  Topological insulator-based methods underpin the topological classification
of gapped bands, including those surrounding semi-metallic nodal defects.
However, multiple bands with gap-closing points can also possess non-trivial
topology. We construct a general wavefunction-based ``punctured-Chern"
invariant to capture such topology. To show its general applicability, we
analyze two systems with disparate gapless topology: 1) a recent
two-dimensional fragile topological model to capture the various
band-topological transitions and 2) a three-dimensional model with a
triple-point nodal defect to characterize its semi-metallic topology with
\emph{half-integers} that govern physical observables such as anomalous
transport. This invariant also gives the classification for Nexus triple-points
($\mathbb{Z}\times\mathbb{Z}$) with certain symmetry restrictions, which is
re-confirmed by abstract algebra.
