  The phase structure of ideal Bose gas system within different boundary
conditions, i.e., the periodic boundary condition and Dirichlet boundary
condition in this work, in an infinite volume, is investigated. It is found
that the ground states of ideal Bose gas within those two boundary conditions
are both topologically nontrivial, which can not be classified by the
traditional symmetry breaking theory. The ground states are different
topological phases corresponding to those two boundary conditions, which can be
distinguished by the off--diagonal particle number susceptibility. Moreover,
this result is universal. The boundary condition may play an important role in
pining the critical endpoint of QCD diagram on the approach of the lattice
simulations and the computation of some solvable statistical models .
