  The classification of gapped phases of non-interacting fermions hinges on the
tenfold symmetries and on the spatial dimension. The notion of dimension leads
to a well defined demarcation between bulk and edge. Here we explore the nature
of topological phases in systems where the distinction between bulk and edge is
nebulous, of which fractal lattices are canonical examples. Our key finding is
that in homogeneous fractal lattices (where every site is equally coordinated),
there are no gapped topological phases. What appears instead is a novel
metallic state -- the fractalized metal -- whose low energy states arrange
hierarchically on the structure of the fractal that hosts them. We study the
properties (such as chiral transport) of this metal and demonstrate its
robustness to disorder. Further, by studying a variety of fractal models we
establish that the homogeneity of the fractal is a key condition for the
realization of such fractalized metallic states.
