The extension problem for complete $UV^{n}$-preimages

Abstract

We investigate the solvability of the extension problem for complete preimages from the given class $\mathscr{F}$ of surjective, perfect mappings of metric spaces, which consists of representing an arbitrary mapping $f_{0}$: $X_{0}\rightarrow Y_{0}\in \mathscr{F}$ as the restriction of another mapping $f$ : $X\rightarrow Y\in \mathscr{F}$, onto the complete preimage $f_{0}^{-1}(Y_{0})=X_{0}$, where $Y$ is an arbitrary metric space, containing $Y_{0}$ as a closed subset. We prove that this problem can be solved for the class of open $UV^{n}$-mappings. Along the way, we also establish that the subset $\exp_{UV^{n}}(\ell_{2}(\tau))$ of the exponent $exp(\ell_{2}(\tau))$ of the Hilbert space $\ell_{2}(\tau)$ of density $\tau$, consisting of $UV^{n}$-compacta, belongs to the class of absolute retracts.