Talk:Normal space

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I changed T4 back to "normal Hausdorff", since I prefer self-explanatory unambiguous terminology. Everybody who sees the term "normal Hausdorff" knows what's going on, no matter when they learned topology. AxelBoldt

I removed this add-on to Tietze:

(If X is normal regular, then we may be able to extend the function when A is not closed; see Extension by continuity.)

The Extension by continuity article requires the target Y to be regular, not the source X. AxelBoldt 19:41 Aug 30, 2002 (PDT)

You're right; it's because R is regular that such an extension may be possible. But in any case, it's really two separate issues; you use extension by continuity (in certain circumstances) to extend to the closure of A, then use the Tietze extension theorem (in any circumstance) to extend to X, and there's really no interaction between these. — Toby 12:38 Sep 4, 2002 (PDT)

PS: I'm looking to see where I got that Stone Cech illustration in Regular space, and how it plugs the gaps in the case when X is indeed the SC compactification. (Or if it was wrong anyway ^_^.) — Toby

Are perfectly normal (Hausdorff) spaces also called T₆? The taxobox certainly implies so; I haven't seen this, but it would make perfect sense. If so, then this fact should be added to Separation axiom and (at the very least!) here. —Toby Bartels 09:32, 18 August 2006 (UTC)Reply

Even I was wondering, T₆ redirects here but the page makes no mention of T₆ anywhere. --Kprateek88(Talk | Contribs) 16:42, 2 November 2006 (UTC)Reply

The Encyclopedia of General Topology (2004, ed. Hart, Nagata, and Vaughan) defines T₆ spaces as perfectly normal Hausdorff spaces (p. 158). I'll add this terminology to the article since it fits in nicely with the other T_i axioms. -- Fropuff 19:09, 8 November 2007 (UTC)Reply

FWIW: Munkres also refers to perfectly normal Hausdorff spaces as T₆. -- Fropuff 06:09, 10 November 2007 (UTC)Reply

The article begins as follows:

"In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space."

This seems to mention two distinct meanings for the term "T4 space".

They seem to imply, without ever addressing this clearly, that a normal space is not necessarily Hausdorff.

If this is right, it would be immensely helpful if the article made this emphatic rather than just casually mentioning it.

To make matters worse, "T4" is used later in the article, without its meaning having been clearly specified in the article.

I hope someone knowledgeable about this topic and also able to write clearly can fix this.

You are right that this may be a little confusing. Axiom T4 is not the same as a T4 space; it's preferable to remove mention of it here. I'll also try to clarify a few other things. PatrickR2 (talk) 05:11, 22 May 2025 (UTC)Reply

Also, note that "T4 space" is clearly defined in the second paragraph of the Definitions section. PatrickR2 (talk) 05:33, 22 May 2025 (UTC)Reply

@TakuyaMurata What is the purpose of having this second alternative characterization of normality? ("For ${\displaystyle X=U_{1}\cup U_{2}}$ with ${\displaystyle U_{1},U_{2}}$ open, there exist disjoint open sets ${\displaystyle V_{1},V_{2}}$ such that ${\displaystyle X=U_{1}\cup V_{1}=U_{2}\cup V_{2}.}$") I just checked it myself and it's easy to see, basically replace disjoint closed sets with their complements being open sets whose union is X, and just rephrase things correspondingly. But what's the point? I don't see this in Engelking for example. It's not a common characterization, maybe just an exercise somewhere. Wikipedia is not meant to be a list of all possible facts about a topic. PatrickR2 (talk) 05:49, 6 August 2026 (UTC)Reply

@PatrickR2: Sorry I should have given a reference. It’s from Johnstone's Stone spaces. Yes, it’s just a rephrasing of the usual definition; the point is that the condition only refers to open sets. Thus, it is a condition on the lattice of open sets; i.e., the topology as opposed to that on a topological space. It is actually useful and important in locale theory and so I thought it’s worth mentioned. It’s not true that every topological property can be stated only for a topology instead of a topological space. So mathematically, it is actually noteworthy if subtle (but the subtlety can be important like in this kind of separation axiom discussion). Taku (talk) 06:13, 10 August 2026 (UTC)Reply

@TakuyaMurata I have streamlined things a little for the Definitions section (and removed mention of Urysohn's lemma, which was uncalled for). One question about "See also Ideal (order theory) § Prime and maximal spectra for a related condition." Is this a condition related to that second bullet or something else? PatrickR2 (talk) 22:59, 10 August 2026 (UTC)Reply

Also, for the Johnstone reference for the second bullet, can you add a specific result number or pager number? PatrickR2 (talk) 23:02, 10 August 2026 (UTC)Reply

(Yes, I will add a specific location for the Johnstone ref. It’s just we are currently in the obon break in Japan, a week-long break and so the university library is closed and that’s why I couldn’t get the book to find the exact location.)

As for Urysohn, I would say that I think it should be mentioned earlier than properties, since it’s perhaps the most standard well-known characterization of normality. (As a comparison, for example, in the connected space article, we give the function characterization immediately.) But, anyway, since this is a matter of editorial judgement, we need a third opinion (and meantime the status quo prevails). Taku (talk) 23:53, 10 August 2026 (UTC)Reply

Happy obon holiday by the way. PatrickR2 (talk) 20:21, 11 August 2026 (UTC)Reply

In fact, I noticed in the discussion of a perfectly normal space, it already refers to Urysohn. That wouldn’t make sense if you didn’t know Urysohn's lemma, a common criticism on math in Wikipedia that an article assumes you know the topic already. Taku (talk) 00:08, 11 August 2026 (UTC)Reply

Aside reminder: Wikipedia is not meant to "teach" a subject. PatrickR2 (talk) 20:20, 11 August 2026 (UTC)Reply