  We introduce continuum-wise hyperbolicity, a generalization of hyperbolicity
with respect to the continuum theory. We discuss similarities and differences
between topological hyperbolicity and continuum-wise hyperbolicity. A shadowing
lemma for cw-hyperbolic homeomorphisms is proved in the form of the L-shadowing
property and a Spectral Decomposition is obtained in this scenario. In the
proof we generalize the construction of Fathi \cite{Fat89} of a hyperbolic
metric using only cw-expansivity, obtaining a hyperbolic cw-metric. We also
introduce cwN-hyperbolicity, exhibit examples of these systems for arbitrarily
large $N\in\mathbb{N}$ and obtain further dynamical properties of these systems
such as finiteness of periodic points with the same period. We prove that
homeomorphisms of $\mathbb{S}^2$ that are induced by topologically hyperbolic
homeomorphisms of $\mathbb{T}^2$ are continuum-wise-hyperbolic and
topologically conjugate to linear cw-Anosov diffeomorphisms of $\mathbb{S}^2$,
being in particular cw2-hyperbolic.
