Talk:Topology

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This article is within the scope of WikiProject Mathematics.
Mathematics grading:	B Class	Top Importance	Field: Geometry and topology
A vital article
"Elementary introduction" needs breaking up into smaller subsections; theorems and "Outline of the deeper theory" need more prose; history needs expansion. Tompw 16:20, 5 October 2006 (UTC)

Contents 1 Useability 2 new intro 3 New To Advanced Math 4 Mandelbrot? 5 Demarcation of branches of topology 6 On being elementary 7 On being famous, on being cited, on being not 8 Stub type for topology 9 String Theory 10 this sentence is tricky 11 Lacan 12 {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} are only homotopic not homeomorphic 13 Topology in psychoanalysis 14 toroid example is odd 15 horribly discouraging intro 16 question 17 Of squares and circles 18 Informal comment in the article 19 Erroneous characterization of toroid graphic

[edit] Useability

Can anyone simplify parts of this article? I can't, reading the intro & history, say that I gained any understanding. ..And I come with B.S. Physics from Carnegie-Mellon. Pitty the more novice! --Ej0c 10:54, 5 June 2006 (UTC)

/Archive1

I think that the sentence "Two figures that can be deformed one into the other are called homeomorphic" in the introduction is somewhat inaccurate. Really this is some kind of homotopy concept, not homeomorphism. On the other hand, the "deformation" concept is far more useful for someone who has little mathematical background. I'm not sure what the best solution to this is. --Dmharvey 03:55, 27 May 2005 (UTC)

[edit] new intro

The articles on Topology, Topological Glossary, and Topological Space currently do not clearly define the areas they cover. I've rewritten the intro here to try to help the beginner find her way around these various articles. Rick Norwood 19:07, 24 October 2005 (UTC)

The idea that topology has usurped the old place of geometry in mathematics, cuckoo-style, was 'big in the sixties'. It reads oddly now. Those guys - well, let's just say POV and have done. The idea that topology is basically the Erlangen program for the homeomorphism group: that probably dates back to the 1930s, but in a sense the functor concept revealed that it was just part of the picture (the category of topological spaces as a whole is the real object of study). There was the division algebraic topology, differential topology, geometric topology that I think dates from differential topology being defined in the 1950s. A palace revolution made geometric topology the most fashionable rather than the least (Thurston, knot polynomials ... late 1970s/early 1980s). It would be good to get an intro that didn't just take the clichés at face value. Charles Matthews 21:21, 24 October 2005 (UTC)

[edit] New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as lattice groups, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

I've offered you a few suggestions over in the Topological Space article. Rick Norwood 22:48, 26 January 2006 (UTC)

[edit] Mandelbrot?

Why does the history have so much from Mandelbrot, who is not a topologist? Charles Matthews 23:14, 31 January 2006 (UTC)

I was sort of wondering that, too. --Trovatore

I don't think that quote adds anything essential. I'm removing it. --Chan-Ho (Talk) 22:26, 15 February 2006 (UTC)

[edit] Demarcation of branches of topology

So I noticed that there were a few places (topology, general topology, Category:Topology, Category:General topology; did I miss any?) where there was text that implied that general topology was an elementary discipline. I think maybe 20 years ago there was a common opinion that general topology had been more or less mined out, but I think it's fair to say that there's been a considerable resurgence since then, and the new stuff is not elementary at all.

However I'm not entirely sure I picked the right dividing line. What I said was that the other topological disciplines needed something like a manifold structure. Is that true? I know only about their applications to manifolds or near-manifolds, but that may just be a reflection of the fact that I don't really know much about those disciplines. --Trovatore 00:07, 1 February 2006 (UTC)

Yes, there's a distinction to clarify. There is the pedagogic subject one has to learn, in which topological spaces are introduced and continuity defined in general. This area, which is foundational for things like functional analysis and manifolds, was pretty much there by the 1940s, with things like filters and uniform spaces as refinements. Then there is the research area also called 'general topology' which grew out of the concerns of the 1930s (dimension theory, Polish school and descriptive set theory, Texas topology), which continues. But there is not a huge cross-over, is there? An old text like Kelley is still OK for graduate students. Charles Matthews 09:45, 1 February 2006 (UTC)

So I don't think I'd include descriptive set theory as part of general topology; descriptive set theory is more about definability than it is about continuous functions. Not sure what you mean by "cross-over". I think the pedagogic subject more or less blends into the research subject and don't see a need to make a demarcation at the interface between those two. My question still stands: Do the other branches need a manifold structure or something close to it? If so, then I think my text is fine.

As an example, a general topologist might find huge distinctions between two totally disconnected spaces and write entire books on different kinds of totally disconnected spaces, but am I right that he's the only sort of topologist who'd be interested in them? To an algebraic topologist, such spaces live in the category of "pathologies that I don't want to know about and have no tools to address if I did", is that fair? --Trovatore 14:46, 1 February 2006 (UTC)

Well, no, totally disconnected spaces are of considerable interest (for example the p-adic integers). As Stone spaces they have a high tendency to be homeomorphic, though. They come to be interesting approximately because they can carry p-adic analytic manifold structure. There are algebraic topologists who think about profinite things. On the distinction, there is an ambiguity in general topology, since general might connote 'of general use', as much as 'of great generality'. Back in the history, there really was a time before the logicians got hold of descriptive set theory. Charles Matthews 15:13, 1 February 2006 (UTC)

Wow, that sounds really interesting—I never knew there was a category "p-adic analytic"; guess I never really thought about it. Still, that fits with my rough and ready distinction of "carrying something like a manifold structure". Of course I'm trying to get at the "of great generality" meaning; I think that's the appropriate one to use to characterize general topology. Do you think my attempt in the four places I listed looks reasonable? --Trovatore 15:52, 1 February 2006 (UTC)

• JA: Kelley is a wonderful classic, not just for Topology alone but as a guide to a Golden Age style of writing, and several generations of mathematicians learned the only formal set theory they ever had out of its appendix. But the standard definitions of several concepts, like the Product Topology, changed after it was written, so it no longer works except as elective reading for graduate courses. Jon Awbrey 13:04, 1 February 2006 (UTC)

Hmmm ... is the thing about the product topology really true? Charles Matthews 13:07, 1 February 2006 (UTC)

No. In addition, the book is still cited as a standard reference and used in graduate courses (although certainly not as popular as it once was). I doubt things have changed very much after the book's publication as the book is not that old and was very modern when it first came out. My readings of the book suggest to me that much of it is still very standard stuff. --Chan-Ho (Talk) 00:19, 15 February 2006 (UTC)

• JA: I will check when the sun comes up in the library. Please be gentle with my memory, I know I have to, I took Topology courses intermittently over a span of 15 years and it was a time of transition, what time isn't, so I may have mixed up the Box and the Product yet again, or some other misremember. Jon Awbrey 13:14, 1 February 2006 (UTC)

[edit] On being elementary

• JA: I've been noticing this widespread misunderstanding here in WikioPolis as to the general meaning of the word "elementary" in mathematics. It all goes back to Euclid's Stoicheia (singular Stoicheion), a Greek word for many things, from the shadow of a gnomon on a sundial, to letters and numbers, to steps of a stair or stepping stones, to the stars, but in geometry having the specific senses glossed below:

Stoicheion. 3. the elements of proof, e.g. in general reasoning the prôtoi sullogismoi, Arist.Metaph.1014b1; in Geometry, the propositions whose proof is involved in the proof of other propositions, ib.998a26, 1014a36; title of geometrical works by Hippocrates of Chios, Leon, Theudios, and Euclid, Procl. in Euc.pp.66,67,68F.: hence applied to whatever is one, small, and capable of many uses, Arist.Metaph.1014b3; to whatever is most universal, e.g. the unit and the point, ib.6; the line and the circle, Id.Top.158b35; the topos (argument applicable to a variety of subjects), ib.120b13, al., Rh.1358a35, al.; stoicheia ta genê legousi tines Id.Metaph.1014b10; (Liddell & Scott, A Greek-English Lexicon)

• JA: See also: Euclid, Elements, Thomas L. Heath (ed.) — Online at Perseus

• JA; Jon Awbrey 04:52, 1 February 2006 (UTC)

[edit] On being famous, on being cited, on being not

• JA: Can't say either way about Munkres, but J.L. Kelley definitely deserves an article. NB. It is necessary to distinguish "works cited" from "standard lit" somehow or other. Most journals, monographs, and surveys that I know mark the distinction as "references" vs. "bibliography", respectively. I've have noticed folks hereabouts have some beef about Bibs, so I've been trying several other terms, like "further reading" or "literature". Jon Awbrey 04:15, 2 February 2006 (UTC)

On Oleg's comment in the edit summary about de-linking Munkres and Kelley as they are not famous enough or something like that...I have to disagree. They more than meet the standard of notability I often see on mathematicians. Munkres' notoriety as an author is probably already enough, but he has done important work, e.g. on smoothing PL-structures (for which I've often seen him cited). Kelley would certainly meet notability criteria just on the basis of his influence in the mathematical community (see obituaries of him for more details). Also, Kelley's book is also very famous, and he proved the theorem that AC is equivalent to Tychonoff's theorem. Articles on these people are certainly welcome and hopefully someone will do one in the future. --Chan-Ho (Talk) 00:28, 15 February 2006 (UTC)

Yes I agree that both authors (especially Kelly), are notable enough for there own article, in fact creating an article on Kelly, has been on my todo list for some time. ;-) Paul August ☎ 18:15, 5 June 2006 (UTC)

And now he's been done, but not by me! Paul August ☎ 20:08, 17 July 2006 (UTC)

Not totally sure about Munkres (others are probably better placed to comment) but I would strongly support keeping Kelly. I believe the influence his book has had on mathematicians of a certain generation (my generation, probably ...) is in itself enough to entitle him to be considered notable. Madmath789 18:41, 5 June 2006 (UTC)

I was in undergrad 88-92 and grad 93-97 and Munkres was the standard book. Kelly was the standard harder reference. They both are clear cut includes. jbolden1517^Talk 18:46, 5 June 2006 (UTC)

[edit] Stub type for topology

Just a heads-up that there's a new stub type, {{topology-stub}}. Assistance in correctly classifying existing articles would be great, and I also wanted everyone to know it's there for new articles. --Trovatore 19:59, 7 February 2006 (UTC)

[edit] String Theory

Doesn't the Donut and coffee cup example represent something in string theory? 70.111.251.203 15:19, 7 March 2006 (UTC)

It is a standard example in topology. And string theory is part of topology, so a book on string theory might mention it. But it is an elementary example, and string theory is anything but elementary. Rick Norwood 21:52, 7 March 2006 (UTC)

[edit] this sentence is tricky

In this sense, a topology is a family of open sets which contains the empty set and the entire space, and is closed under the operations union and finite intersection.

Maybe finite intersection means Finite intersection property, but operations union definitely is not Operation Union. Can someone try to explain operations union? --Abdull 19:10, 8 June 2006 (UTC)

The word "operations" in this sentence is designed to introduce the names of the two operations that a topology must be closed under. They are the operations of union and finite intersection. A similar wording is used in the statement that the set of integers is closed under the operations addition and multiplication. It is clear, now, or should that sentence be rewritten? Rick Norwood 19:18, 8 June 2006 (UTC)

[edit] Lacan

Should Lacan's "topology" really be in this article? It doesn't seem any more related to topology than the new age "energy" has to do with energy in physics. The physics energy article doesn't have a "Energy in Reiki" section does it? That would belong in the Reiki article not the energy (physics) article, likewise I'm inclined to think that Lacan's "topology" belongs in the article about Lacanian psychoanalysis, not this article. Brentt 12:09, 16 August 2006 (UTC)

[edit] {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} are only homotopic not homeomorphic

The article mentions that "topological equivalence" formally means homeomorphity, and then as an example gives that {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} are in one class of equivalence. May be too pedantic a notice, but {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} are not homeomorphic, but only homotopic. Shouldn't this better be fixed somehow? Tamokk 20:47, 20 August 2006 (UTC)

It has been fixed with the assumption that the lines of the letters are assumed to have non-zero width. --TheVelho 02 January 2007

You can't fix a bad idea by doing the bad idea well. The idea of using letters to represent surfaces with boundary -- as this section does -- is a colossally bad idea if your goal is to communicate to the non-topologist what topology is all about. The inclusion of the phrase "the lines of the letters are assumed to have non-zero width" does keep this colossally bad idea from being technically false, but it is a huge didactical mistake. It would be far, far more useful for communicating the idea of topological equivalence to classify uppercase, sans-serif letters according to topological type.Daqu 13:41, 27 February 2007 (UTC)

[edit] Topology in psychoanalysis

I'm curious as to why this section is even here. So a psychologist has borrowed some words from mathematics. Mathematicians do that all the time. This seems to be the case of Undue weight, is your average reader of this article interested? No. The correct weighting for this is 0 - delete. --Salix alba (talk) 14:33, 12 September 2006 (UTC)

As my last edit summary shows, not fifteen minutes ago I was of the same opinion, but a bit too cautious to just get rid of it myself. It is, after all, bollocks. (And I didn't notice that after Anti-Vandal Patrol's revert of the IP's edits, the IP himself removed the paragraph, and my supposed "revert to IP" actually had the effect of restoring it).

However, on the other hand, a good paragraph which tells the reader precisely this (that Lacan's topology is bollocks) might be in order. Don't worry, I'm not going to start any revert wars over this, but I do think that there is something to be said for a five line paragraph mentioning Lacan's ramblings and pointing out to the reader that they are not to be confused with the actual subject of topology, but are merely the pretentious maunderings of yet another postmodernist. Just to alleviate any confusion that might exist - and after all, a reader might be drawn to this article to find precisely that fact out... Byrgenwulf 14:50, 12 September 2006 (UTC)

It might be better under the Jacques Lacan page, or Borromean rings, or maybe a new page Mathematical models in psychoanalysis, or maybe we could broden the section to topological methaphors, I'm sure there are many other instances where the languare of topology has been borrowed in other fields. Off the top of my head we have the Tokamak torus in physics, which works because of the Hairy ball theorem. Thom's work comes to mind, while not strictly topology he was very keen on aplying mathematical ideas to diverse phenomena. I saw a book yesterday which had a chapter on postmodern mathematics which this stuff probably fits.

It's actually not complete bolocks! What lucan is saying is that the Real, the Imaginary, and the Symbolic are interlinked and if you remove one the other two fall apart. He's used pictorial means to represent this, theres quite a few similar instances in the psychological field where they use some form of pictorial representation to illustrate relationships of abstract concepts. By the methodology of psychology its about as valid as much of the other work and by the subjects nature they need to use a very different methodological framework. We in mathematics are not equipped to judge, apart from say this is not mathematics. --Salix alba (talk) 15:37, 12 September 2006 (UTC)

Saying that the real, imaginary and symbolic are interlinked is fine, as would be using a diagram to illustrate the idea. However, saying that "the phallus is the square root of minus one" (as Lacan did) because of the result of a so-called "calculation" is out and out pretentiousness...in other words, using terminology to sound posh but actually not saying very much of any import. I certainly appreciate the benefits of using analogies to and from maths, it's just that Lacan didn't use his "method" as analogy: he seemed to think he was actually doing mathematics. So do his disciples. But he wasn't: he was using mathematical words. There is one Hell of a difference.

But fine, the paragraph can stay out. I suppose the article is better without mention of him... Byrgenwulf 15:59, 12 September 2006 (UTC)

"What should we make of Lacan's mathematics? Commentators disagree about Lacan's intentions: to what extent was he aiming to 'mathematicize' psychoanalysis? We are unable to give any definitive answer to this question — which, in any case, does not matter much, since Lacan's 'mathematics' are so bizarre that they cannot play a role in any serious psychological analysis.

"To be sure, Lacan does have a vague idea of the mathematics he invokes (but not much more). It is not from him that a student will learn what a natural number or a compact set is, but his statements, when they are understandable, are not always false. On the other hand, he excels (if we may use this word) at the second type of abuse listed in our introduction: his analogies between psychoanalysis and mathematics are the most arbitrary imaginable, and he gives absolutely no empirical or conceptual justification for them (neither here nor elsewhere in his work).

"[...] Lacan's defenders (as well as those of the other authors discussed here) tend to respond to these criticisms by resorting to a strategy that we shall call 'neither/nor': these writings should be evaluated neither as science, nor as philosophy, nor as poetry, nor ... One is then faced with what could be called a 'secular mysticism': mysticism because the discourse aims at producing mental effects that are not purely aesthetic, but without addressing itself to reason; secular because the cultural references (Kant, Hegel, Marx, Freud, mathematics, contemporary literature ...) have nothing to do with traditional religions and are attractive to the modern reader."

Quoted from Alan Sokal and Jean Bricmont, Intellectual Impostures (2nd edition, 2003), p. 34.

There are so many more important things which should be discussed in a general overview of topology than Lacan's misappropriation of its terminology. Anville 18:22, 12 September 2006 (UTC)

Good point Anville. Its just not appropriate subject matter for this article period. Its not about whether its bull or incredibly insightful, its simply that Lacan has not contributed anything of note to the study of topology, which this article is about. Brentt 19:46, 12 September 2006 (UTC)

[edit] toroid example is odd

It seems to me that the "toroid" shown has many holes! Either the picture should be improved, or the reader should be instructed that the picture is meant to be interpreted as something like a wire mesh under the surface of the real toroid. --69.70.139.251 01:19, 20 October 2006 (UTC)

[edit] horribly discouraging intro

that intro is full of jargon that nobody is going to understand

Your criticism is entirely apt. I'm going to try to work on the intro, and see what happens. By the way, sign your posts with four tildes. Rick Norwood 14:19, 24 October 2006 (UTC)

[edit] question

quote: "......For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside......"

I always thought square and circle are 2 dimensional (length and width) if someone could enlighten me please. thank you

The circle and the square are the outline alone, not the area they contain. They're infinitelly thin objects, and that's why it was said they divide the plane in two parts "the inside and outside". At least this is what I understood from that paragraph. ☢ Ҡi∊ff⌇↯ 06:49, 11 November 2006 (UTC)

Laypersons use the word "circle" in two ways, so that a hoop is a circle but so is a plate. Mathematicians need to distinguish between these two objects, and so the hoop is a circle, the plate a disk. In technical terms, a circle is the set of all ordered pairs (x,y) such that the sum of the squares of the variables is constant.Rick Norwood 13:17, 11 November 2006 (UTC)

[edit] Of squares and circles

Those who have this page on their watchlist will have seen various edits relating to what dimension the sqaure and the circle are. I will try and put this matter to bed:

1. Points (and finite collections thereof) are zero-dimensional.
2. Lines of any length (and finite unions thereof) are one-dimensional.
3. In plane geometry, "solid" (i.e. filled-in) regions of the plane of any size are two dimensional.
4. In geometry of three or more dimensions, planes of any size (and finite unions thereof) are two dimensional
5. The square and the circle are not points, and are therefore, not zero dimensional.
6. The square and the circle refered to in the article "both separate the plane into two parts, the part inside and the part outside"
7. The shapes been reffered to are *not* the filled in versions; therefore they are not two dimensional.
8. The shapes are in fact both a short line bent into the relevant shape, which then has its ends joined togther.
9. Therefore a square and a circle are of the same dimensionality as a line
10. Therefore a square and a circle are one dimensional

There are various other arguments I could give (for instance, I can descrie a square or a circle using a function of one variable). I'd also like to point that a filled in circle is a disc (which is a 2D object).

If you don't understand or disput any part of my reasoning, please say indicate which point number is the problem, and I will do my best to explain. Tompw (talk) 23:41, 25 December 2006 (UTC)

This is basically what I argued on a comment in this very page a few months ago. I wonder if we could changed it from:

For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.

To:

For example, the square and the circle have many properties in common: they are both, topologically, one dimensional objects and both separate the plane into two parts, the part inside and the part outside.

The confusion arises because people have in mind the geometric view of a circle. As you probably know, in geometry the number of coordinates is used to give it a dimension, but in topology, we are talking about the dimension of the object itself. So, for example, we have a geometric 2-sphere being the same as the topologic 1-sphere.

I took the liberty to add this bit into the article. Feel free to make it clearer (we could add a little note at the bottom of the article, but I don't quite know the notation for those) — Kieff 00:30, 26 December 2006 (UTC)

An excellent addition :-) (To be precise, the circle and square are one-dimensional objects that are generally seen as a 2D immersion). (More geomtetric, but less rigrourous way to see that a cirlce is 1D: you can describe it with an equation of one variable). Tompw (talk) 22:28, 26 December 2006 (UTC)

---

Just a side note here: Claim (1) at the top of the section is not quite right. There are lots of zero-dimensional spaces other than finite collections of points. For example Cantor space is zero-dimensional --Trovatore 06:13, 26 December 2006 (UTC)

Yes, you're absolutly right. However, I felt that would be irrelevant to the argument I was making. (I've removed the word "only" in point 1... also the Cantor set is a countably infinte collection of points. Of the top of my head, I think any countably infinite collection of points is 0D, but I definately didn't want to get into coutnable vs. uncountable infinites, because a line is an (uncountable) infinte collection of points). Tompw (talk) 22:28, 26 December 2006 (UTC)

Cantor space is actually uncountable -- in fact it has the same cardinality as the reals.

I think a countable subset of R (or more generally Rⁿ is always zero-dimensional. The result certainly does not hold for topological spaces in general, though. Consider a topological space with just three points; call them a, b, and c. Let the open sets be:

• The empty set
• {a}
• {a,b}
• {a,c}
• {a,b,c}

You can easily check that this is a topology, and that the space is not zero-dimensional (for example the open cover given by {a,b} and {a,c} has no refinement into open sets that do not meet -- actually, it has no proper refinement at all). --Trovatore 08:20, 28 December 2006 (UTC)

That's neat :-) What is the dimension of that one? Actually, doesn't it depend on what sort of dimension you use? (Perhaps I should have said I was talking about subsets of Rⁿ) Tompw (talk) 18:42, 2 January 2007 (UTC)

[edit] Informal comment in the article

Is the clause "(Of course, the topologist must be an American. British doughnuts don't have a hole)" really necessary. It seems very informal and irrelevant to me. Davw 11:30, 16 January 2007 (UTC)

You're right, it's irrelevant. I've decided to remove it from the article by commenting it out, just in case someone comes in the future trying to add it in. This sort of nitpicking is pretty common over here. — Kieff 12:12, 16 January 2007 (UTC)

[edit] Erroneous characterization of toroid graphic

The caption beneath the (Mathematica?) graphic of a "toroid" reads as follows: A toroid in three dimensions; A coffee cup with a handle and a donut are both topologically indistinguishable from this toroid.

This is blatantly untrue. The "toroid" as shown is a depiction of a torus -- a 2-manifold, i.e., a surface. But the coffee cup and the doughnut are 3-manifolds with boundary. (Yes, indeed, the coffee cup and doughnut are classical examples of two different shapes that are topologically equivalent: each one is topologically a solid torus. But neither one is a torus per se.) This is not a minor point; this error will be rather confusing to a newcomer. It needs to be corrected.Daqu 13:12, 27 February 2007 (UTC)Daqu 13:20, 27 February 2007 (UTC)