  The purpose of this paper is to study the relation between the $C^0$-topology
and the topology induced by the spectral norm on the group of Hamiltonian
diffeomorphisms of a closed symplectic manifold. Following the approach of
Buhovsky-Humili\`ere-Seyfaddini, we prove the $C^0$-continuity of the spectral
norm for complex projective spaces and negative monotone symplectic manifolds.
The case of complex projective spaces provides an alternative approach to the
$C^0$-continuity of the spectral norm proven by Shelukhin. We also prove a
partial $C^0$-continuity of the spectral norm for rational symplectic
manifolds. Some applications such as the Arnold conjecture in the context of
$C^0$-symplectic topology are also discussed.
