  Using the Sierpi\'nski carpet and gasket, we investigate whether fractal
lattices embedded in two-dimensional space can support topological phases when
subjected to a homogeneous external magnetic field. To this end, we study the
localization property of eigenstates, the Chern number, and the evolution of
energy level statistics when disorder is introduced. Combining these
theoretical tools, we identify regions in the phase diagram of both the carpet
and the gasket, for which the systems exhibit properties normally associated to
gapless topological phases with a mobility edge.
