Extension properties of Stone-Čech coronas and proper absolute extensors

Abstract

We characterize, in terms of $X$, extensional dimension of the Stone-Čech corona $\beta X\setminus X$ of locally compact and Lindelöf space $X$. The non-Lindelöf case 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 Tychonov cube $I^{\tau }$ we find necessary and sufficient condition, in terms of $I^{\tau }\setminus X$, for $X$ to be in the class $\mathrm{AE}([L])$. 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.

Accepted for publication - minor revisions (including numbering of certain statements)