Extension properties of Stone–Čech coronas and proper absolute extensors

A. Chigogidze¹

¹ Department of Mathematics College of Staten Island, CUNY 2800 Victory Blvd. Staten Island, NY 10314, U.S.A.

Fundamenta Mathematicae, Tome 222 (2013) no. 2, pp. 155-173

Résumé

We characterize, in terms of $X$, the extensional dimension of the Stone–Čech corona $\beta X \setminus X$ of a locally compact and Lindelöf space $X$. The non-Lindelöf case is also settled in terms of extending proper maps with values in $I^{\tau }\setminus L$, where $L$ is a finite complex. Further, for a finite complex $L$, an uncountable cardinal $\tau $ and a $Z_{\tau }$-set $X$ in the Tikhonov cube $I^{\tau }$ we find a necessary and sufficient condition, in terms of $I^{\tau }\setminus X$, for $X$ to be in the class $\operatorname {AE}([L])$. We also introduce a concept of a proper absolute extensor and characterize the product $[0,1)\times I^{\tau }$ as the only locally compact and Lindelöf proper absolute extensor of weight $\tau > \omega $ which has the same pseudocharacter at each point.

Détail

DOI : 10.4064/fm222-2-3

Keywords: characterize terms extensional dimension stone ech corona beta setminus locally compact lindel space non lindel settled terms extending proper maps values tau setminus where finite complex further finite complex uncountable cardinal tau tau set tikhonov cube tau necessary sufficient condition terms tau setminus class operatorname introduce concept proper absolute extensor characterize product times tau only locally compact lindel proper absolute extensor weight tau omega which has pseudocharacter each point