Talk:Regular space

Untitled

I will remove the following paragraph because it is not correct.

Suppose that A is a set in a topological space X and f is a continuous function from A to some space Y.

Suppose that, whenever a net or filter in A converges to a point in X (say x = lim_n a_n), then f(a_n) converges to a point y in Y.

Then we would like to be able to extend the domain of definition of f to the closure of A, by letting f(x) = y, and we would like the extension to be continuous as well.

If Y is a regular space, then this is always possible.

If Y is regular Hausdorff, then such a continuous extension will not only exist but will be unique.

Note that if A is a dense set, then f will be extended to all of X.

This is called extension by continuity, since the extension of f is defined (uniquely, in the Hausdorff case) by the :requirement that it be continuous.

Here is a counterexample: let the space X = { 0, 1, 2 } have the topology { ∅, { 0 }, { 1 }, { 0, 1 }, X }, and let Y be a (regular) T₁ space. The set A = { 0, 1 } is dense in X. Let f: A → Y be an injective function. f is continuous, but cannot be extended continuously to X. Tohuwaboho (talk) 13:27, 26 August 2008 (UTC)[reply]

I removed the following paragraph:

Note that if A is a dense set, then f will be extended to all of X. Consider the special case where X is the Stone Cech compactification of A and Y is a compact Hausdorff space. Because Y is compact, the limit required for the theorem will exist. Because a compact Hausdorff space is regular, the condition on Y is met. Thus, f will have a unique extension to X. This proves the universal property of the Stone Cech compactification.

The argument appears suspect: it only uses the fact that A is dense in X and shows that there's always a unique continuous extension to every compact Hausdorff space Y. But in general there isn't, even if X is compact Hausdorff. AxelBoldt 19:31 Aug 30, 2002 (PDT)

What doesn't go through in this general case is the existence of a limit; we might have only a cluster point. Of course, in the case of the Stone Cech compactification, we do have a limit, but I don't see how to argue for this any more simply than using the universal property to begin with, and I can't find where I got the idea to use this example. So the uproperty of the SC compactification is an example of extension by continuity, but not one that seems relevant to the theorem as regards regular spaces. I will put back the first sentence, however. — Toby 19:27 Sep 4, 2002 (PDT)

This is something that perhaps be brought up on the separation axiom or topology glossary page, but I don't understand the comment "the terms 'regular', 'completely regular', etc. are better than the less clear T(i) axioms, so the T(i) terminology won't be used" (my emphasis). Why should the terms regular, completely regular, etc. be any more clear? The way the terms are actually defined on wikipedia it seems that the choice has been made that regular not include T(0), e.g. There are 2 main choices, one can either

Define the terms so that regular, completely regular, etc. are given the additional weak separation axiom, so that one has the convenience of the nicer-sounding (and more evocative) terms at one's fingertips, since TBOMK most people don't really consider e.g. regular spaces that aren't T(0). Without this, one has to either use the terms T(i) (which apparently people here think is bad, for whatever reason) or constantly say "regular and T(0)", etc. which is inconvenient. The disadvantage to this choice is that of course the strength of the T(i) axioms is not an increasing function of i.

The other choice is to define the terms so that the strength of the T(i) axioms IS an increasing function of i. This has the mathematical/notational elegance of not having to remember chains of conjunctions and so on, but now you need to use the T(i) consistently, and not slip into using "regular", "completely regular", etc. because by this way of defining things, "regular" isn't usually enough.

But why are the terms "regular" and "completely regular" any "more clear" than T(i) on the surface? Only if you have already decided on a convention!! If someone is confused on what the convention is, reading "regular" or "completely regular" isn't going to help them, (unless it has some extra term added, but that doesn't always seem to be the case). One can just as easily ask, "what does regular mean?" as "what does T(3) mean?" I don't see how one is "more clear" than the other.

So, my question is, there seems to be a disagreement on the wiki about standards. On the one hand, several articles say, "T(i) notation and definitions are bad, we shouldn't use them. On the other hand, the separation axiom article freely uses the T(i) terminology and definitions.

My own opinion is that there are tradeoffs either way. And since it appears that the choice has been made to define it so that T(i) is an "increasing function of i", I think it should probably stay that way. But then it doesn't make any sense for this article (regular space) to say that the T(i) notation or defintion is "bad", since the main reason to define it this way is so that the logical implications are notationally elegant... Revolver

Okay, reading on, I see I misinterpreted the statement slightly. Yes, the T(i) notation and terms are less clear when compared to unambiguous terms like "Tychonoff" or "regular Hausdorff" or "normal Hausdorff", which are unambiguous. However, I guess what my point is, is that this convention which does eliminate ambiguity, is only useful or accurate if it is stuck to religiously. I think in several places in some of the articles I've read the author(s) have slipped into the habit of saying "normal" or "regular" when they really mean "normal Hausdorff" or "regular Hausdorff", because they're used to dropping the "Hausdorff" part in practice, (say, when writing a journal article, you can say, "for this article, all spaces are Hausdorff"), and so they start "dropping" the Hausdorff unconsciously. Normally (no pun) this isn't a problem, but their are some cases where there really is a difference between the two, and especially on pages where the essential logical implications among terms and what not is supposed to be presented completely clearly, it should really matter. I think the convention "normal Hausdorff", "regular Hausdorff", etc. is actually BETTER for this purpose, but only if people make an effort to follow it religiously. After all, what's the point in making precise conventions if they're not followed everywhere? Revolver

My reasoning is this:

The axioms T₃ and stronger have no unambiguous name in common usage consisting of a single term (at least not until you get to metrisable). Nevertheless, they do all have names that are at least unambiguous and in commong usage, and these are most important. Those unambiguous names are the ones that include the phrase "Hausdorff", and those are the names that I advocate in this case.
The axioms that are incomparable to T₂, OTOH, have no unambiguous names in common usage. So it's less clear here that we should use an ambiguous but common (and short) name like "regular" instead of an unambiguous but uncommon name like "R₂" (at least short) or "with T₃ Kolmogorovification (not even short!). But looking at this example, I at least believe that "regular" is the best one here -- and indeed, I go with the short and common but ambiguous names. This still leaves the choice between "regular" and "T₃" as the name for the weak axiom; and it seems clear to me that "regular" is winning out in the literature now (probably because people like the feature of increasing strength in the index).
Having made those choices, however, how do we enforce them? Well, we can't enforce them perfectly, especially not on a wiki. But at least in a wiki, it becomes easy to correct any errors. So if you read an article that says "regular" but requires T₀ to make it correct, then please do change it to "regular Hausdorff"! Conversely, if somebody writes "Not all T₃ spaces are necessarily Hausdorff.", then please change that to "regular". That is how conventions are enforced on a wiki.

Now, there is more to your objection than this. Since enforcement of the convention is so weak, aren't we better off with no convention at all? I don't believe so, as long as we handle the convention intelligently. If we had no convention, then changing "regular" to "regular Hausdorff" (when that is what is meant) would still be a good idea, because it makes things unambiguous. And changing "T₃" to "regular" when that is what is meant would also be a good idea, simply becuase the latter term is now more common than the former, in the non-Hausdorff context. All the same, a statement like "Not all regular spaces are necessarily Hausdorff." could be confusing -- and it might even be confusing without an official convention. So if it adds to clarity by adding a parenthetical comment like "(according to the more premissive modern definition of the term)", then feel free to do so -- convention or no! I think that it's wrong for Wikipedia to expect our conventions to clarify things when the article text alone is unclear, so we should make the article text as clear as possible, regardless of conventions. (In most situation, however, a link to regular space will probably work -- but not for all.)

It's also worth noting that, most of the time that "regular" is said instead of "regular Hausdorff", things are still basically OK. If I say that a certain space is regular when it's actually regular Hausdorff -- well, it's still regular! And most general theorems (like "Any completely regular space can be made into a uniform space.") remain valid, so long as both terms (in this case, "completely regular" and "uniform") are interpreted the same way (either with T₂ or without) -- that's much of the motivation for people that try not to restrict attention to Hausdorff spaces.

-- Toby Bartels 00:19, 15 Feb 2004 (UTC)

Haussdorff non-regular space

I edited the given example of a Haussdorff space that is not regular. The indicated topology was in fact the discrete topology on R (which is regular). The problem was that the countable subset to be removed from the open sets should be a fixed one. Moreover, for the example to work, it should also be nonclosed. Instead of 'countable and nonclosed', I proposed the more general 'nonclosed with empty interior'. Juan Cappa (talk) 18:43, 9 February 2024 (UTC)[reply]