The cantor set and inverse limits of upper semi-continuous functions

In this paper, we study inverse limits with a single upper semi-continuous function F such that it is the union of mappings defined from a compact metric space X into itself. We prove that if Dom(F) is a totally disconnected space, then \(\displaystyle \lim _{\longleftarrow }(X,F)\) ${\displaystyle \underset{⟵}{lim}(X,F)}$ is homeomorphic to the Cantor set. This gives a partial answer to a problem posed by Ingram (Topol Appl 299:1–11, 2021), and it answers a question asked by Capulín et al. (Topol Proc 60:71–80, 2022).