F-2 - Continuum Theory (General)

Abstract

A compactum is compact Hausdorff space while a continuum is a connected compactum. Continuua are classified by their local and global connectivity properties. A locally connected continuum has a base of connected open sets. In metric continua, local connectedness is equivalent to local arc connectedness. Among the continua that are everywhere—far from being locally connected—are the indecomposable continua. It is not the union of two of its proper subcontinua. A compactum is hereditarily indecomposable if each of its subcontinua is indecomposable. The nice compacta are those with very strong local connectivity properties such as polyhedra, manifolds, and absolute neighborhood retracts (ANRs). At the opposite end of the spectrum are the hereditarily indecomposable compacta. It is fascinating that these opposite ends of the spectrum are never very far apart. Hereditarily indecomposable compacta are initially very counterintuitive but because they contain few internal connections, they are often easy to work with. Another classification of continua is obtained by looking at finite open covers. If Ρ is a class of compact, connected polyhedra, a continuum X is said to be Ρ -like if for each open cover ▪ of X there is a continuous surjection : X→P;∈ Ρ with fibers refining ▪.