Murat Tuncali - On generalizations of the Hahn-Mazurkiewicz theorem

The Hahn-Mazurkiewicz theorem characterizes the Hausdorff continuous images of [0,1] as the class of locally connected metric continua (Peano continua). A theorem of Alexandroff gives a characterization of the Hausdorff continuous images of the Cantor ternary set as the class of compact metric spaces. Following these theorems, it was natural to ask whether one could obtain generalizations of these results in the category of Hausdorff spaces.

Nikiel (1988) obtained a characterization of locally connected continuous images of compact ordered spaces. Bula and Turzanski (1986) gave a characterization of conitnuous images of compact ordered spaces. Since these characterizations were obtained, there has been a considerable amount of development in the study of continuous images of ordered continua (arcs) and compact ordered spaces. In this talk, we will give a survey of the results concerning the classes of spaces that are continuous images of ordered continua and compact ordered spaces, and related classes.