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.

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

A map (= continuous function) is of order k if each of its point-inverses has at most k elements. Following (4), maps of order 2 are called simple. Which maps are compositions of simple closed (open, clopen) maps? How many simple maps are really needed to represent a given map? It is proved herein that every closed map of order k defined on an n-dimensional metric space is a composition of (n + 1)k 1 simple closed maps (with metric domains). This theorem fails to be true for non-metrizable spaces. An appropriate map on a Cantor cube of uncountable weight is described.

/pdf/compositions-of-simple-maps-x1v8ln7h7e.pdf

Compositions of simple maps

/pdf/conditions-which-ensure-that-a-simple-map-does-not-raise-mn53g1hpyk.pdf

J. Mioduszewski

Papers

Conditions which ensure that a simple map does not raise dimension