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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Induced universal maps and some hyperspaces with the fixed point property

Author: Sam B. Nadler
Journal: Proc. Amer. Math. Soc. 100 (1987), 749-754
MSC: Primary 54B20; Secondary 54C10, 54F20
MathSciNet review: 894449
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Abstract: For a (metric) continuum $ Z$, let $ {2^Z}$ (resp., $ C(Z)$) denote the space of all nonempty compacta (resp., continua) in $ Z$ with the Hausdorff metric. We prove: (1) If $ f$ is a monotone map of a continuum $ X$ onto a Peano continuum $ Y$, then, for any maps $ g:{2^X} \to {2^Y}$ and $ h:C(X) \to C(Y)$, there is $ A \in {2^X}$ and $ B \in C(X)$ such that $ f(A) = g(A)$ and $ f(B) = h(B)$. We use (1) to prove: (2) If $ X$ is an inverse limit of dendrites with quasi-monotone bonding maps, then $ {2^X}$ and $ C(X)$ have the fixed point property. Thus, we have a proof that for certain indecomposable continua $ X,{2^X}$ has the fixed point property.

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Keywords: CE-map, continuum, dendrite, fixed point property, hyperspaces, indecomposable continuum, inverse limit, monotone map, Peano continuum, quasi-monotone map, universal map
Article copyright: © Copyright 1987 American Mathematical Society