  Hyperbolic metamaterials (HMMs), an unusual class of electromagnetic
metamaterials, have found important applications in various fields due to their
distinctive properties. A surprising feature of HMMs is that even continuous
HMMs can possess topological edge modes. However, previous studies based on
equal-frequency surface (analogy of Fermi surface) may not correctly capture
the topology of entire bands. Here we develop a topological band description
for continuous HMMs that can be described by a non-Hermitian Hamiltonian
formulated from Maxwell's equations. We find two types of three dimensional
non-Hermitian triply-degenerate points with complex linear dispersions and
topological charges $\pm 2$ and 0 induced by chiral and gyromagnetic effects.
Because of the photonic nature, the vacuum band plays an important role for
topological edge states and bulk-edge correspondence in HMMs. The topological
band results are numerically confirmed by direct simulation of Maxwell's
equations. Our work presents a general non-Hermitian topological band treatment
of continuous HMMs, paving the way for exploring interesting topological phases
in photonic continua and device implementations of topological HMMs.
