On compact fibered spaces
by
J. Gerlits
Math. Inst. Hungarian Academy
Coauthors: Z. Szentmiklóssy

An old open problem by D. H. Fremlin is: Is it consistent that any hereditarily Lindelöf compact T₂-space has a two-to-one continuous map onto a metrizable space? (A Souslin-continuum is a counter-example.)

Definition (V. V. Tkachuk) X is metrizably fibered if it can be mapped continuously and with metrizable fibers onto a metrizable space.

Theorem (V. V. Tkachuk) A compact T₂-space X is metrizably fibered iff there exist a countable cover F consisting of closed G_\delta subsets of X such that

KerxF= \cap {F in F: x in F}

is metrizable for any x in X.

Definition (V. V. Tkachuk) X is fibered if there is a countable closed cover F of X with Ker_xF metrizable for any x in X.

In this lecture we answers some problems left open by V. V. Tkachuk in his cited paper and raise some new problems about compact fibered spaces.

Reference

V. V. Tkachuk: A glance on compact spaces ..., Topology Proceedings 19 (1994), 321-333

Date received: June 30, 1998