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.
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