Talk:Separation axiom

Old talk (formerly at "Talk:Separation axioms")

I have a request: the explanation of the different usages of the terms in the literature is useful, but we also need a standard usage guide for Wikipedia of those terms. I would suggest that we adopt the usage from the Topology glossary, make that the most prominent in this article, and then, in a separate section, explain how some people use the terms differently, and for what reasons.

Right now, as the article stands, I can not unambiguously use the term "regular topological space" in any Wikipedia article. AxelBoldt, Wednesday, April 3, 2002

The Topology Glossary deliberately doesn't define T₃ and T₄ because of this problem. I would prefer to use the stronger definitions for these, but I'm not sure what other people think. --Zundark, Thursday, April 4, 2002

Whatever terminology you think is the most standard, let's go with it. As long as we explain the other terminologies in this article, we should be fine. AxelBoldt

I'm pretty sure that what I've called the "modern convention"s in the article are the more standard today, as reflected in Axel's comment in the article's Glossary. So everything should be cool now. -- Toby Bartels (2002 April 24)

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Axel, I don't like having the word "space" in the entries to the table in the glossary, for two reasons:

1. It clutters the table.
2. I've noticed that even people that don't mind saying "Hausdorff space" or "Let X be Fréchet." still avoid saying "Fréchet space" (in this context), to avoid confusion with the other, more common meaning (which needs its own article ^_^).

If you think that it's unclear that the column labelled "Space" refers to the names given to spaces while the columns labelled "Axioms" refer to the names given to axioms, then I'd rather explain this with text before the table. — Toby Bartels, Wednesday, June 12, 2002

the reason I put in the "space" was because the table is so long that one usually doesn't see the table header, one only sees a row like T4, T4+T0, T4. And I see the crucial point of the article to be the distinction between T4, the axiom, and T4, the space. How about if we put "space" in only for those entries where "X, the space" is different from "X, the axiom"? But it's your baby, so do what you think is best. AxelBoldt, Wednesday, June 12, 2002

Although I appreciate the thought, the "your baby" sentiment doesn't seem very Wikiish, does it? In any case, I think that your idea is good, so I'll implement it. — Toby Bartels, Thursday, June 13, 2002

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If anybody is watching this article: I intend to do a major rewrite for purposes of clarity of exposition. This will banish all talk of historical and variant meanings of terminology, fixing a specific language for all time, except for warnings and a link at the beginning. This link will be to a new page, History of the separation axioms, that will explain how the axioms were originally numbered and how the terminology has changed. (I even hope to look at some primary sources to research this!) In contrast, this page, Separation axiom, will have no ambiguities, not even between "T_i axiom" and "T_i space", and will agree with both Topology Glossary and (hopefully) usage in the rest of Wikipedia. — Toby 17:35 Jul 23, 2002 (PDT)

OK, this is done. I still have to write History of the separation axioms, which will incorporate some material from the old page that didn't make it into the new page. (I also have that on my own machine, so it is not lost!) Note that the page has been renamed following naming conventions to a singular noun, but I'm not going to move this talk page, since it's largely irrelevant to the new version. — Toby 20:54 Aug 5, 2002 (PDT)

Note about old talk

Note that an older version of this page existed at Separation axioms, but the page was completely redesigned for the move. There is old talk at Talk:Separation axioms, but this is largely irrelevant to the new page, so I haven't moved it. -- Toby 20:56 Aug 5, 2002 (PDT)

I've moved and history merged it to line up with the history merge I did on the main article. Graham87 06:22, 31 July 2013 (UTC)

Removal of alternative definitions

I removed both of the alternative definitions (one put in by me, one put in by Axel) in order to keep the emphasis on the relationships to various forms of separation -- what it is that joins all of these together as separation axioms. These alternative characterisations (and many more) may still be found on the axioms' individual pages. (I made it more clear before the definitions that this is so.) -- Toby Bartels 13:13 21 May 2003 (UTC)

Removed "Continuous

I removed the word "continuous" from "separated by a function" in the section giving the definitions, since that word is not used in the introductory section on separation, nor in the phrase as defined on Separated sets. I'm not absolutely opposed to adding the word "continuous" throughout, although I would argue that it's unnecessary in a topological context (especially since, if you remove continuity from the definition given on Separated sets, the notion becomes trivial). -- Toby Bartels 23:45, 2005 Jan 31 (UTC)

Error

Fully normal does not imply regular. On the other hand, for a Hausdorff space, full normality and paracompactness are equivalent. Manta 21:51, 19 December 2006 (UTC)

Do you know of a specific counterexample (a space that's fully normal but not regular)? But to be honest, I doesn't strike me as at all likely that full normality should imply regularity, and since I appear to be the one that originally put that word in, I think that I'd better remove it. Still, if there is a specific counterexample that inspired you to make this comment, it would be nice if you mentioned it here.

And while I'm giving you advice: as long as you're sure of your facts (well, subject to NPOV and verifiability), then you should be bold and edit the page yourself! ^_^

—Toby Bartels 19:12, 9 May 2007 (UTC)

Ha!, I see that you gave an example on Talk:Paracompact space: Sierpinski space. So I am quite satisfied; you are entirely correct! —Toby Bartels 19:23, 9 May 2007 (UTC)

T_1/2 spaces?

I have a book "Topological Methods in Chemistry" by Merrifield and Simmons that discusses T_1/2 spaces. By their definition, a space is a T_1/2-space if each point of the space is either open or closed or both. This notion was used for a discussion of finite topologies. It is claimed that T_1/2 spaces are truly intermediate between T₀ and T₁ spaces in the sense that all T₁ spaces are T_1/2 and all T_1/2 spaces are T₀ and there exist weaker topologies that do that satisfy the stronger. However this separation axiom is not reflected in this article. Jason Quinn 00:30, 2 June 2007 (UTC)

Then please, be bold and put it in! Cite Merrifeld and Simmons as your reference in the Sources section. If you don't know how it fits in to the diagram or the discussion on T₀ vs non-T₀, then just place it in the section on Other separation axioms. This article should be comprehensive, so the more the merrier! --Toby Bartels 22:33, 14 June 2007 (UTC)

Senseless sentence?

Domenicozambella, in an edit summary, wrote:

I erased a sentence that does not make much (any?) sense.

I put the sentence back. It makes sense to me, but I wrote it, so that doesn't mean much. Can it be improved? —Toby Bartels (talk) 19:11, 12 December 2010 (UTC)

Why these definitions?

I'm a little confused why these are the definitions on Wiki. Are these the commonly accepted ones? I feel like since Counterexamples in Topology is such a famous book, all the axioms should match that book to avoid confusion. — Preceding unsigned comment added by 2601:646:C900:3210:9C20:C011:B89F:6786 (talk) 00:12, 21 May 2022 (UTC)

If you think that common terminology is missing, go ahead and add it. If you think some terminology should be removed, that too is possible; either be bold and remove it, or discuss it here first. FWIW, despite some amount of exposure to topology, I haven't heard of Counterexamples in Topology. —Quantling (talk | contribs) 22:01, 24 May 2022 (UTC)

Counterexamples in Topology is a great book, but unfortunately the terminology is no longer current. The definitions here (particularly T₃ vs regular and T₄ vs normal, but also completely Hausdorff vs Urysohn) are the ones commonly accepted today (at least in English), largely under the influence of the book by Willard (see the references). There is some discussion of this at History of the separation axioms, but it's hard to source that, so it's not very complete. ―Toby Bartels (talk) 04:40, 10 October 2022 (UTC)

The definitions on the wiki are mostly the currently used ones. Counterexamples in Topology was written in 1970 and some of its terminology is a little dated, incompatible with more modern usage. See in particular History of the separation axioms for the rationale of the definitions used in Wikipedia. I would also recommend Willard as a reference, as well as taking a look at https://topology.pi-base.org. PatrickR2 (talk) 04:43, 10 October 2022 (UTC)

Diagram of relationships between the axioms

It seems that the beautiful diagram that indicates the relationships between the various axioms has a problem at the top. Perfectly normal spaces are always completely regular, and hence R0 (see https://math.stackexchange.com/questions/4525139 for example). So in the diagram PN should be paired up with T6 and not a separate node. PatrickR2 (talk) 04:47, 10 October 2022 (UTC)

Major occurrence of strong separation axioms

I personally encounter seldomly separation axioms stronger than Hausdorffness other than in the context of general topology. But still, such axioms are involved in some common facts listed as follows. The T numbers refer to the theorem number of the pi-base (only main theorems are listed).

· A σ-compact and locally compact Hausdorff space is T₄ (T26).

· A T₃ Lindelöf space is T₄ (T30).

· A nontrivial connected T₄ space has at least cardinality of continuum (T80 and T309).

· A linearly ordered space is T₅ (T273).

· A metrizable space is T₆ (T268). 129.104.241.193 (talk) 02:04, 1 June 2024 (UTC)