Spectrum of weighted composition operators. Part VIII. Lower semi-Fredholm and Fredholm spectra of weighted compositions on $C(K)$, case of non-invertible surjections

We characterize lower semi-Fredholm and Fredholm of weighted composition operators on $C(K)$ in the case when the corresponding map is an open surjection of the compact space $K$ onto itself. The obtained criterions involve the notion of almost homeomorphisms of compact spaces.