  We suppose that the doping of the 2D hard-core boson system away from
half-filling may result in the formation of multi-center topological
inhomogeneity (defect) such as charge order (CO) bubble domain(s) with Bose
superfluid (BS) and extra bosons both localized in domain wall(s), or a {\it
topological} CO+BS {\it phase separation}, rather than an uniform mixed CO+BS
supersolid phase. Starting from the classical model we predict the properties
of the respective quantum system. The long-wavelength behavior of the system is
believed to remind that of granular superconductors, CDW materials, Wigner
crystals, and multi-skyrmion system akin in a quantum Hall ferromagnetic state
of a 2D electron gas. To elucidate the role played by quantum effects and that
of the lattice discreteness we have addressed the simplest nanoscopic
counterpart of the bubble domain in a checkerboard CO phase of 2D hc-BH square
lattice. It is shown that the relative magnitude and symmetry of
multi-component order parameter are mainly determined by the sign of the $nn$
and $nnn$ transfer integrals. In general, the topologically inhomogeneous phase
of the hc-BH system away from the half-filling can exhibit the signatures both
of $s,d$, and $p$ symmetry of the off-diagonal order.
