A 5D view of the Alpha Per, Pleiades, and Praesepe clusters
  Aims: Our scientific goal is to provide revised membership lists of the Alpha
Per, Pleiades, and Praesepe clusters exploiting the second data release of Gaia
and produce five-dimensional maps (ra, dec, parallax, pmRA, pmDEC) of these
clusters.
  Methods: We implemented the kinematic method combined with the statistical
treatment of parallaxes and proper motions to identify astrometric member
candidates of three of the most nearby and best studied open clusters in the
sky.
  Results: We cross-correlated the Gaia catalogue with large-scale public
surveys to complement the astrometry of Gaia with multi-band photometry from
the optical to the mid-infrared. We identified 517, 1248, and 721 bona-fide
astrometric member candidates inside the tidal radius of the Alpha Per, the
Pleiades, and Praesepe, respectively. We cross-matched our final samples with
catalogues from previous surveys to address the level of completeness. We
update the main physical properties of the clusters, including mean distance
and velocity as well as core, half-mass, and tidal radii. We infer updated ages
from the white dwarf members of the Pleiades and Praesepe. We derive the
luminosity and mass functions of the three clusters and compare them to the
field mass function. We compute the positions in space of all member candidates
in the three regions to investigate their distribution in space.
  Conclusions: We provide updated distances and kinematics for the three
clusters. We identify a list of members in the Alpha Per, Pleiades, and
Praesepe clusters from the most massive stars all the way down into the
hydrogen-burning limit with a higher confidence and better astrometry than
previous studies. We produce complete 5D maps of stellar and substellar
bona-fide members in these three regions. ABRIDGED
