A topological version of Bertini's theorem

We give a topological version of a Bertini type theorem due to Abhyankar. A new definition of a branched covering is given. If the restriction ${\displaystyle {\pi }_V:V\to Y}$ of the natural projection π: Y × Z → Y to a closed set V ⊂ Y × Z is a branched covering then, under certain assumptions, we can obtain generators of the fundamental group π₁((Y×Z)\V).