Talk:Curve

• any three of its elements are ordered by a betweenness relation (which also distinguishes curves as open, or closed);
• for any two distinct elements, their distance d to each other is measured being non-zero;
• for any two elements A and Q holds that for each element B between A and Q there exists an element N between B and Q (where N is neither between A and B, nor between A and Q) such that for each element P between N and Q (where P is neither between B and N, nor between B and Q) holds that

d( P, Q ) < d( B, N ) < d( B, Q ).

since the statements took me a while to figure out, and I'm still not sure that it's a correct definition of a 1-manifold (for example, it seems like we can have points p and q which have no point "between" them). More complete defintions referred to manifold. Chas zzz brown 22:29 Mar 8, 2003 (UTC)

As for: the statements took me a while to figure out

Nice try.

It would have been easier if you had made separate clauses. I might have written the above as:

• For any two elements A and Q, the following holds for each element B between A and Q: There exists an element N with the following properties:
  • N is between B and Q;
  • N is not between A and B
  • N is not between A and Q
  • For each element P between N and Q which is neither between B and N, nor between B and Q, the following relationship holds:

d( P, Q ) < d( B, N ) < d( B, Q ).

This makes it easier to see that, for example, this is problematic for the real numbers; let A = 0, B = 1, and Q = 2. Then if we use the usual definition of "between", there is no N which is between B and Q which is not between A and Q - thus, the real number line is not a curve; is that what you wanted? I assumed that what you meant was that N is between B and Q, but 'not bewteen A and B; but there are further problems.

As for: it seems like we can have points p and q which have no point "between" them

In stating the "betweenness relation", existence of at least three distinct elements is assumed; say A, B, and Q.

As required, therefore exists an element N between B and Q; and so on, further elements between N and Q. Similarly there are also elements between B and A.

Let S = R union {p, q} where p and q are elements. Given distinct a, b, c in S, define Btw(a,b,c) as a if a is between b and c, b if b is between a and c, and c if c is between a and b.

For all distinct x, y, z in R, define Btw(x, y, z) = y iff x<y<z or z<y<x (using the usual ordering on R). For all distinct x, y in R, define Btw(p, x, y) = x if x<y, and y if y<x. For all distinct x, y in R, define Btw(q, x, y) = x if x<y, and y if y<x. For all x in R, define Btw(p, q, x) = q.

Then there is no element in S which is between p and q, but S seems to fit your requirements otherwise. Is S a curve?

As for: More complete definitions referred to manifold

Geometric relations between physical entities may not be representable as an n-manifold.

Can you give a concrete example (from physics specifically) that is not either a manifold or a pseudomanifold? I grant that in toplogy we want to consider all sorts of general definitions of curve, but in physics, we usually have some continuous physical system in mind.

For example, every subset of n + 1 elements might "look like it fits in no less than Rⁿ" (to paraphrase the article on homeomorphism) based on their distance ratios, while ignoring any order between elements. But it may nevertheless be found (based on just these distance ratios, but taking the order between elements into account, as described) that such a set represents a continuous 1-dimensional topological space, i.e. a curve.

If you want to define a curve as a "continuous 1-dimensional topological space", then that's not neccessarily a particularly "physics" based definition. Also, you need to define what "continuous 1-dimensional" means. It seems like what you want is a canonical toplogical space over some infinite set with (1) a total order, (2) some metric on the space that supports the total order, and (3) a homeomorphism between your curve and that space. However, your use of "continuous" and "physics" makes me think that your further want the resulting space to be differentiable; and that is another whole kettle of fish!

I honestly don't think that's what you've defined here; and that's what worries me. Chas zzz brown 23:42 Mar 9, 2003 (UTC)

I'm restoring the article accordingly, attempting to integrate what you pointed out.

Regards, Frank W ~@) R 09:47 Mar 9, 2003 (UTC).

I'm reverting it - please address the following:

• any three of its elements are ordered by a betweenness relation (which also distinguishes curves as open, or closed);

Exactly how does the betweeness relation distinguish between an open curve and a closed one? How are "open" and "closed" defined in this sense?

• for any two distinct elements, their distance d to each other is measured being non-zero;

Are you requiring d to be strictly positive? If so, say so.

• for any two elements A and Q holds that for each element B between A and Q there exists an element N between B and Q (where N is neither between A and B, nor between A and Q) such that for each element P between N and Q (where P is neither between B and N, nor between B and Q) holds that

• d( P, Q ) < d( B, N ) < d( B, Q ).

As noted above, address what type of space requires that there exists an element between B and Q which is not between A and Q. Likewise for P between N and Q and not between B and Q.

Finally, can you give some justification that this approach is indeed related the definition of a curve "as used by physicists"?

Cheers - Chas Chas zzz brown 08:10 Mar 10, 2003 (UTC)

As for: [...] let A = 0, B = 1, and Q = 2. Then if we use the usual definition of "between", there is no N which is between B and Q which is not between A and Q

Allright -- in my attempt to avoid an explicit distinction between open and closed curves, the notion

not between

was used ambiguously. The requirement should be more concretely:

N is between B and Q (which by itself is sufficient for open curves), and A does not belong to the ordered set between B and Q which also contains N (which distinguishes the two segments between any two elements of a closed curve).

(This also addresses the further point your trying to make with the definition of Btw, AFAIU.)

Thanks for pointing this out. I'll restate the definition of curve in physics accordingly, and I'll retain a reference to Manifold there.

As for: Exactly how does the betweeness relation distinguish between an open curve and a closed one?

(For completeness:) A curve is closed if for every element N between two distinct elements B and Q there exists another element A which is also between B and Q but which does not belong to the ordered set between B and Q which contains N.

As for: If you want to define a curve as a "continuous 1-dimensional topological space", then that's not neccessarily a particularly "physics" based definition. Also, you need to define what "continuous 1-dimensional" means.

I didn't initially introduce the notions

1-dimensional version of [...] special topological spaces

and

a curve can be continuously mapped onto.

But -- I have to take credit/blame for using these two notions together anyways; and since apparently

Again, the idea here is [...]

... to use notions from physics in order to express a particular instance of a "continuous 1-dimensional topological space", therefore I'll try to express the idea of a curve by means of notions from topology in the first place.

As for: [... want ...] a canonical topological space over some infinite set with (1) a total order, (2) some metric on the space that supports the total order, and (3) a homeomorphism between your curve and that space

As far as physics is concerned, We may not want the claim or requirement that a homeomorphism exists as part of any definition, but at most arising as a relation between selfstanding definitions.

Therefore, in the instance of physics, metric(s) have to provide pretty much everything that makes a set a curve.

As for: [...] want the resulting space to be differentiable

Certainly not explicitly.

But should any curve not be differentiable wrt itself anyways? Or are you referring to various homeomorphisms to certain ever so "canonical" other sets?

As for: Can you give a concrete example (from physics specifically) that is not either a manifold or a pseudomanifold?

First of all, neither Wikipedia nor myself seem presently to have command of the notion "pseudomanifold[?]"; I've only referred to n-manifold.

If you'd like Us to use the notion of "pseudomanifold" of curve (presumably at least in mathematics), then you should define "pseudomanifold" in the first place.

Further, yes, I can give concrete examples for how to measure and compare distances and (importantly, quite separately) how to establish betweenness. With these inputs, the example remains:

[... a set of which] every subset of n + 1 elements might "look like it fits in no less than Rⁿ" (to paraphrase the article on homeomorphism) based on their distance ratios, while ignoring any order between elements. But it may nevertheless be found (based on just these distance ratios, but taking the order between elements into account, as described) that such a set represents a continuous 1-dimensional topological space [...]

(namely by pointing out a suitable assignment of which subsets are to be called open sets, and which are not, based on comparing the pairwise distances).

Finally, if you insist on particular physical labels being attached to the elements in this example, then why not call them "those who belonged together to one particular frame, and who met (one after the other) some particular baseball, coming through".

As for: Are you requiring d to be strictly positive?

"Positivity" seems to be an attribute of (real) numbers ...

In physics, distance values are exclusively to quantify whether two are meeting (i.e. distance value Zero, which was exlcuded), or how far they are apart.

However, not to overload the inequality symbol <, I'll use it to express inequality between numbers, i.e. for distance ratios.

Best regards, Frank W ~@) R 23:52 Mar 10, 2003 (UTC).

Frank, no offense, but your changes are incorrect. Your attempt at a definition of a topological curve includes the idea of open sets as subsets of C, but does not give any clue what space C is embedded in; thus, any closed subset of any space is a curve by your definition. A Google search shows no hits for the term "toplogical skin". We need to provide known material; see Wikipedia:What Wikipedia is not, #10. Original Research. Regards, Chas zzz brown 02:00 Mar 11, 2003 (UTC)

Right on! A definition of some particular object as a topological space is precisely that the definition is given in terms of Set theory, and the designation of certain sets as open sets, as a primitive notion of topology. Correspondingly, there is no mentioning of "embed*" anywhere in the present Wikipedia articles on Topology or Set (or related to those, as far as I've checked); not even in the listings of "What links there".

As for: any closed subset of any space is a curve by your definition.

No offense, but -- Counterexample:

Topological Space (please feel free to check that it is one):

set A == { a, b, c, d, e, f ..., 0, 1, 2, 3, 4, 5, 6 ... } of infinitely many elements, with the following sets declared open:

• A itself,
• {},
• { b, d, f, h, k, m, p, r ..., 0, 2, 4, 6, 8 ... } == B,
• { c, e, g, j, l, n, q, s ..., 1, 3, 5, 7, 9 ... } == C,
• { d, h, m, r ..., 0, 4, 8, 12 ... } == D,
• { e, j, n, s ..., 1, 5, 9 ... } == E,
• { f, k, p ..., 2, 6, 10 ... } == F,
• { g, l, q ..., 3, 7 ... } == G
• { h, r ..., 0, 8, 16 ... } == H and so on;
• and the various unions, and intersections of all these open sets.

By construction, any subsets with only one element, such as { 0 }, or only a finite number of elements, such as { 0, 2 }, are not open.

Noting that A = Union( { a }, B, C ), and Intersect( B, C ) = {}, therefore set $ == Union( { a }, B ) is a (proper) closed subset (of the set/space A).

Is set $ a curve, according to the suggested definition ??:

Consider $ 's subset $$ == Union( { 0, 4, 8 }, F ).

Since Intersect( D, F ) = {}, and Intersect( H, F ) = {}, etc., and because neither { 0 }, nor { 0, 4 }, nor { 0, 8 }, nor { 0, 4, 8 } are open sets, therefore element 0 does not belong to the Interior of set $$. Similar arguments hold for 4, and 8. No element of the set { 0, 4, 8 } belongs to the interior of set $$; and the interior of set $$ is set F.

However: The set { 0, 4, 8 } is a subset of set D; and since Intersect( D, F ) = {}, therefore set D constitutes an open set which contains (at least) two elements of { 0, 4, 8 }, but no other element of the interior of set $$.

Therefore set $ is no curve. Q. e. ... darn it, of course I had missed the other in no 'other' element; and of course the closed curves raise their ugly special-cased head again, requiring consideration of 'three' elements instead of two.

So -- thanks again (but surely, asking: Which set not a curve, but this definition? might have sufficed, too). I'll restate the definition of a topological curve accordingly; to the best of my knowledge, for which don't claim completeness, of course -- who does ? I can barely keep track of all that's not known (and/or which presumptions become outdated) in physics, let alone topology.

Anyways: which definition of a topological curve would you propose to be known, since AFAIR you raise these notions ?

As for: A Google search shows no hits for the term "toplogical skin".

1. Google suggests alternative spellings, and

2. I used that term as an abbreviation for a more lengthy expression -- can you find a more appropriate one in the Topology glossary?

3. Let'S be more explicit anyways ...

As for: We need to provide known material; see Wikipedia:What Wikipedia is not, #10. Original Research.

There's nothing original about the notion of Curve (even according to the 1911 Encyclopedia Britannica, nor Topology (though the 1911 might disagree), nor Distance (as a notion in Physics) to begin with. The research to establish what's therefore known involves more than hitting on Wippycadia:what Wikipedia is not.

Regards, Frank W ~@) R 11:31 Mar 11, 2003 (UTC).

p.s. Did you consider any particular closed subsets of a more specific space, when you suggested that any closed subset of any space is a curve by your definition (to which I tried to give a "counterexample for the right reasons"); or really just any closed subset of any old (topological) space ? Frank W ~@) R 17:35 Mar 11, 2003 (UTC)

Frank, please don't include what you think seems like a good defintion of a curve. 4 incorrect definitions (including the one I just reverted) should make you pause, and look up a reference which gives the definition you seek. We want to try to collect known information here - why not seek that information out and include it, rather than trying to re-invent the wheel? Cheers - Chas zzz brown 21:28 Mar 11, 2003 (UTC)

As for: incorrect definitions (including the one I just reverted)

As far as you're claiming "incorrectness" based on an assertion to which a counterexample was suggested without addressing this suggestion itself, nor stating explicitly just what does seem (exclusively) "correct" to you, you're excluding yourself from referencing what's known.

And if the wheel couldn't be re-invented equally (within the GFDL), it was not known to begin with.

As for: look up a reference which gives the definition you seek

Continually, of course; approaching already about a dozen times for this this particular topic (/w/wiki.phtml?title=Curve&action=history) alone. However, some present references merely give reversions (exsponges &) in response to the vista one sought to share -- after all, the reference on would ideally consult is still under construction.

Cheerios[?], Frank W ~@) R 09:42 Mar 13, 2003 (UTC)

See what I mean? Cat-pee. *sigh* -- Tarquin 10:48 Mar 13, 2003 (UTC)

My brain hurts just trying to read his response. Reverting again. Frank, please provide a reference external to wikipedia - for example, a definition from a book on topology, or from a physics textbook - that uses any of your given definitions of "curve".

Applying your current definition, let C = R with the usual topology (open sets are unions of open intervals). This is the real number line - we certainly want this to be a curve.

Let S = (0,1) union {3,4,5}; i.e., union of an open interval and a set of three points. S is a proper subset of R, with interior I = (0,1) and boundary B = {3,4,5}. The open subset O = (2,6) contains the three points in B, but does not intersect I, thus O and I are disjoint; therefore, the real number line R is not a curve by your definition; nor is any figure topologically equivalent to R, such as parabolas, lines, etc. - pretty much anything that anyone else would call a curve.

Even if your definition were correct, think about your audience here. The motivation behind the definition of a curve as "homeomorphic to R" (the commonly used one) is that we want things that are "like" the real number line. If one has to wade through a thicket of statements just to see whether the real line actually satisfies the definition, then the definition isn't particularly useful, especially to people who don't already know the material (our likely readers). If you know of another commonly used definition, include a reference. If you don't know what "homeomorphic to R" means, I applaud your attempts to learn; but in that case don't write an article defining what a curve is in mathematics! Chas zzz brown 21:21 Mar 13, 2003 (UTC)

As for: let C = R with the usual topology (open sets are unions of open intervals). [...] Let S = (0,1) union {3,4,5} [...]

Good point; and we know that your example depends on (me) having omitted a requirement for S to be a closed set (wrt C), with an actually nonempty interior I (where the other conditions assure existence of such sets S). Thanks for the reference.

Requiring S to be closed doesn't change the situation. Let S = [0,1] union {3,4,5}. S is a closed set - it is the union of a finite number of closed sets. It has boundary B = {0,1,3,4,5}, and interior I = (0,1). As before, O = (2,6) is an open set containing 3 points of B and is disjoint from I, so the real line is not a curve according to this. Chas zzz brown 21:26 Mar 14, 2003 (UTC)

(Btw., in the above counterexample with $ == Union( { a }, B ), a suitable closed subset is then $$ == { a, b, d, f, k, m, p ..., 2, 4, 6, 10, 12 ... } = Complement( A, Union( C, H ) ) = Union( F, { a, b, d, m, ..., 4, 12 ... } ), where the elements 4, 12, 20 and so on belong together to D but not to F, i.e. not to the interior of $$.)

I don't know if your counterexample meets all your requirements for a curve; I was incorrect to state that every closed set was a curve by that defintion. But I was correct in stating that your definition doesn't meet a major requirement - namely that things we would naturally call a curve, like the real number line, are not classified as curves by your definition. Chas zzz brown 21:26 Mar 14, 2003 (UTC)

As for: [...] if that case don't write an article defining what a curve is in mathematics!

I don't presume to give definitions in mathematics, and my only modification in the corresponding section had been adding n-Manifold as a generalization to Rⁿ.

... and several attempts at a mathematical definition, via distance functions and open sets, of what a curve is defined as. Chas zzz brown 21:26 Mar 14, 2003 (UTC)

I merely wish Us not to omit the point(s) of view on the notion of curve which are not represented by the present section In mathematics, a curve is typically [...]. (And surely any incidental articles on Peano curve[?], Trefoil curve[?], Lemniscate etc. practically write themselves.)

The present statement that a (simple) curve is to be one-dimensional, continuous, and not self-intersecting may be at least for the moment sufficiently inclusive, especially in reference to physics and the notion of metric, and pending further definition and reference of these topological notions in Wikipedia.

Basically, I agree (although I might change "continuous" to to "connected").

As for: The motivation behind the definition of a curve as "homeomorphic to R" (the commonly used one) is that we want things that are "like" the real number line.

As emphasized above already: Is there any motivation to limit the notion of curve in mathematics to certain subsets of n-Manifolds, or even Rⁿ??

Well, that's somewhat awkwardly put; I'd say there is no mathematical motivation to extend the notion of "curve" beyond certain limits. For example, there are sets one could call "1-dimensional" subsets of a vector space; but if the vector space is over a ring which doesn't have a total order (for example, over C), then it isn't very much like our intuitive notion of what a curve is. It would be more natural to call such a subset a surface, because it is more like our intuitive notions of what a surface is.

And if I can make my point concerning "homeomorphism" any clearer still:

If you do command reference of R 's usual topology, then why should the article not state this directly and explicitly, with or without reference to open intervals in turn; rather than having to put "like" in shudder-quotes ??

Because the homeomorphism link provides a better place to include the more detailed analysis; introducing the definition again here just confuses people who have no idea what open and closed sets are. And after all, "homeomorphic" is just a fancy latin way of saying "has the same shape as" - equally vague as "like", without further definition. I put "like" in quotes to indicate that in this case, "like" has a very specific meaning - see homeomorphism.

As for: If one has to wade through a thicket of statements just to see whether the real line actually satisfies the definition, then the definition isn't particularly useful, especially to people who don't already know the material (our likely readers).

I have no prejudice towards Our readers, however likely or unlikely they may appear...

We'll just have to take their existence as axiomatic.

... (they've taught me already to take greater care in distinguishing closed sets from sets without any further qualification -- thanks again). Nor do I judge anyone's ability to perceive e.g. the Sierpinski carpet "like" the real number line, or not.

However, We may want not (only) to appeal to arbitrary perception, but (also) represent what can be equally known and understood.

I don't get your point here. An article about the term "curve" in mathematics should be about precisely that - what mathematicians, as a group, mean by the term "curve". Sure, mathematicians could call a "cup of coffee" a "curve" - but they don't. So why include "cup of coffee" as a possible mathematical definition of "curve"? Chas zzz brown 21:26 Mar 14, 2003 (UTC)

As for: commonly used definition

I don't presume to enforce what's common or not; considering all Our readers, any presently known definitions may all to soon become mere historical curiosities. Instead, and still, I'd like Us to present what's known to Us, and such that it can remain known unambiguosly.

We are editors, not researchers. It's not what's known to us, it's what known by us to be known in general.

As for: please provide a reference external to wikipedia

Not for definitions, as a matter of principle: namely in order to support the implementation of Wikipedia as an Encyclopedia, i.e. a self-containted circle (or rather: [[Karl Menger|sponge &).

That seems a rather silly stance to me. Supporting definitions using other incorrect or incomplete definitions in the name of self-referentiality serves no one. "Garbage in, garbage out". Chas zzz brown 21:26 Mar 14, 2003 (UTC)

Insofar, I consider the present statement of what a (simple) curve is (outside the more or less severely limiting "typical definition in mathematics" which follows) already an achievement.

Yes - it helps clarify what is meant by "like" the real line. Chas zzz brown 21:26 Mar 14, 2003 (UTC)

Regards, Frank W ~@) R 10:25 Mar 14, 2003 (UTC).

As for: Requiring S to be closed doesn't change the situation. [...] boundary B = {0,1,3,4,5}, and interior I = (0,1).

Correct -- and quite obviously so. Yet ... evidently I was under the impression that it does, for the condition I was considering, and which I have so far expressed only very poorly; namely that (roughly, trying to be more concise, and thus less error-prone):

If two closed subsets S and T of a simple curve C have the same border (of three or more elements), then they must share some of their interior.

Evidently this condition applies only to such closed subsets with (individually) nonempty interior.

Just as evidently (and as indicated earlier) this condition is "wrong" as well; for instance: a loop C can be represented as four consecutive closed "sectors [0, 90], [90, 180], [180, 270], [270, 0]" such that the closed Union( [0, 90], [180, 270] ) and the closed Union( [90, 180], [270, 0] ) have the same four boundary points, but nevertheless disjoint interiors.

Accounting for all this, I'd consider the following condition presently a pretty precise representation of which topological spaces are known as "simple curves":

For any three elements of a simple curve C there are at least two distinct closed subsets of C (both with nonempty interior) whose boundaries have exactly these three elements in common; and then these two subsets must have some interior in common, too.

(This alone might even imply "connectedness"; which otherwise needs to be required separately.)

IOW: I'd surely appreciate an encyclopedic entry to list some or all topological spaces of which certain instances (i.e. at least in certain fields of knowledge) would be called "simple curves" despite failing this condition.

As for: attempts at a mathematical definition, via distance functions

Not at all: I was careful to distinguish the definition of "curve" in physics; based on measured distance values.

As for: attempts at a mathematical definition, via [...] open sets

Hardly: None of my attempts to express a topological "curve" definition used the symbol "n", which appears indispensible for the "typical notion of curve in mathematics" as presently stated.

As for: I'd say there is no mathematical motivation to extend the notion of "curve" beyond certain limits.

Of course I don't wish to disrespect this POV. However, the n in n-Manifolds, or even Rⁿ is obviously in general too limiting for the description of one-dimensional, connected, and not self-intersecting relations that may be derived from (infinitely many) distance values, or ratios.

As for: An article about the term "curve" in mathematics should be about precisely that - what mathematicians, as a group, mean by the term "curve"

If the mathematically inclined wish to present precisely and exclusively what "curve" is to be in mathematics, then they may need to create a new and separate article, e.g. Curve (mathematics)[?]. As it presently stands, the introduction of the article to be talked about exceeds the limit presented by the n which still appears there to define "(typical) curve in mathematics".

As for: might change "continuous" to "connected"

Good point: in this article (or thereby even in general), the notion of a curve "being connected" should take precedence over the capability to be "continuously mapped to R". (Since you had used the word "continuous" first, I had tried to be "less upsetting" by retaining it.)

With the same motivation I suggest that the explicit and separate definitions of curve or R 's usual topology as topological spaces ought to have precedence over mentionings of a homeomorphism between them.

As for: We are editors, not researchers. It's not what's known to us, it's what known by us to be known in general.

We are encyclopedists first; not copiers, nor poets. The contents of articles must be thereby equally known to anyone with the capacity to edit, wikify, and follow links. Otherwise the contents can surely be altered beyond recognition, or disappear alltogether.

As for: Supporting definitions using other incorrect or incomplete definitions in the name of self-referentiality serves no one.

Sure (except of course as examples for "incorrectness" and/or "incompleteness".)

However, a free and ongoing encyclopedic framework cannot deny to point out whatever can be understood to follow from the persistent knowledge already presented there; even if it was theretofore barely recognizable.

As for: "Garbage in, garbage out".

"Give the dudes a beer and they drink five minutes. Show them the fridge and they drink all night."

Regards, Frank W ~@) R 22:09 Mar 15, 2003 (UTC)