  A piecewise Tolman-Bondi-Lemaitre (TBL) cell-model for the universe
incorporating local collapsing and expanding inhomogeneities is presented here.
The cell-model is made up of TBL underdense and overdense spherical regions
surrounded by an intermediate region of TBL shells embedded in an expanding
universe. The cell-model generalizes the Friedmann as well as Einstein-Straus
swiss-cheese models and presents a number of advantages over other models, and
in particular the time evolution of the cosmological inhomogeneities is now
incorporated within the scheme. Important problem of gravitational collapse of
a massive dust cloud, such as a cluster of galaxies or even a massive star, in
such a cosmological background is examined. It is shown that the collapsing
local inhomogeneities in an expanding universe could result in either a black
hole, or a naked singularity, depending on the nature of the set of initial
data which consists of the matter distribution and the velocities of the
collapsing shells in the cloud at the initial epoch from which the collapse
commences.
