Sigma-fragmentable spaces, sigma-scattered networks and tri-quotient maps
by
Roger W. Hansell
Mathematics Dept., University of Connecticut, Storrs, CT 06269 U.S.A.

All topological spaces are assumed to be regular and Hausdorff. A study is made of the relationships between spaces that are sigma-fragmented by a finer metric and spaces having a sigma-scattered network. A number of equivalencies are established which extend to general topological spaces results previous given by the author in the context of normed (or function) spaces with the weak (resp. pointwise convergence) topology [H]. Exact conditions are given under which the metric topology will be complete or analytic. The equivalencies enable us to give comparatively straightforward proofs of recent results by Kenderov and Moors [KM], and to prove that both types of spaces are preserved by tri-quotient maps (see [M] for the definition) with compact fibers, thus any perfect map. In particular, it follows that perfect maps preserve sigma-scattered spaces, a question raised by the author several years ago. The assumption on the fibers cannot be omitted as R. Pol has shown that any metric space is a continuous open (hence tri-quotient) image of a sigma-discrete (equivalently, sigma-scattered) metric space.

References

[H] R. W. Hansell, Descriptive sets and the topology of non-separable Banach spaces (original manuscript, circulated in 1989-90, to appear in Serdica Math. J.).

[KM] P. S. Kenderov and W. B. Moors, Fragmentability and sigma-fragmentability of Banach spaces, J. London Math. Soc. (2) 60 (1999) 203-223.

[M] E. Michael, Partition-complete spaces and their preservation by tri-quotient and related maps, Topoplogy Appl. 73 (1996) 121-131.

Date received: June 21, 2000

Copyright © 2000 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 # caeh-31.