The normal Reed space problem

Friday, March 22, 2019 - 10:00 to 10:45

Thackeray 321

Speaker Information
Peter Nyikos
University of South Carolina

Abstract or Additional Information

A Reed space is a regular space that is the union of
countably many open metrizable subspaces. A Reed space is monotonic if
the union can be ascending.

One of the most remarkable unsolved problems in all of topology is:

The Normal Reed Space Problem. Is every normal Reed space metrizable?

The concepts in this problem are taught in almost every course in
topology, except perhaps for "normal, " and that can be explained in a
minute to anyone who knows what a topology is. On the other hand, it
has resisted the efforts of some of the best researchers in the three
decades since it was posed by G. M. Reed.  Even set-theoretic
consistency results for it are lacking.

As with the Normal Moore Space Problem, normality makes a huge
difference. There are easy familiar examples of spaces which are both
Moore and Reed, and nonmetrizable. Some of them are monotonic and
others are the union of two open metrizable subspaces.

The Normal Reed Space Problem is made tremendously difficult by the
fact that that a counterexample has to be a Dowker space with a
σ-disjoint base. We lack consistency results even for this more
general problem, which defied strenuous efforts by both Mary Ellen
Rudin and Zoltán Balogh, two of the greatest researchers in the
history of set-theoretic topology.

This talk is focused on monotonic Reed spaces where each of the open
metrizable subspaces is strongly zero-dimensional. There is a nice
structure theory for these spaces that may pave the way for either a
counterexample or to a Yes answer for the subclass of these Reed
spaces. In any event, it makes further progress on the problem
feasible for many graduate students and other researchers in general

Research Area