Absolute Endpoints of Chainable Continua

Abstract

ABSTRACT. An endpoint of chainable continuum is a point at which it is always possible to start chaining that continuum.Some endpoints appear to have the property that one is almost "forced" to start (or finish) the chaining at these points. This paper characterizes these "absolute endpoints", and this characterization is used to show that in a chainable continuum locally connected at p is equivalent to connected im kleinen at p. Introduction.Roughly speaking, an endpoint of a chainable continuum is a point at, which it is always possible to start chaining that continuum. R. H. Bing used this notion to study chainable continua in general and the pseudo-arc in particular in several of his landmark papers [2,3,4]. Some endpoints of chainable continua appear to have the additional property that, not only is it possible to start chaining there, but that a person is practically "forced" to start (or finish) the chaining at those points. The main purpose of this paper is to characterize these "absolute endpoints".Along the way, we will use their characterization to show that, in a chainable continuum, locally connected at p is equivalent to connected im kleinen at p. This answers a question R. H. Bing asked me in conversation. This paper is affectionately dedicated to the memory of Professor Bing.

Absolute end points of irreducible continua

“…The equivalence of conditions (l)-(4) has been stated in Rosenholtz's paper [9] as Proposition 1.3, p. 1309 for arc-like continua. However, the same proof is in fact valid for our more general setting, with the only change that the unique irreducible subcontinua between the points considered (the uniqueness is a consequence of the condition that each subcontinuum of an arc-like continuum is unicoherent, see [9], Theorem 0.0, p. 1306 and Lemma 1.2, p. 1308) are replaced by arbitrary ones which do exist by Theorem 1 of §48,1, p. 192 of the Kuratowski monograph [7]. The rest of the conclusion is a consequence of Theorem 20 of Thomas' paper [10], p. 28, where it is shown that local connectedness at a point, semi-local connectedness at a point, and aposyndeticity at a point with respect to any other point of a continuum coincide provided the continuum is irreducible.…”

“…A point p of an arc-like continuum X is called an absolute end point of X if p and X satisfy any of conditions (l)-(7) above. It is known from Rosenholtz's paper ( [9], Remark, p. 1310) that each absolute end point is an end point.…”

“…Rosenholtz has proved ( [9], Proposition 1.3, p. 1309) an equivalence of certain four conditions related to end points of arc-like continua in the sense of Bing ([2], p. 660) and defined a concept of an absolute end point as a point of an arc-like continuum that satisfies any one of them. It is observed in the paper that each of the conditions under consideration is also equivalent to two other conditions and that the scope of application of the definition can be extended to arbitrary irreducible continua (not necessarily arc-like ones).…”

“…for arbitrary (irreducible) continua, we need a lemma, which generalizes Lemma 1.4 of Rosen hoi tz's paper[9], p. 1310. Let a point p of an irreducible continuum X be given such that X is connected im kleinen at p and hereditarily unicoherent at p. Then either p is an absolute end point of X, or X is the union of two subcontinua having only p in common and each having p as an absolute end point.…”