Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Invariant arcs, Whitney levels, and Kelley continua


Author: M. van de Vel
Journal: Trans. Amer. Math. Soc. 326 (1991), 749-771
MSC: Primary 54H12; Secondary 52A01, 54B20
DOI: https://doi.org/10.1090/S0002-9947-1991-1010415-8
MathSciNet review: 1010415
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: As an application of convexity in spaces of arcs, three results of a somewhat different nature have been obtained. The first one gives some simple conditions under which an arc of a semilattice is mapped back into itself by an order-preserving function. The second result states that certain Whitney levels are absolute retracts. Finally, Kelley continua are characterized by what we call approximating coselections.


References [Enhancements On Off] (What's this?)

  • [C] D. W. Curtis, Application of a selection theorem to hyperspace contractibility, Canad. J. Math. 37 (1985), 747-759. MR 801425 (86m:54014)
  • [ENN] C. Eberhart, S. B. Nadler, Jr., and W. O. Nowell, Jr., Spaces of order arcs in hyperspaces, Fund. Math. 112 (1981), 111-120. MR 619487 (82k:54011)
  • [E] R. Engelking, General topology, PWN-Polish Scientific Publishers, Warszawa, 1977. MR 0500780 (58:18316b)
  • [G&] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove and D. S. Scott, A compendium of continuous lattices, Springer-Verlag, Berlin, 1980, xx+371 pp. MR 614752 (82h:06005)
  • [GN] J. T. Goodykoontz and S. B. Nadler Jr., Whitney levels in hyperspaces of certain Peano continua, Trans. Amer. Math. Soc. 274 (1982), 671-694. MR 675074 (84h:54010)
  • [J1] R. E. Jamison, A general theory of convexity, Dissertation, University of Washington, Seattle, Washington, 1974.
  • [J2] -, Tietze's convexity theorem for semilattices and lattices, Semigroup Forum 15 (1978), 357-373.
  • [M] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 0042109 (13:54f)
  • [vMV] J. van Mill and M. van de Vel, Equality of the Lebesgue and the inductive dimension functions for compact spaces with a uniform convexity, Colloq. Math. 50 (1986), 187-200. MR 857852 (88f:54070)
  • [N] S. B. Nadler, Hyperspaces of sets, Dekker, New York, 1978, 707 pp. MR 0500811 (58:18330)
  • [P] A. Petrus, Contractibility of Whitney continua in $ C(X)$, General Topology Appl. 9 (1978), 275-288. MR 510909 (80a:54010)
  • [V1] M. van de Vel, Pseudo-boundaries and pseudo-interiors for topological convexities, Dissertationes Math. 210 (1983), 1-72. MR 695220 (85c:52002)
  • [V2] -, A selection theorem for topological convex structures, (to appear).
  • [V3] -, A Helly property of arcs, Arch. Math. 52 (1989), 298-306. MR 989886 (90c:06008)
  • [W] L. E. Ward Jr., A note on Whitney maps, Canad. Math. Bull. 23 (1980), 373-374. MR 593400 (81k:54056)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54H12, 52A01, 54B20

Retrieve articles in all journals with MSC: 54H12, 52A01, 54B20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1010415-8
Keywords: Absolute retract, approximating coselection, arc, continuous selection, convex set, Kelley continuum, Lawson semilattice, Whitney map
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society