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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Arc-smooth continua


Authors: J. B. Fugate, G. R. Gordh and Lewis Lum
Journal: Trans. Amer. Math. Soc. 265 (1981), 545-561
MSC: Primary 54F20; Secondary 54F50
MathSciNet review: 610965
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0610965-3
PII: S 0002-9947(1981)0610965-3
Keywords: Continuous selection, contractibility, convex metric, dendrite, dendroid, fixed point set, hyperspace of subcontinua, order preserving mapping, partially ordered space, set function $ T$, smooth continuum, smooth dendroid, thread action, topological semigroup
Article copyright: © Copyright 1981 American Mathematical Society