Hyperspaces of Peano continua of euclidean spaces

Helma Gladdines; Jan van Mill

ArticleOriginal scientific text

Title

Hyperspaces of Peano continua of euclidean spaces

Authors ¹, ¹

Affiliations

Faculteit Wiskunde en Informatica, Vrije Universiteit de Boelelaan, 1081a Amsterdam, The Netherlands

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^{\infty}

, where B denotes the pseudo-boundary of the Hilbert cube Q.

Keywords

ENG

Hilbert cube, Hilbert space, absorbing system, Z-set,

F_{σ δ}

, hyperspace, Peano continuum,

ℝ^{n}

J. Baars, H. Gladdines, and J. van Mill, Absorbing systems in infinite-dimensional manifolds, Topology Appl., to appear.
C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa 1975.
M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite dimensional absolute retracts, Michigan Math. J. 33 (1986), 291-313.
R. Cauty, L'espace des fonctions continues d'un espace métrique dénombrable, Proc. Amer. Math. Soc. 113 (1991), 493-501.
R. Cauty, L'espace des arcs d'une surface, Trans. Amer. Math. Soc. 332 (1992), 193-209.
D. W. Curtis, Hyperspaces of finite subsets as boundary sets, Topology Appl. 22 (1986), 97-107.
D. W. Curtis and R. M. Schori, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19-38.
J. J. Dijkstra and J. Mogilski, The topological product structure of systems of Lebesgue spaces, Math. Ann. 290 (1991), 527-543.
J. J. Dijkstra, J. van Mill, and J. Mogilski, The space of infinite-dimensional compacta and other topological copies of ${(l^{2}_f)}^{ω}$ , Pacific J. Math. 152 (1992), 255-273.
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.
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.
S. Mazurkiewicz, Sur l'ensemble des continus péaniens, ibid., 273-274.
S. B. Nadler, Hyperspaces of Sets, Marcel Dekker, New York 1978.
J. van Mill, Infinite-Dimensional Topology: prerequisites and introduction, North- Holland, Amsterdam 1989.

Additional information

http://matwbn.icm.edu.pl/ksiazki/fm/fm142/fm14226.pdf

Title

Hyperspaces of Peano continua of euclidean spaces

Affiliations

Abstract

Keywords

Bibliography

Additional information