# Topology: reference for “Great Wheel of Compactness”

This seems to be a very informative diagram showing the relationship between four forms of compactness in a general topological space. Prior to finding this I was trying to make sense of a seemingly countless (now seen to be countable = 12) collection of theorems relating one to another. The 12 relations are seen to simply to 6 proofs (A - F) and 6 corollaries by transitivity.

I found a version of this here https://pantherfile.uwm.edu/ancel/www/OLD%20COURSES/MATH%20752%20SPRING%202011/CHAPTER%20III/751.F10.IIIB-C.pdf

I haven't seen it anywhere else and would be interested if anyone has information about it.

• By the way, there is a lot of compact-like properties and relations between them. See, for instance, the diagrams in the surveys by M. Matveev, A Survey on Star Covering Properties and E.K. van Douwen, G.M. Reed, A.W. Roscoe, I.J. Tree, Star covering properties. Topology Appl, 39:1 (1991), 71-103 or a small draft diagram for Hausdorff spaces from a draft paper “On sequentially pseudocompact and pracompact spaces” by Oleg Gutik and me. – Alex Ravsky Dec 29 '15 at 16:06
• The link you provided seems pretty complete. You could also check out other textbooks such as Munkres "Topology". Otherwise, do you have a specific question? – Lee Mosher Dec 29 '15 at 16:31
• From the URL, it looks like this document comes from the web site of Ric Ancel at the U. Wisconsin Milwaukee math department. So he would be the natural person to ask. – Lee Mosher Dec 29 '15 at 18:36
• I’ve not seen the diagram in any textbook, though all of the information in it was in my first undergraduate topology course 50 years ago. Truth to tell, it does not strike me as a very natural way of organizing the information. – Brian M. Scott Dec 30 '15 at 0:21
• @LeeMosher Thanks for the lead. Ric Ancel at the U. Wisconsin Milwaukee math department confirms he created the original in the 1980's. – Tom Collinge Jan 3 '16 at 13:34

From: Fredric D Ancel


Sent: 02 January 2016 22:08 To: Tom Collinge Subject: Re: Your (?) notes

Tom,

The diagram is a suped-up modification of one that I created sometime in the 1980’s for a Moore-method introductory graduate level topology I was teaching at University of Wisconsin-Milwaukee (or possible at the University of Oklahoma). It was in my notes for the class. The goal of the diagram was to help students keep track of the relations between various forms of compactness. Some students memorized it and would quote it back to me when I couldn’t remember these relations myself. The “great wheel” terminology is a humorous and probably politically incorrect reference to a Buddhist construct I read about is some Kipling novel.

The original diagram was created in a primitive version of Word (or possibly MacPaint) - no curved lines and very basic typography. Someone else has improved its aesthetics in the interim. I have attached an early version of the diagram below.

• Ric Ancel

https://pantherfile.uwm.edu/ancel/www/OLD%20COURSES/MATH%20752%20SPRING%202011/CHAPTER%20III/751.F10.IIIB-C.pdf

(Professor Fredric D Ancel at the University of Wisconsin Milwaukee).