Hurewicz theorem for extension dimension

Abstract

We prove a new selection theorem for multivalued mappings of C-space. Using this theorem we prove extension dimensional version of Hurewicz theorem for a closed mapping $f\colon X\to Y$ of $k$-space $X$ onto paracompact $C$-space $Y$: if for finite $CW$-complex $M$ we have $\ed Y\le [M]$ and for every point $y\in Y$ and every compactum $Z$ with $\ed Z\le [M]$ we have $\ed(f^{-1}(y)\times Z)\le [L]$ for some $CW$-complex $L$, then $\ed X\le [L]$.