Continuous images of linearly ordered continua and compacta

摘要

Mary Ellen Rudin has enriched general topology with many remarkable theorems and intricate examples. One of her last contributions was the proof of Nikiel's conjecture that Hausdorff compact monotonically normal spaces are continuous images of linearly ordered compacta. Previously, L. Bruce Treybig and Jacek Nikiel proved that connected locally connected images of linearly ordered compacta are also images of linearly ordered continua. Therefore, images of linearly ordered continua coincide with monotonically normal locally connected continua. Since metric compacta are monotonically normal, this deep result is an extension of the celebrated Hahn-Mazurkiewicz theorem, giving a topological characterization of continuous images of the unit interval of real numbers as locally connected metrizable continua. The purpose of the present paper is to pay tribute to Mary Ellen Rudin by surveying the research in this area realized by a number of topologists during the past hundred years, i.e., since 1914, the year of the publication of the Hahn-Mazurkiewicz theorem. (C) 2015 Elsevier B.V. All rights reserved.