Turn a continuous function on a compact space into a homeomorphism If \(\phi:X \to X\) is a continuous map of a compact space, it can be converted to a homeomorphism by an “inverse limit” construction. Let \(X^\mathbb{Z}\) denote the sapce of 2-sided sequences of elements of \(X\), with the product topology, and let \(\tilde{X}\) denote the subset of \(X^\mathbb{Z}\) consisting of those sequences \(x=(x_{1})\) for which \(\phi(x_{i+1}) = x_{i} ~\forall i \in \mathbb{Z}\). It follows easily from the compactness of \(X\) that \(\tilde{X}\) is a non-empty compact space. \cite{auslander:minimal}