  We investigate the fate of topological states on fractal lattices. Focusing
on a spinless chiral p-wave paired superconductor, we find that this model
supports two qualitatively distinct phases when defined on a Sierpinski gasket.
While the trivial phase is characterized by a self-similar spectrum with
infinitely many gaps and extended eigenstates, the novel "topological" phase
has a gapless spectrum and hosts chiral states propagating along edges of the
graph. Besides employing theoretical probes such as the real-space Chern
number, inverse participation ratio, and energy-level statistics in the
presence of disorder, we develop a simple physical picture capturing the
essential features of the model on the gasket. Extending this picture to other
fractal lattices and topological states, we show that the p+ip state admits a
gapped topological phase on the Sierpinski carpet and that a higher-order
topological insulator placed on this lattice hosts gapless modes localized on
corners.
