596:) 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,
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552:) 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.
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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
Knowledge 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
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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
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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
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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
515:, 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(
732:(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'.
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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
630:). 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.
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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.
652:(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 "T
499:) 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
519:) (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(
656:" 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).
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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 "R
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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 T
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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. —
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544:) consistently, and not slip into using "regular", "completely regular", etc. because by this way of defining things, "regular" isn't usually enough.
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586:) notation or defintion is "bad", since the main reason to define it this way is so that the logical implications are notationally elegant...
540:. 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(
709:.") remain valid, so long as both terms (in this case, "completely regular" and "uniform") are interpreted the same way (either with T
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I edited the given example of a
Haussdorff space that is not regular. The indicated topology was in fact the discrete topology on
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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(
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So, my question is, there seems to be a disagreement on the wiki about standards. On the one hand, several articles say, "T(
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is compact, the limit required for the theorem will exist. Because a compact
Hausdorff space is regular, the condition on
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spaces are necessarily
Hausdorff.", then please change that to "regular". That is how conventions are enforced on a wiki.
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or without) -- that's much of the motivation for people that try not to restrict attention to
Hausdorff spaces.
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to make it correct, then please do change it to "regular
Hausdorff"! Conversely, if somebody writes "Not all T
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page, but I don't understand the comment "the terms 'regular', 'completely regular', etc. are better than the
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578:", I think it should probably stay that way. But then it doesn't make any sense for this article (
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on Knowledge. If you would like to participate, please visit the project page, where you can join
600:. After all, what's the point in making precise conventions if they're not followed everywhere?
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can't find where I got the idea to use this example. So the uproperty of the SC compactification
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559:) notation and definitions are bad, we shouldn't use them. On the other hand, the
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But why are the terms "regular" and "completely regular" any "more clear" than T(
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Okay, reading on, I see I misinterpreted the statement slightly. Yes, the T(
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The other choice is to define the terms so that the strength of the T(
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Having made those choices, however, how do we enforce them? Well, we
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be a good idea, because it makes things unambiguous. And changing "T
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not include T(0), e.g. There are 2 main choices, one can either
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I will remove the following paragraph because it is not correct.
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but only if people make an effort to follow it religiously
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is continuous, but cannot be extended continuously to
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This is something that perhaps be brought up on the
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340:Here is a counterexample: let the space
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536:) axioms IS an increasing function of
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626:term (at least not until you get to
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567:) terminology and definitions.
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359:= { 0, 1 } is dense in
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