  The Cu-doped topological insulator Bi$_2$Se$_3$ has recently been found to
undergo a superconducting transition upon cooling, raising the possibilities
that it is the first known "topological superconductor" or realizes a novel
non-Abelian superconducting state. Its true nature depends critically on the
bulk and surface state band topology. We present the first photoemission
spectroscopy results where by examining the band topology at many different
copper doping values we discover that the topologically protected spin-helical
surface states remain well protected and separate from bulk Dirac bands at the
Fermi level where Copper pairing occurs in the optimally doped topological
insulator. The addition of copper is found to result in nonlinear electron
doping and strong renormalization of the topological surface states. These
highly unusual observations strongly suggest that superconductivity on the
topological surface of Cu$_x$Bi$_2$Se$_3$ cannot be of any conventional type in
account of the general topological theory. Characteristics of the three
dimensional bulk Dirac band structure are reported for the first time with
respect to the superconducting doping state and topological invariant
properties which should help formulate a specific theory for this novel
superconductor.
