  We introduce the notion of topological electronic states on random lattices
in non-integer dimensions. By considering a class $D$ model on critical
percolation clusters embedded in two dimensions, we demonstrate that these
topological random fractals exhibit a robust mobility gap, support quantized
conductance and represent a well-defined thermodynamic phase of matter. The
finite-size scaling analysis further suggests that the critical properties are
not consistent with the class $D$ systems in two dimensions. Our results
establish topological random fractals as the most complex systems known to
support nontrivial band topology with their distinct unique properties.
