Thackeray 321

### 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

topology.