The Peano problem, related problems and solutions
by
Peter Nyikos
University of South Carolina

The celebrated Hahn-Mazurkiewicz theorem gave an internal characterization of continuous Hausdorff images of the closed unit interval: they are the compact, connected, locally connected, metrizable spaces. Such spaces are now known as Peano continua. Partly parallelling the well-known "metrization problem" was what might be called the "Peano problem": give an internal characterization of the images of compact, connected ordered spaces that is a natural generalization of the Hahn-Mazurkiewicz theorem.

It took almost a century after the publication of the Hahn-Mazurkiewicz theorem before this less-known problem was solved through the joint work of Jacek Nikiel and Mary Ellen Rudin. And until this year, very few people were aware of their elegant result:

THEOREM. A Hausdorff space is the continuous image of a compact, connected, ordered space if, and only if, it is compact, connected, locally connected, and monotonically normal.

This talk surveys the history of this theorem, and gives some recent related results and open problems.

Date received: May 8, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cawr-05.