Talk:Curve/Archive 1

message: The Hahn-Mazurkiewicz theorem is the following
characterization of general continuous curves:
A Hausdorff topological space is a continuous image of the
unit interval if and only if it is a compact, connected,
locally connected metrizable space.
So presumably the simple curve theorem is a toy version of this? Anyway, I presume these belong together, in some sense.
Charles Matthews 20:50, 14 Feb 2004 (UTC)
thread_title:  The Hahn-Mazurkiewicz Theorem 

message: Such examples do exist. Please do not remove "simple" from here!!!
:I will remove it unless you can give me a specific "example".  Just saying "examples do exist" isn't enough.  To clarify, I assume you are talking about 2-dimensional Lebesgue measure in the plane.  Can you give a specific example of a simple curve whose image in the plane has positive 2-dimensional Lebesgue measure?  I don't know of any, and since you didn't bother to say what your "examples" are, I'll assume there aren't any.  BTW, the Peano curve is not simple, the Koch snowflake IS simple, but its image has zero 2-dim Lebesgue measure. Revolver 22:59, 15 Feb 2004 (UTC)
thread_title: Simple curve with positive Lebesgue measure

message: In general if you do not know something please do not make changes!!!
One can construct such example by simple varying Peano curve construction making it simple (so you get a closed nowhere dense set of positive measure)
I was trying to search interenet for such examples but found nothing, in a few days I will provide a reference.
Tosha
Sorry, let me explain my specific reasons for making the change:
*Your wording was very unclear; the way you had worded it, it sounded as if you were saying that the Peano curve (or a Peano curve) was an example of a simple curve with positive measure. Since the term "Peano curve" has this conventional meaning, that's how I took it, you didn't say anything about "varying the construction", so from the way I understood it, you were saying that the Peano curve was a simple curve with positive Lebesgue measure, which it is not. When I read something I know is untrue, I change it. THAT was the reason I took simple out. Now I see that wasn't what you meant, but can you see how I read it that way??
In general if you do not know something please do not make changes!!!
:Okay, but there is a flip side to this -- in general, if you're going to make unusual claims in an article (and a simple curve whose image has positive measure seems unusual to me), then don't be surprised if people question if it's correct and want examples. Maybe among certain researchers, these kinds of curves are well-known, but I think if you took a survey of mathematicians in general, most would be surprised (or at least unaware) of these curves (in fact, out of a dozen or so grad students I asked, most "guessed" such curves didn't exist). I'm just saying, this fact is not part of the common knowledge. It's part of mathematical nature not to believe something until given an example, esp. when you misunderstand the example, this doesn't help, either.
Apologies,
Revolver 09:07, 17 Feb 2004 (UTC)
thread_title: Do not remove "Simple" in "Simple curve with positive Lebesgue measure"

message: I disagree strongly with the trend this article is taking under the edits of Tosha, who is consolidating the differential geometry of curves in one article. Simply because a topic can be consolidated does not mean that it should be.  It makes Wikipedia into a nice mathematical treatise, but it is not, in my opinion, the way a hyperlinked encyclopedia on this subject should be organized. For example, rectifiable curve is now redirects here without definition at all, and "regular curve" gets a single line, buried in the next. These are important definitions which should have articles to themselves which at least present the definitions, and perhaps links to theorems regarding their usage. In my opinion, Wikipedia suffers when they are lumped too much together, in the style of a mathematics monograph. -- Decumanus 19:58, 15 Mar 2004 (UTC)
thread_title:  disagreement with trend of article