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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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Keywords: Continuous selection, contractibility, convex metric, dendrite, dendroid, fixed point set, hyperspace of subcontinua, order preserving mapping, partially ordered space, set function <IMG WIDTH="20" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$T$">, smooth continuum, smooth dendroid, thread action, topological semigroup
Article copyright: © Copyright 1981 American Mathematical Society