Mappings of Terminal Continua

Various kinds of nonseparating subcontinua were studied by a number of authors, see, for example, the expository paper [2], where a large amount of information on this subject is given. In the topological literature, or in continuum theory (to be more precise), the term “terminal,” when applied either to subcontinua of a given continuum or to points, and the same name “terminal” was assigned to several concepts defined in quite different ways, see, for example, definitions of terminal points or terminal subcontinua of a given continuum in [2, Definition 1.1, page 7], [10, page 461], [12, page 458], [14, page 17], [15, page 190], and [16, Definition 1.54, page 107]. See [2, page 35] for a discussion on relations to some other concepts for which the name “terminal” (or a similar one) is used. In the present paper we deal with terminal continua as defined by Gordh Jr. in [12]. To avoid any confusion or misunderstanding in the terminology, we have to use another name for the considered concept. Since Gordh restricts his considerations to subcontinua of hereditarily unicoherent continua only, we rename this concept, following [5, Section 3, page 380], as HU-terminal. To formulate the concept and to prove its properties, we have to recall some needed definitions and auxiliary results. A continuum means a compact, connected Hausdorff space. A subcontinuum I of a continuum X is said to be irreducible about a subset S ⊂X provided that S ⊂ I and no proper subcontinuum of I contains S. A continuum I is said to be irreducible provided that there are two points a and b in I such that I is irreducible about {a,b}. Then I is said to be irreducible between a andb or froma tob. Each continuum, containing some two points, contains a continuum which is irreducible between them. A continuum is said to be hereditarily unicoherent provided that the intersection of any two of its subcontinua is connected. A continuum X is hereditarily unicoherent if and only if for each subset S of X there is in X exactly one subcontinuum I(S) irreducible about S (this was shown in [4, T1, page 187] for metric continua only, but the proof works in the nonmetric case as well; compare [12, Remark, page 458]).