Non-Hermitian band theory distinguishes between line gaps and point gaps. While point gaps can give rise to intrinsic non-Hermitian band topology without Hermitian counterparts, line-gapped systems can always be adiabatically deformed to a Hermitian limit. Here, we show that line-gap topology and point-gap topology can be intricately connected: topological line-gapped systems in
d
dimensions induce nontrivial point-gap topology on their
(
d
−
1
)
-dimensional boundaries when suitable internal and spatial symmetries are present. Since line-gapped systems essentially realize Hermitian topological phases, this establishes a correspondence between Hermitian bulk topology and intrinsic non-Hermitian boundary topology. For the correspondence to hold, no non-Hermitian perturbations are required in the bulk itself, so that the bulk can be purely Hermitian. Concomitantly, the presence of non-Hermitian perturbations in the bulk does not affect any results as long as they do not close the bulk line gap. On the other hand, non-Hermitian perturbations are essential on the boundary to open a point gap. The non-Hermitian boundary topology then further leads to higher-order skin modes, as well as chiral and helical hinge modes, that are protected by point gaps and hence unique to non-Hermitian systems. We identify all the internal symmetry classes where bulk line-gap topology induces boundary point-gap topology as long as an additional spatial symmetry is present and establish the correspondence between their topological invariants. There also exist some symmetry classes where the Hermitian edge states remain stable, in the sense that even a point gap cannot open on the boundary.