Exactly k-to-1 maps: from pathological functions with finitely many discontinuities to well-behaved covering maps

Abstract

Many mathematicians encounter k-to-1 maps only in the study of covering maps. But, of course, k-to-1 maps do not have to be open. This paper touches on covering maps, and simple maps, but concentrates on ordinary k-to-1 functions (both continuous and finitely discontinuous) from one metric continuum to another. New results, old results and ideas for further research are given; and a baker's dozen of questions are raised.