# Etymology of "regular" and "normal" spaces

• I
• dkotschessaa
In summary, the conversation discusses the confusion between regular and normal spaces in topology, and the linguistic and historical justification for their names. The conversation also mentions various separation properties in German and the lack of a name for completely regular spaces. It ends with a discussion on the importance of remembering names and word origins in mathematics.
dkotschessaa
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...

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:
dextercioby
fresh_42 said:
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.

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.

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.

Fantastic! Now I will remember.

-Dave K

dkotschessaa said:
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.
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.
I am also very interested in word origins in general, not least of all in mathematics.
In this case you might be interested in the following website which I frequently use (out of interest):
http://www.etymonline.com/index.php?allowed_in_frame=0&search=normal

dkotschessaa
fresh_42 said:
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):
http://www.etymonline.com/index.php?allowed_in_frame=0&search=normal

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

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.

dkotschessaa
fresh_42

## 1. What is the difference between regular and normal spaces?

The terms "regular" and "normal" are used to describe different properties of topological spaces. A regular space is one in which every point has a neighborhood that is topologically equivalent to a Euclidean space. A normal space is one in which any two disjoint closed sets can be separated by disjoint open sets. In simpler terms, a regular space is "well-behaved" locally, while a normal space is "well-behaved" globally.

## 2. How are regular and normal spaces related?

All normal spaces are also regular, but the converse is not always true. This means that every normal space is also regular, but there are regular spaces that are not normal. In other words, normality is a stronger condition than regularity.

## 3. Can you provide examples of regular and normal spaces?

Some examples of regular spaces include Euclidean spaces, spheres, and tori. Normal spaces include metric spaces, compact spaces, and Hausdorff spaces. An example of a space that is both regular and normal is a cube.

## 4. What is the significance of regular and normal spaces in topology?

Regular and normal spaces are important concepts in topology because they help to classify and distinguish different types of spaces. They also have practical applications in fields such as geometry, physics, and computer science.

## 5. How do the concepts of regularity and normality apply to the real world?

In the real world, regular and normal spaces can be used to describe and analyze various phenomena, such as the shape of objects, the behavior of fluids, and the structure of networks. They also have practical applications in fields such as engineering, architecture, and urban planning.

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