  Superconductors with topological surface or edge states have been intensively
explored for the prospect of realizing Majorana bound states, which obey
non-Abelian statistics and are crucial for topological quantum computation. The
traditional routes for making topological insulator/superconductor and
semiconductor/superconductor heterostructures suffer fabrication difficulties
and can only work at low temperature. Here, we use angle-resolved photoemission
spectroscopy to directly observe the evolution of a topological transition of
band structure nearby the Fermi level in two-dimensional high-T$_{c}$
superconductor FeTe$_{1-x}$Se$_{x}$/SrTiO$_{3}$(001) monolayers, fully
consistent with our theoretical calculations. Furthermore, evidence of edge
states is revealed by scanning tunneling spectroscopy with assistance of
theoretical calculations. Our study provides a simple and tunable platform for
realizing and manipulating Majorana states at high temperature.
