F-1 - Continuum Theory

Abstract

Continuum theory includes the study of all continua (that is, all compact and connected spaces) and the maps between them. The vitality of an area is often determined by the existence of major open problems that are either natural or connected to other areas of mathematics. This chapter focuses on these problems with a brief description of each, and is restricted mainly to metric continua. All continua naturally belong to one of the two important types. A decomposable continuum is the union of two proper subcontinua and is indecomposable otherwise. All locally connected continua are decomposable while all indecomposable continua are nowhere locally connected. Hence, decompasable continua include the class of continua with characteristic local properties, while indecomposable continua include the class with complicated local structure. It is well known that all locally connected continua are arcwise connected and hence between every pair of points there is a very simple irreducible subcontinuum containing them. Each one-dimensional continuum can be represented as an inverse limit on graphs and are called graph-like. Special classes of one dimensional continua are the tree-like, arc-like, and circle-like continua.