  The fermion doubling theorem plays a pivotal role in Hermitian topological
materials. It states, for example, that Weyl points must come in pairs in
three-dimensional semimetals. Here, we present an extension of the doubling
theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional
non-Hermitian systems without any symmetry constraints, which can host two
different types of topological point nodes, namely, (i) Fermi points and (ii)
exceptional points. We show that these two types of protected point nodes obey
doubling theorems, which require that the point nodes come in pairs. To prove
the doubling theorem for exceptional points, we introduce a generalized winding
number invariant, which we call the discriminant number. Importantly, this
invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with
exceptional points of arbitrary order, and moreover can also be used to
characterize non-defective degeneracy points. Furthermore, we show that a
surface of a three-dimensional system can violate the non-Hermitian doubling
theorems, which implies unusual bulk physics.
