Abstract
A continuum is a compact connected metric space. A continuum X is a triod if it contains a subcontinuum N such that X - N is the sum of three mutually separated sets. A continuum which contains no triod is atriodic. A continuum X is unicoherent if whenver X = A (UNION) B, where A and B are subcontinua, then A (INTERSECT) B is a continuum. A continuum is hereditarily unicoherent if every subcontinuum is unicoherent. A continuum is indecomposable if it is not the union of two proper subcontinua.An atriodic, hereditarily unicoherent continuum X has special properties. R. H. Sorgenfrey proved it is irreducible between some two points in X, say p and q. For each x in X there is a subcontinuum K(,x) which is the intersection of all continua containing x in their interior. If K(,x) contains p or q then the complement of K(,x) is connected, otherwise the complement has exactly two components. Also each K(,x) intersects only a finite number of distinct K(,x)'s. The K(,x) structure of atriodic hereditarily unicoherent continua is thoroughly investigated.Other subcontinua studied are continua of continuity which are continua that contain or are contained in any other subcontinuum that intersects them. The properties of continua of continuity are used to prove a result that gives a characterization of continua of semispan zero. The semispan of a continuum X is the least upper bound of the set of numbers (epsilon) for which there is a subcontinuum Z(,(epsilon)) of X x X such that one projection of Z(,(epsilon)) contains the other, and for each point (x,y) in Z(,(epsilon)) the distance from x to y is greater or equal to (epsilon).The other main results are as follows: Atriodic, hereditarily unicoherent continua are shown to have semispan zero if and only if every indecomposable subcontinuum has semispan zero. The properties of indecomposable continua are used to show that semispan zero is preserved under open mappings having the finite to one property.