  Photonic topological insulators exhibit bulk-boundary correspondence, which
requires that boundary-localized states appear at the interface formed between
topologically distinct insulating materials. However, many topological photonic
devices share a boundary with free space, which raises a subtle but critical
problem as free space is gapless for photons above the light-line. Here, we use
a local theory of topological materials to resolve bulk-boundary correspondence
in heterostructures containing gapless materials and in radiative environments.
In particular, we construct the heterostructure's spectral localizer, a
composite operator based on the system's real-space description that provides a
local marker for the system's topology and a corresponding local measure of its
topological protection; both quantities are independent of the material's bulk
band gap (or lack thereof). Moreover, we show that approximating radiative
outcoupling as material absorption overestimates a heterostructure's
topological protection. As the spectral localizer is applicable to systems in
any physical dimension and in any discrete symmetry class, our results show how
to calculate topological invariants, quantify topological protection, and
locate topological boundary-localized resonances in topological materials that
interface with gapless media in general.
