Hyperspaces of Peano continua of euclidean spaces

Helma Gladdines; Jan van Mill

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 2, page 173-188
  • ISSN: 0016-2736

Abstract

top
If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space L ( ℝ n ) is homeomorphic to B ∞ , where B denotes the pseudo-boundary of the Hilbert cube Q.

How to cite

top

Gladdines, Helma, and van Mill, Jan. "Hyperspaces of Peano continua of euclidean spaces." Fundamenta Mathematicae 142.2 (1993): 173-188. <http://eudml.org/doc/211980>.

@article{Gladdines1993,
abstract = {If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space $L(ℝ^n)$ is homeomorphic to $B^∞$, where B denotes the pseudo-boundary of the Hilbert cube Q.},
author = {Gladdines, Helma, van Mill, Jan},
journal = {Fundamenta Mathematicae},
keywords = {Hilbert cube; Hilbert space; absorbing system; Z-set; $F_\{σδ\}$; hyperspace; Peano continuum; $ℝ^n$; Vietoris topology; Peano-continua},
language = {eng},
number = {2},
pages = {173-188},
title = {Hyperspaces of Peano continua of euclidean spaces},
url = {http://eudml.org/doc/211980},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Gladdines, Helma
AU - van Mill, Jan
TI - Hyperspaces of Peano continua of euclidean spaces
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 2
SP - 173
EP - 188
AB - If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space $L(ℝ^n)$ is homeomorphic to $B^∞$, where B denotes the pseudo-boundary of the Hilbert cube Q.
LA - eng
KW - Hilbert cube; Hilbert space; absorbing system; Z-set; $F_{σδ}$; hyperspace; Peano continuum; $ℝ^n$; Vietoris topology; Peano-continua
UR - http://eudml.org/doc/211980
ER -

References

top
  1. [1] J. Baars, H. Gladdines, and J. van Mill, Absorbing systems in infinite-dimensional manifolds, Topology Appl., to appear. Zbl0794.57005
  2. [2] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa 1975. 
  3. [3] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite dimensional absolute retracts, Michigan Math. J. 33 (1986), 291-313. Zbl0629.54011
  4. [4] R. Cauty, L'espace des fonctions continues d'un espace métrique dénombrable, Proc. Amer. Math. Soc. 113 (1991), 493-501. 
  5. [5] R. Cauty, L'espace des arcs d'une surface, Trans. Amer. Math. Soc. 332 (1992), 193-209. Zbl0762.54012
  6. [6] D. W. Curtis, Hyperspaces of finite subsets as boundary sets, Topology Appl. 22 (1986), 97-107. Zbl0575.54009
  7. [7] D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19-38. Zbl0409.54044
  8. [8] J. J. Dijkstra and J. Mogilski, The topological product structure of systems of Lebesgue spaces, Math. Ann. 290 (1991), 527-543. Zbl0734.46013
  9. [9] J. J. Dijkstra, J. van Mill, and J. Mogilski, The space of infinite-dimensional compacta and other topological copies of ( l f 2 ) ω , Pacific J. Math. 152 (1992), 255-273. Zbl0786.54012
  10. [10] T. Dobrowolski, W. Marciszewski, and J. Mogilski, On topological classification of function spaces C p ( X ) of low Borel complexity, Trans. Amer. Math. Soc. 328 (1991), 307-324. Zbl0768.54016
  11. [11] C. Kuratowski, Evaluation de la classe borélienne ou projective d'un ensemble de points à l'aide des symboles logiques, Fund. Math. 17 (1931), 249-272. Zbl57.0092.05
  12. [12] S. Mazurkiewicz, Sur l'ensemble des continus péaniens, ibid., 273-274. Zbl0003.10601
  13. [13] S. B. Nadler, Hyperspaces of Sets, Marcel Dekker, New York 1978. Zbl0432.54007
  14. [14] J. van Mill, Infinite-Dimensional Topology: prerequisites and introduction, North- Holland, Amsterdam 1989. Zbl0663.57001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.