Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
extending generalized Whitney map; hyperspace
extending generalized Whitney map; hyperspace
Summary:
For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^{X}$ can be extended to a Whitney map for $2^{X}$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.
References:
[1] Charatonik, J.J., Charatonik, W.J.: Whitney maps—a non-metric case. Colloq. Math. 83 (2) (2000), 305–307. MR 1758323 | Zbl 0953.54013
[2] Engelking, R.: General Topology. PWN Warszawa, 1977. MR 0500780 | Zbl 0373.54002
[3] Illanes, A., Nadler, Jr., S.B.: Hyperspaces: Fundamentals and Recent advances. Marcel Dekker, New York-Basel, 1999. MR 1670250 | Zbl 0933.54009
[4] Jones, F.B.: Aposyndetic continua and certain boundary problems. Amer. J. Math. 63 (1941), 545–553. DOI 10.2307/2371367 | MR 0004771 | Zbl 0025.24003
[5] Kelley, J.L.: Hyperspaces of a continuum. Trans. Amer. Math. Soc. 52 (1942), 22–36. DOI 10.1090/S0002-9947-1942-0006505-8 | MR 0006505 | Zbl 0061.40107
[6] Lončar, I.: Non-metric rim-metrizable continua and unique hyperspace. Publ. Inst. Math. (Beograd) (N.S.) 73 (87) (2003), 97–113. DOI 10.2298/PIM0373097L | MR 2068242 | Zbl 1054.54026
[7] Michael, E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152–182. DOI 10.1090/S0002-9947-1951-0042109-4 | MR 0042109 | Zbl 0043.37902
[8] Nadler, S.B.: Hyperspaces of sets. Marcel Dekker, Inc., New York, 1978. MR 0500811 | Zbl 0432.54007
[9] Nadler, S.B.: Continuum theory. Marcel Dekker, Inc., New York, 1992. MR 1192552 | Zbl 0757.54009
[10] Smith, M., Stone, J.: On non-metric continua that support Whitney maps. Topology Appl. 170 (2014), 63–85. DOI 10.1016/j.topol.2014.02.007 | MR 3200390 | Zbl 1296.54037
[11] Stone, J.: Non-metric continua that support Whitney maps. Dissertation. Zbl 1296.54037
[12] Ward, L.E.: Extending Whitney maps. Pacific J. Math. 93 (1981), 465–470. DOI 10.2140/pjm.1981.93.465 | MR 0623577 | Zbl 0457.54008
[13] Whitney, H.: Regular families of curves, I. Proc. Nat. Acad. Sci. 18 (1932), 275–278. DOI 10.1073/pnas.18.3.275 | Zbl 0004.07503
[14] Whitney, H.: Regular families of curves. Ann. of Math. (2) 34 (1933), 244–270. DOI 10.2307/1968202 | MR 1503106 | Zbl 0006.37101
[15] Wilder, B.E.: Between aposyndetic and indecomposable continua. Topology Proc. 17 (1992), 325–331. MR 1255815 | Zbl 0788.54041
Search
Partner of
