# Definition talk:Tychonoff Separation Axioms

If these are actually Axioms, we should put them in the Axioms namespace. --Cynic (talk) 20:44, 10 January 2009 (UTC)

- They're not. It's just what they're called, sort of like Europe is called a continent. They are called the Separation Axioms because they are usually assumed in the statement of a proof about a space of a certain sort, but they aren't axioms in the sense that a=b implies b=a is an axiom. Zelmerszoetrop 06:59, 11 January 2009 (UTC)

Turns out that putting a LaTeX formula in a heading doesn't work in a contents page. Shame, but there you go ... so I adjusted the heading names. --prime mover (talk) 22:17, 23 February 2009 (UTC)

- This is no longer an issue. MathJax allows $\LaTeX$ in titles. So the titles have been amended appropriately. --prime mover 00:29, 23 April 2011 (CDT)

Is the last line of the definition correct? I would expect it to say "normal" there. --prime mover (talk) 22:27, 23 February 2009 (UTC)

- Sure, a mistake, I was just copying from above... Headings are ok like this, it is better to have a readable table of contents.--Cañizo 00:20, 24 February 2009 (UTC)

Further notes ... I've googled and can't find reference to this set of axioms being named for Kolmogorov. T0 appears to be, but collectively they seem to be named for Tychonoff (or Tikhonov, spelling varies), and even that's not universal. Should this page be renamed just "Separation Axioms"? --prime mover (talk) 06:41, 24 February 2009 (UTC)

- I agree, just leaved the title because it was already there. I'll do it.--Cañizo 19:00, 24 February 2009 (UTC)

- Yep, just wanted confirmation, I know little about this area of maths beyond the basics of Hausdorff spaces. --prime mover (talk) 20:37, 24 February 2009 (UTC)

## Rethink of Naming Convention

I understand I'm the only one on ProofWiki exploring this area at the moment, so no doubt I'm the only one who (currently) cares. But I'm writing this section just in case there are eacgle-eyed watchers out there who do have an interest in this.

There are different conventions for the naming of these axioms. We currently have them in as matching Wikipedia's take on the subject.

However, I am going to change the pages and their contents so as to match the work in (for example) 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: *Counterexamples in Topology* (2nd ed.), as the advantage of relying directly on the T number to directly specify the axiom itself (rather than using the numbers in the artificial way of ensuring that the chain of inclusions is a neat sequence of numbers) makes it easier to define the precise properties of a space by its combination of numbers.

So rather than saying a space is "$T_3$ iff it's regular and $T_0$", we would say that "a space is regular iff it fulfil the $T_3$ and $T_0$ properties".

This appears to be the way the subject evolved. The fact that $T_3$-ness without $T_0$-ness does not imply $T_2$-ness is a fact of mathematics, and twisting the nomenclature around so as to make the implications neater holds the danger of the details being obscured.

Feel free to comment and object before I actually start work on this.

--prime mover 07:04, 27 April 2011 (CDT)

- ...good, no comments, no objections. Job done. Now I can get back to work. --prime mover 17:50, 27 April 2011 (CDT)

## Other conventions

What do you think about $R$-axioms (see here)?

- They have to be included in ProofWiki but unless I find them in a reputable work of mathematics I won't touch them. Wikipedia is all very well but unless it's for something trivial I won't use it as a source work. --prime mover (talk) 17:28, 27 December 2012 (UTC)

- BTW it's expected that you sign your posts on talk pages. Please check the FAQ page: What do you mean "sign your post"?. --prime mover (talk) 17:39, 27 December 2012 (UTC)