  Moir\'e superlattices created by the twisted stacking of two-dimensional
crystalline monolayers can host electronic bands with flat energy dispersion in
which interaction among electrons is strongly enhanced. These superlattices can
also create non-trivial electronic band topologies making them a platform for
study of many-body topological quantum states. Among the moir\'e systems
realized to date, there are those predicted to have band structures and
properties which can be controlled with a perpendicular electric field. The
twisted double bilayer graphene (TDBG), where two Bernal bilayer graphene are
stacked with a twist angle, is such a tunable moir\'e system, for which partial
filling of its flat band, transport studies have found correlated insulating
states. Here we use gate-tuned scanning tunneling spectroscopy (GT-STS) to
directly demonstrate the tunability of the band structure of TDBG with an
electric field and to show spectroscopic signatures of both electronic
correlations and topology for its flat band. Our spectroscopic experiments show
excellent agreement with a continuum model of TDBG band structure and reveal
signatures of a correlated insulator gap at partial filling of its isolated
flat band. The topological properties of this flat band are probed with the
application of a magnetic field, which leads to valley polarization and the
splitting of Chern bands that respond strongly to the field with a large
effective g-factor. Our experiments advance our understanding of the properties
of TDBG and set the stage for further investigations of correlation and
topology in such tunable moir\'e systems.
