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(When) can you embed a closed map with finite discrete fibers into a (branched) cover?

Assume all spaces are topological manifolds. A branched cover is a continuous open map with discrete fibers. A finite branched cover is one with finite fibers. Questions. Given closed map $X\to S$ ...
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What is definition of branched covering?

What is definition of branched covering in the page 10 of following paper ? In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. ...
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