Topics in Continuum Theory

Abstract

Continuum Theory is the study of compact, connected, metric spaces. These spaces arise naturally in the study of topological groups, compact manifolds, and in particular the topology and dynamics of one-dimensional and planar systems, and the area sits at the crossroads of topology and geometry. Major contributors to its development include, but are not limited to, Urysohn, Borsuk, Moore, Sierpinski, Menger, Mazurkiewicz, Ulam, Hahn, Whyburn, Kuratowski, Knaster, Moise, Cech and Bing. After the 1950's the area fell out of style, primarily due to the increased interest in the then-developing field of algebraic topology. Current efforts in complex and topological dynamics, including many open problems, often fall within the purview of continuum theory. This paper covers a selection of the standard topics in continuum theory, as well as a number of topics not yet available in book form, e.g. Kelley continua and some classical results concerning the pseudo-arc. As well, many known results are given their first explicit proofs, and two new results are obtained, one concerning slc continua and the other a broad generalization of a result of Menger.