  Bound states in the continuum (BICs) are counter-intuitive localized states
with eigenvalues embedded in the continuum of extended states. Recently,
nontrivial band topology is exploited to enrich the BIC physics, resulted in
topological BICs (TBICs) with extraordinary robustness against perturbations or
disorders. Here, we propose a simple but universal mirror-stacking approach to
turn nontrivial bound states of any topological monolayer model into TBICs.
Physically, the mirror-stacked bilayer Hamiltonian can be decoupled into two
independent subspaces of opposite mirror parities, each of which directly
inherits the energy spectrum information and band topology of the original
monolayer. By tuning the interlayer couplings, the topological bound state of
one subspace can move into and out of the continuum of the other subspace
continuously without hybridization. As representative examples, we construct
one-dimensional first-order and two-dimensional higher-order TBICs, and
demonstrate them unambiguously by acoustic experiments. Our findings will
expand the research implications of both topological materials and BICs.
