Indecomposable Continua

This chapter discusses concepts related to indecomposable continua. A continuum is indecomposable if it is not the union of two proper sub-continua. It is hereditarily indecomposable if each of its sub-continua is indecomposable. The best-studied hereditarily indecomposable continuum is the pseudo-arc. Moise has shown that the pseudo-arc is hereditarily equivalent, that is, homeomorphic to each of its non-degenerate sub-continua. He gave it its name because it shares this property with the arc. Any non-degenerate hereditarily equivalent continuum, other than the arc, must be hereditarily indecomposable and tree-like. Mohler and Oversteegen constructed examples of non-metric decomposable hereditarily equivalent continua, including one that is not a Hausdorffarc. Smith has constructed a non-metric hereditarily indecomposable hereditarily equivalent continuum, obtained as an inverse limit of ω₁ copies of the pseudo-arc. Oversteegen and Tymchatyn have shown that any planar hereditarily equivalent continuum must be close to being chainable, that is, must be weakly chainable and have symmetric span zero. The chapter explains the concepts of hereditary equivalence and homogeneity. It also explains about fixed points, maps of products, and homeomorphism groups.