  Topological materials hosting metallic edges characterized by integer
quantized conductivity in an insulating bulk have revolutionized our
understanding of transport in matter. The topological protection of these edge
states is based on symmetries and dimensionality. However, only
integer-dimensional models have been classified, and the interplay of topology
and fractals, which may have a non-integer dimension, remained largely
unexplored. Quantum fractals have recently been engineered in metamaterials,
but up to present no topological states were unveiled in fractals realized in
real materials. Here, we show theoretically and experimentally that topological
edge and corner modes arise in fractals formed upon depositing thin layers of
bismuth on an indium antimonide substrate. Scanning tunneling microscopy
reveals the appearance of (nearly) zero-energy modes at the corners of
Sierpi\'nski triangles, as well as the formation of outer and inner edge modes
at higher energies. Unexpectedly, a robust and sharp depleted mode appears at
the outer and inner edges of the samples at negative bias voltages. The
experimental findings are corroborated by theoretical calculations in the
framework of a continuum muffin-tin and a lattice tight-binding model. The
stability of the topological features to the introduction of a Rashba
spin-orbit coupling and disorder is discussed. This work opens the perspective
to novel electronics in real materials at non-integer dimensions with robust
and protected topological states.
