A Decomposition Theorem for Topological Semi-small Maps

A continuous surjection $f: X \rightarrow Y$ is an almost covering map if there exists a nowhere dense closed set $R \subset Y$ with a finite decomposition $R = \bigsqcup_β T_β$ such that the restriction $f\vert_{f^{-1}(Y \setminus R)}$ is a covering map and, for each $β$, the restriction $f\vert_{f