Basic definitions and properties of topological branched coverings (Q1380910)

A topological branched covering is defined as a continuous surjection \(p: E\to B\) such that the restriction of \(p\) to the preimage of the complement of a nowhere dense singular set in \(B\) is a covering (as opposed to the usual definition of branched coverings in the PL or simplicial category). In the present paper, some basic theory and notation for topological branched coverings is developed. In particular, the Absolute Covering Homotopy Property and the Arc Lifting Property are discussed, and graphs are interpreted as branched coverings of the circle. As an application, a topological version of a theorem (``Bertini's theorem'') from analytic geometry is proved.