The space of ANR’s in $ℝ^n$

Volume 146 / 1994

Tadeusz Dobrowolski, Leonard R. Rubin Fundamenta Mathematicae 146 (1994), 31-58 DOI: 10.4064/fm-146-1-31-58

Abstract

The hyperspaces $ANR(ℝ^n)$ and $AR(ℝ^n)$ in $2^{ℝ^n} (n ≥ 3)$ 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 $G_{δσ δ}$-spaces and that, indeed, they are not $F_{σ δσ }$-spaces. The main result is that $ANR(ℝ^n)$ is an absorber for the class of all absolute $G_{δσ δ}$-spaces and is therefore homeomorphic to the standard model space $Ω_3$ of this class.

Authors

  • Tadeusz Dobrowolski
  • Leonard R. Rubin

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