The space of ANR’s in $ℝ^n$

Tadeusz Dobrowolski; Leonard Rubin

ArticleOriginal scientific text

Title

The space of ANR’s in $ℝ^{n}$

Authors ¹, ¹

Affiliations

Department of Mathematics, The University of Oklahoma, 601 Elm Avenue, Room 423, Norman, Oklahoma 73019-0315, U.S.A.

Abstract

The hyperspaces

A N R (ℝ^{n})

and

A R (ℝ^{n})

in

2^{ℝ^{n}} (n \geq 3)

consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute

G_{δ σ δ}

-spaces and that, indeed, they are not

F_{σ δ σ}

-spaces. The main result is that

A N R (ℝ^{n})

is an absorber for the class of all absolute

G_{δ σ δ}

-spaces and is therefore homeomorphic to the standard model space

Ω_{3}

of this class.

Keywords

ENG

hyperspace, absolute neighborhood retract, absolute retract,

G_{δ σ δ}

-set, absorber

[BGvM] J. Baars, H. Gladdines and J. van Mill, Absorbing systems in infinite-dimensional manifolds, Topology Appl. 50 (1993), 147-182.
[BP] C. Bessaga and A. Pełczyński, Selected Topics in Infinite-Dimensional Topology, PWN, Warszawa, 1975.
[BM] M. Bestvina and J. Mogilski, Characterizing certain incomplete infinite-dimensional retracts, Michigan Math. J. 33 (1986), 291-313.
[Bor] K. Borsuk, On some metrizations of the hyperspace of compact sets, Fund. Math. 41 (1954), 168-202.
[C1] R. Cauty, L'espace des pseudo-arcs d'une surface, Trans. Amer. Math. Soc. 331 (1992), 247-263.
[C2] R. Cauty, L'espace des arcs d'une surface, ibid. 332 (1992), 193-209.
[C3] R. Cauty, Les fonctions continues et les fonctions intégrables au sens de Riemann comme sous-espaces de $L^{1}$ , Fund. Math. 139 (1991), 23-36.
[CDGvM] R. Cauty, T. Dobrowolski, H. Gladdines et J. van Mill, Les hyperespaces des rétractes absolus et des rétractes absolus de voisinage du plan, preprint.
[Cu1] D. W. Curtis, Hyperspaces of finite subsets as boundary sets, Topology Appl. 22 (1986), 97-107.
[Cu2] D. W. Curtis, Hyperspaces of non-compact metric spaces, Compositio Math. 40 (1986), 139-152.
[CN] D. W. Curtis and N. T. Nhu, Hyperspaces of finite subsets which are homeomorphic to $ℵ_{0}$ -dimensional linear metric spaces, Topology Appl. 19 (1985), 251-260.
[DvMM] 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.
[DM] T. Dobrowolski and W. Marciszewski, Classification of function spaces with the pointwise topology determined by a countable dense set, preprint.
[DR] T. Dobrowolski and L. R. Rubin, The hyperspaces of infinite-dimensional compacta for covering and cohomological dimension are homeomorphic, Pacific J. Math. 164 (1994), 15-39.
[GvM] H. Gladdines and J. van Mill, Hyperspaces of Peano continua of euclidean spaces, Fund. Math. 142 (1993), 173-188.
[Kur1] K. Kuratowski, Sur une méthode de métrisation complète de certains espaces d'ensembles compacts, ibid. 43 (1956), 114-138.
[Kur2] K. Kuratowski, Topology I, Academic Press, New York and London, 1966.
[MS] S. Mardešić and J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.
[vM] J. van Mill, Infinite-Dimensional Topology: prerequisites and introduction, North-Holland, Amsterdam, 1989.
[SR] J. Saint Raymond, Fonctions boréliennes sur un quotient, Bull. Sci. Math. (2) 100 (1976), 141-147.
[Tor] H. Toruńczyk, Concerning locally homotopy negligible sets and characterization of $l_{2}$ -manifolds, Fund. Math. 101 (1978), 93-110.

Additional information

http://matwbn.icm.edu.pl/ksiazki/fm/fm146/fm14613.pdf