TOPCOM, Ask a Topologist by M. Henriksen

Topology Atlas Document # topc-69.htm | Production Editor: Thomas M. Zachariah

Ask a Topologist

Query and Response from Volume 4, #1, of TopCom

This query from a graduate student was answered by M. Henriksen and is originally posted in: Ask a Topologist Bulletin Board

Query by Darin Brown Hi, I am a 3rd-year grad student (just transfered to UCSB), which means it's getting closer and closer to choosing an area of research and advisor. I have a good background in analysis, and we have a strong PDEs group here at UCSB, so I'm seriously considering this area. But, I've also always found general point-set topology to be a fun distraction. I seem to be rather alone in this liking, at least among most mathematicians I've known so far. The phrases "ugly", "pathological", and "useless" are often thrown in my face. There are topologists here, of course, but algebraic topologists whose research is far removed from the questions or methods of general topology. Mainly, I just seem to get strange looks if I mention interest in some kind of set-theoretic area (topology, real analysis, measure), because it's so "pathological", while I try to remind them that most scientists (and humans) consider most of math to be "pathological".

Another concern I have is that I always hear "point-set topology is dead" or "general topology is dead" as an active area of research. I'm not sure how much to trust these judgments. Certainly my knowledge of what's current and new in general topology (or areas which might have grown out of it in the meantime) is lacking. I don't know too much past Dugundji or Counterexamples in Topology (fun book, although most of my profs seemed proud to have "never heard of that space"), so I have no idea if there are any new fields or new questions which have developed since, although I know that the further one goes in general topology, the more one encounters deep issues in set theory and logic.

I've been trying to find some decent NON-TRIVIAL resources or expositions about what may or may not be going on. By "non-trivial" I mean: please, no undergrad textbooks, or lengthy motivations of why metric spaces or topologies are so important, something that assumes some experience and understanding of the basics, but at the same time is accessible enough for a grad student or general mathematician to understand without "joining the club". I have found a couple more advanced books, but unfortunately they were exercises in symbol-pushing, with no comments about anything. Thanks, any information is seriously appreciated.

Darin

Response by Melvin Henriksen

Paraphrasing what Mark Twain wrote to the editor of a newspaper that had printed a story announcing his death, any obituary of general topology is premature. Of course, when its critics call it "dead" they mean that it is uninteresting, useless, and no longer a viable field for research. In the article "There are too many B.A.D. mathematicians", I commented on the tendency of otherwise capable research mathematicians to confuse fact with opinion in a destructive way. (B.A.D. abbreviates Bigoted and Destructive.) It appeared first in The Mathematical Intelligencer in 1992, and is posted with some additional commentary in Topology Atlas.

Perhaps the best survey of the much but not all of general topology is the book "General Topology" by Richard Engelking (Warsaw 1977). There is a later edition published by Handelmann if your library has it. Another more specialized survey is "Extensions and Absolutes of Hausdorff Spaces" by J. Porter and R.G. Woods, Springer-Verlag, New York 1988. Mastering the contents of these two excellent books still leaves the reader unacquainted with large parts of the area but does convey its flavor. Getting the feel of a grown-up version of "Counterexamples in Topology" may be obtained by looking at "Open Problems in Topology" edited by J. van Mill and G. M. Reed, North-Holland, Amsterdam 1990. This book does an excellent job of conveying the scope of general topology and its frontiers. Perhaps the best way to see that this field is alive and well is to browse through "Topology Atlas" on the Internet. I hope that these references will help you to decide whether you should do research in an area related to general topology. Above all, I hope that your healthy skepticism about conventional wisdom concerning "dead" fields will enable you to do research in areas about which you feel enthusiastic instead of just following the latest fads.