  Topological superconductivity is an exotic quantum phenomenon with coupled
nontrivial topological order and superconductivity together. A direct idea for
producing topological superconductors is to create superconductivity based on
the well recognized topological insulators. The topological insulating states
in highly efficient thermoelectric materials Bi$_2$Te$_3$ and Bi$_2$Se$_3$ and
their alloy Bi$_{2}$Te$_{3-x}$Se$_{x}$ have been established from
angle-resolved photoemission and transport experiments. Superconductivity was
also observed based on these popular topological insulators by the application
of pressure chemical dopant and heterostructures. However, the experiments
mainly focusing on Bi$_{2}$Se$_3$ doped by metals have not provided the
consistent evidence to support the topological superconductivity. Here we carry
out a systematic high-pressure study on a topological insulator
Bi$_{2}$Te$_{2.7}$Se$_{0.3}$ to provide the convincing evidence for the
expected topological superconductivity. Four phases with different structures
are found upon compression. The topological surface state is identified in the
entire initial phase, while superconductivity is found to coexist with such a
state of the compressed material after its passing the electronic topological
transition, followed by three other superconducting phases without topological
character. For these superconducting phases, we observe that the upper critical
field follows with the temperature in the critical exponent ${2/3}$ for the
first one with the topological surface state and $1$ for the left. These
observations support the realization of the topological superconductivity in
the initial phase according to the theoretically proposed critical field
measure. This work also points out a big pool and new direction for finding
topological superconductors from topological thermoelectric materials.
