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I Etymology of "regular" and "normal" spaces

  1. Dec 12, 2016 #1
    This is kind of a silly question. I always get "regular" spaces and "normal" spaces confused in topology. This will be a problem if I am asked on a qualifier to prove something about one of these spaces. Is there any linguistic or historical justification to why a regular space deals with a point and a closed set, and a normal space deals with two closed sets?

    Otherwise I'm going to need a mnemonic...
  2. jcsd
  3. Dec 12, 2016 #2


    Staff: Mentor

    In German, we number the separation properties:

    ##T_0## Kolmogoroff
    ##T_1## Fréchet
    ##T_2## Hausdorff
    ##T_{2\frac{1}{2}}## Urysohn
    ##T_3## Vietoris - regular, if also ##T_1##
    ##T_{3\frac{1}{2}}## Tychonoff
    ##T_4## Tietze - normal, if also ##T_1##

    Only for completely regular I haven't found a name.
    (T stands for "Trennungsaxiom" which is the German word for separation axiom.)

    What makes me wonder is, that we have these properties named after topologists, whereas I got the impression, that in English far more theorems, methods and so on are named. I've read about many theorems here, I didn't even know they have a name. As a result I refuse to write Abelian with a small letter. It would be unfair when Lagrange, Legendre, Hamilton and many more are always written with caps.

    I have the feeling that there is no easy-to-remember way in this jungle, which becomes even more complicated, if we add examples and counterexamples to the list, including additional names like Niemytzki or Mysior. At least I'm looking for such a method since I've first read about the separation axioms. I have a similar problem with Banach and (pre-) Hilbert spaces.

    Jean Dieudonné writes that in the beginning, being Hausdorff has been widely regarded as the minimum requirement, until Zariski came up with his topology on algebraic varieties. In general the problem arose, when topology became its own field of research and people searched for a minimal system of axioms, such that theorems are still meaningful. As such the concept as a whole is relatively young. Dieudonné also notices that similar questions can be found in dimension theory.

    How's that as mnemonic?
    Normally, we want to have on a regular basis at least Hausdorff spaces.
    Last edited: Dec 12, 2016
  4. Dec 12, 2016 #3
    Lovely response! Danke. I proudly pronounce, in what I think is a pretty good accent, any German words I know. In particular I do like saying "Trennungsaxiom" at least to myself.

    I knew ##T_2## was Hausdorff but I was not aware of the others. I agree about names. We pass over theorems and names so quickly in textbooks sometimes and miss the whole story. I am also very interested in word origins in general, not least of all in mathematics.

    Fantastic! Now I will remember.

    -Dave K
  5. Dec 12, 2016 #4


    Staff: Mentor

    In the book from J. Dieudonné I mentioned are a lot of short biographies assembled. Only a few lines each. Hausdorff's is one that I won't forget anymore. Although not directly affected, the shoah has been the reason for his death. The meaning of words change tremendously when they get a face. Until I read about his faith, I used Hausdorff as I used the words continuous or smooth.
    In this case you might be interested in the following website which I frequently use (out of interest):
  6. Dec 12, 2016 #5
    This is great. Thanks.

    Also, I have a dumb joke.. I saw a poster at my school advertising the Hausdorff school: https://www.hcm.uni-bonn.de/events/eventpages/hausdorff-school/

    My first thought was "At a Hausdorff school, does every unique student get his or her own room?"

    -Dave K
  7. Dec 12, 2016 #6


    Staff: Mentor

    My first thought was kind of funny, too:

    How can you name a school that teaches science, which is alongside music (and evolution of course) the only subject, that really units us, how can you name it by a person who is associated with separation?

    But I'm glad they did, regarding Felix.
  8. Dec 21, 2016 #7
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