Totally regular curves as inverse limits

Abstr•.ct. P•ecently J. Nikiel showed that if a continuum X is home-oreorphic to the inverse limit of connected graphs under monotone bonding surjections, then X is a totally regular curve. Here it is shown that the converse is also true. This theorem is then applied to give a simple proof of the following old result of O. G. Harrold: each totally regular curve admits a convex metric such that the resulting metric space is of finite linear measure. A continuum is a compact, connected metric space. A curve is a one-dimensional continuum. A curve X is regular provided there exists a basis B for X so that each B • B has finite boundary. Each subcontinuum of a regular curve is locally connected [6, Theorem 2, p. 283]. Moreover, in a regular curve, each sequence of pairwise disjoint subcontinua forms a null-sequence: that is, the diameters of the subcontinua converge to zero. A continuum X is said to be completely regular provided X contains no nowhere dense subcontinuum. Following [8], we say that a continuum X is totally regular provided that, for each countable set P C X, there is a basis /3 of open sets for X so that for each B • B, PCibd(B) = 0 and B has finite boundary. Each completely regular curve is totally regular [8, (3.2)], and trivially, each totally regular curve is regular. Totally regular continua have been studied under the name of continua of finite degree-see, for example, [10] and [5]. A graph is a compact, connected 1-dimensional polyhedron. In what follows, maps are always continuous functions. A map f: X- • Y is monotone provided f-•(y) is connected for each y C Y. A point x of a connected space X is said to be a separating point of X if X•{x} i