Factorization of set-valued mappings
by
Stoyu Barov
The University of Alabama

Let X be a Hausdorff space with a point-countable base, Y = \cup _i Y_i a regular space with each Y_i G_\delta-compact in Y. Let F:(K(Y), \tau) --> 2^{(K(Y), \tau)} be a lower semicontinuous map, where K(Y) = { K : K is compact subset of Y }, and \tau is the Tychonoff topology on the hyperspaces. Having factorized F through a metric space by an l.s.c. map G and an u.s.c. map H with compact images we find an u.s.c. selection G' for G with compact images such that

(H o G')(y) subset È y in K  F(K).

Moreover, we have |G'(y)| <= 2^i-1 whenever y in Y_i. This theorem can be applied for approaching the following problem:

(*) If f: X --> Y is continuous, under which conditions f is an inductively perfect map.

Regarding F = f ^-1 in certain situations, we can apply the above result. Thus, we show that the problem (*) could be approached by selection theorems.

Date received: March 4, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-58.