Finite-to-One Factor

A Map $\psi:M\to M$, where $M$ is a Manifold, is a finite-to-one factor of a Map $\Psi:X\to X$ if there exists a continuous Onto Map $\pi:X\to M$ such that $\psi\circ\pi =\pi\circ\Psi$ and $\pi^{-1}(x)\subset X$ is finite for each $x\in M$.