Induced universal maps and some hyperspaces with the fixed point property
HTML articles powered by AMS MathViewer
- by Sam B. Nadler PDF
- Proc. Amer. Math. Soc. 100 (1987), 749-754 Request permission
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.References
- Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
- Karol Borsuk, Theory of shape, Monografie Matematyczne, Tom 59, PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0418088
- D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), no. 1, 19–38. MR 512241, DOI 10.4064/fm-101-1-19-38
- W. Holsztyński, On the composition and products of universal mappings, Fund. Math. 64 (1969), 181–188. MR 243491, DOI 10.4064/fm-64-2-181-188
- W. Holsztyński, On the product and composition of universal mappings of manifolds into cubes, Proc. Amer. Math. Soc. 58 (1976), 311–314. MR 407832, DOI 10.1090/S0002-9939-1976-0407832-0
- W. Holsztyński, Universal mappings and fixed point theorems, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 15 (1967), 433–438 (English, with Russian summary). MR 221493
- W. Holsztyński, Universality of mappings onto the products of snake-like spaces. Relation with dimension, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 161–167 (English, with Russian summary). MR 230294
- J. Krasinkiewicz, Curves which are continuous images of tree-like continua are movable, Fund. Math. 89 (1975), no. 3, 233–260. MR 388358, DOI 10.4064/fm-89-3-233-260
- K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
- A. Y. W. Lau, A note on monotone maps and hyperspaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), no. 2, 121–123 (English, with Russian summary). MR 410699
- Sam B. Nadler Jr., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 49, Marcel Dekker, Inc., New York-Basel, 1978. A text with research questions. MR 0500811
- Sam B. Nadler Jr., Universal mappings and weakly confluent mappings, Fund. Math. 110 (1980), no. 3, 221–235. MR 602888, DOI 10.4064/fm-110-3-221-235
- Sam B. Nadler Jr. and J. T. Rogers Jr., A note on hyperspaces and the fixed point property, Colloq. Math. 25 (1972), 255–257. MR 377829, DOI 10.4064/cm-25-2-255-257
- David R. Read, Confluent and related mappings, Colloq. Math. 29 (1974), 233–239. MR 367903, DOI 10.4064/cm-29-2-233-239
- J. Segal, A fixed point theorem for the hyperspace of a snake-like continuum, Fund. Math. 50 (1961/62), 237–248. MR 139144, DOI 10.4064/fm-50-3-237-248
- Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 100 (1987), 749-754
- MSC: Primary 54B20; Secondary 54C10, 54F20
- DOI: https://doi.org/10.1090/S0002-9939-1987-0894449-4
- MathSciNet review: 894449