A continuum whose hyperspace of subcontinua is not $g$-contractible
HTML articles powered by AMS MathViewer
- by Alejandro Illanes PDF
- Proc. Amer. Math. Soc. 130 (2002), 2179-2182 Request permission
Abstract:
A topological space $Y$ is said to be $g$-contractible provided that there exists a continuous onto function $f:Y\rightarrow Y$ such that $f$ is homotopic to a constant function. Answering a question by Sam B. Nadler, Jr., in this paper we construct a metric continuum $Z$ such that its hyperspace of subcontinua $C(Z)$ is not $g$-contractible.References
- David P. Bellamy, The cone over the Cantor set-continuous maps from both directions, Topology Conference (Proc. General Topology Conf., Emory Univ., Atlanta, Ga., 1970) Dept. Math., Emory Univ., Atlanta, Ga., 1970, pp. 8–25. MR 0341439
- R. Engelking and A. Lelek, Cartesian products and continuous images, Colloq. Math. 8 (1961), 27–29. MR 131263, DOI 10.4064/cm-8-1-27-29
- Jack T. Goodykoontz Jr., More on connectedness im kleinen and local connectedness in $C(X)$, Proc. Amer. Math. Soc. 65 (1977), no. 2, 357–364. MR 451188, DOI 10.1090/S0002-9939-1977-0451188-5
- Iwona Krzemińska and Janusz R. Prajs, A non-$g$-contractible uniformly path connected continuum, Topology Appl. 91 (1999), no. 2, 151–158. MR 1664453, DOI 10.1016/S0166-8641(97)00194-6
- Sam B. Nadler Jr., Some problems concerning hyperspaces, Topology Conference (Virginia Polytech. Inst. and State Univ., Blacksburg, Va., 1973) Lecture Notes in Math., Vol. 375, Springer, Berlin, 1974, pp. 190–197. MR 0370465
- 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., Continuum theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR 1192552
Additional Information
- Alejandro Illanes
- Affiliation: Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, México 04510, D.F. México
- Email: illanes@matem.unam.mx
- Received by editor(s): April 24, 2000
- Received by editor(s) in revised form: February 19, 2001
- Published electronically: February 12, 2002
- Communicated by: Alan Dow
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2179-2182
- MSC (2000): Primary 54B20
- DOI: https://doi.org/10.1090/S0002-9939-02-06307-4
- MathSciNet review: 1896056