  A hallmark of certain topology, including the Chern number, is the
obstruction to constructing exponentially localized Wannier functions in the
bulk bands. Conversely, other types of topology do not necessarily impose
Wannier obstructions. Remarkably, such Wannier-localizable topological
insulators can host boundary states that are detachable from the bulk bands. In
our accompanying Letter (D. Nakamura et al., arXiv:2407.09458), we demonstrate
that non-Hermitian topology underlies detachable boundary states in Hermitian
topological insulators and superconductors, thereby establishing their tenfold
classification based on internal symmetry. Here, using $K$-theory, we elucidate
the relationship between Wannier localizability and detachability of
topological boundary states. From the boundary perspective, we classify
intrinsic and extrinsic non-Hermitian topology, corresponding to nondetachable
and detachable topological boundary states, respectively. From the bulk
perspective, on the other hand, we classify Wannier localizability through the
homomorphisms of topological phases from the tenfold Altland-Zirnbauer symmetry
classes to the threefold Wigner-Dyson symmetry classes. Notably, these two
approaches from the boundary and bulk perspectives lead to the same
classification. We clarify this agreement and develop a unified understanding
of the bulk-boundary correspondence on the basis of $K$-theory.
