  To better understand mutation-invariant and hereditary properties of quivers
(and more generally skew-symmetrizable matrices), we have constructed a
topology on the set of all mutation classes of quivers which we call the
mutation class topology. This topology is the Alexandrov topology induced by
the poset structure on the set of mutation classes of quivers from the partial
order of quiver embedding. The closed sets of our topology -- equivalently, the
lower sets of the poset -- are in bijective correspondence with
mutation-invariant and hereditary properties of quivers. We show that this
space is strictly $T_0$, connected, non-Noetherian, and that every open set is
dense. We close by providing open questions from cluster algebra theory in the
setting of the mutation class topology and some directions for future research.
