricted mainly to the study of the tachyonic
instability and stationary scalarized black holes and neutron stars. Up to date
there has been proposed no realistic physical mechanism for the formation of
isolated scalarized black holes and neutron stars. We study for the first time
the stellar core collapse to a black hole and a neutron star in
scalar-Gauss-Bonnet theories allowing for a spontaneous scalarization. We show
that the core collapse can produce scalarized black holes and scalarized
neutron stars starting with a non-scalarized progenitor star.
