Non-polyhedral proof of the Michael Finite-dimensional Selection Theorem
by
Sergei Ageev
University of Saskatchewan

We suggest a new method of the proof of the Finite-dimensional Selection Theorem of E.Michael free of using a passage to nerves of covers. This approach permits to simplify the known proofs of this theorem and to find some new selection theorems. In particular, we get a generalization of the Filtered Finite-dimensional Selection Theorem due to Schepin-Brodskij: no elements of the filtration are assumed to be complete.

Theorem. Let Y be a complete metric space, X a (n+1)-dimensional paracompact space. Let L_i, i=0, 1, ..., n+1, be an equi-LC^i-1-family of subsets in Y such that L₀, L₁, ..., L_n, L_n+1 is an increasing sequence. Let also F₀, F₁, , ..., F_n, F_n+1 be an increasing sequence of lower semicontinuous selections of a lower semicontinuous mapping F:X --> Y with closed values. If { F_i(x)|x from X} lies in L_i for each i=0, 1, ..., n+1, and the embedding e_i of F_i(x) into F_i+1(x) is i-aspherical for each x from X and for each i=0, 1, ..., n+1, then there exists a selection s:X --> Y of F.

Date received: August 31, 2002