  Classification of matter through topological phases and topological edge
states between distinct materials has been a subject of great interest
recently. While lattices have been the main setting for these studies, a
relatively unexplored realm for this physics is that of continuum fluids. In
the typical case of a fluid model with a point spectrum, nontrivial topology
and associated edge modes have been observed previously. However, another
possibility is that a continuous spectrum can coexist with the point spectrum.
Here we demonstrate that a fluid plasma model can harbor nontrivial topology
within its continuous spectrum, and that there are boundary modes at the
interface between topologically distinct regions. We consider the ideal
magnetohydrodynamics (MHD) model. In the presence of magnetic shear, we find
nontrivial topology in the Alfv\'{e}n continuum. For strong shear, the Chern
number is $\pm 1$, depending on the sign of the shear. If the magnetic shear
changes sign within the plasma, a topological phase transition occurs, and
bulk-boundary correspondence then suggests a mode localized to the layer of
zero magnetic shear. We confirm the existence of this mode numerically.
Moreover, this reversed-shear Alfv\'{e}n eigenmode (RSAE) is well known within
magnetic fusion as it has been observed in several tokamaks. In examining how
the MHD model might be regularized at small scales, we also consider the
electron limit of Hall MHD. We show that the whistler band, which plays an
important role in planetary ionospheres, has nontrivial topology. More broadly,
this work raises the possibility that fusion devices could be carefully
tailored to produce other topological states with potentially useful behavior.
