Arc-smooth continua

Fugate, J. B.; Gordh, G. R.; Lum, Lewis

doi:10.1090/S0002-9947-1981-0610965-3

Arc-smooth continua
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by J. B. Fugate, G. R. Gordh and Lewis Lum PDF

Trans. Amer. Math. Soc. 265 (1981), 545-561 Request permission

Abstract:

Continua admitting arc-structures and arc-smooth continua are introduced as higher dimensional analogues of dendroids and smooth dendroids, respectively. These continua include such spaces as: cones over compacta, convex continua in ${l_2}$, strongly convex metric continua, injectively metrizable continua, as well as various topological semigroups, partially ordered spaces, and hyperspaces. The arc-smooth continua are shown to coincide with the freely contractible continua and with the metric $K$-spaces of Stadtlander. Known characterizations of smoothness in dendroids involving closed partial orders, the set function $T$, radially convex metrics, continuous selections, and order preserving mappings are extended to the setting of continua with arc-structures. Various consequences of the special contractibility properties of arc-smooth continua are also obtained.

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