  Through techniques afforded by $C^*$-algebras and Hilbert modules, we study
the topology of spaces which parametrize families of instanton gauge fields on
noncommutative Euclidean four-spheres $S^4_\sigma$. By deforming the ADHM
construction of instantons on the classical sphere $S^4$, we obtain families of
instantons on the quantum sphere which are naturally parametrized by
noncommutative topological spaces. Using the internal gauge theory of
$S^4_\sigma$ determined by the inner automorphisms of its function algebra, we
find that one may always recover a classical parameter space by making a
suitable choice of internal gauge.
