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A great variety of topological phases have been classified as a consequence of discovery of the quantum Hall effect (QHE), but this work has recently led to discovery of topological phases in lattice models understood in terms of a generalisation of the QHE framework, to that of the quantum skyrmion Hall effect. In this generalisation, point charges are generalised to compactified Landau levels, or p-branes more generally. For compactification via fuzzification, these compactified p-branes carrying charge are necessarily expressed in terms of angular momentum as quantum skyrmions. Within this framework, however, (d+1)-dimensional systems, based on the number of Cartesian coordinates they possess, can also have D additional dimensions encoded in (pseudo)spin degrees of freedom, even for very small (pseudo)spin such as (pseudo)spin 1/2. Such systems can therefore realise intrinsically (D+d+1)-dimensional topological states with signatures observed in lattice tight-binding models. We discuss the implications of these findings for quantum field theories, as well as experimental proposals for observing this physics.