Talk:Topology/Archive 1

Initial discussion
(Discussion moved from main Topology entry)

Just to be clear, a Moebius strip can be modeled in 3D space (we made them as kids all the time), it's the Klein bottle that cannot.

Its been a while since I've had topology, but I seem to remember there being a theorem indicating that there were only some specific number (6 is it?) of different topological shapes in 3-space. Does anyone remember this?

I've been out of it for far too long...

I don't know what you mean by "different topological shapes in 3-space", but I can't imagine a meaning for which there would only be 6 of them. Even if we restrict ourselves to nice objects such as compact surfaces (which seems to be what you have in mind) there are infinitely many pairs that are non-homeomorphic. There is a representation theorem that characterizes them in very simple terms, however.

As far as 2-manifolds in three space go, you can embed any of them except for non-orientable surface without boundaries. That leaves the sphere with any number of handles and holes, the MoebiusBand with any number of handles and extra holes, and a pair of Moebiusbands glued partway edge-to-edge with any number of handles and extra holes. This sort of thing should go on Manifold.

OK - fair enough, but generally, the purpose of this whole exercise is to use a mathematical model to further understand something that exists, not bring into existance things have the exact characteristics of any given mathematical model. The Klein bottle is interesting because there is nothing physical that it is modelling, but we can analyze it anyway.

''which exercise is that? if this is the enclyopedia page on topology (the initial page, anyway, which should expand ultimately into many), the exercise is not primarily about modelling the physical world, but about doing mathematics''

As far as your "paper is not continuous" argument, does this mean that we should be using Toplogical models that have CantorSet qualities in order to make our models more like the real world?

3-D Euclidean space itself is as much a "theoretical object" as a Mobius strip, and the physical analogues we build of Mobius strips are certainly closer to actual Mobius strips than to tori. Tori are not three dimensional objects -- they are two dimensional objects -- compact connected 2-manifolds. That is, a torus is not the topological equivalent of a donut, for it is hollow.

When one is studying the topology of surfaces, there is no escaping the fact that surfaces are 2-dimensional. A torus's surface is not 2-dimensional and a torus does not have guts as part of its identity, because a torus is a surface. That we cannot build these objects in our physical 3-space because of physical limitations is irrelevant. We cannot exhibit a straight line in our 3-space either, or a flat surface, or even a point for that matter. These are all mathematical objects and this is an encyclopedia page about mathematics.

However, it is useful to point out that we can build a Mobius strip exactly in 3-space. That is, we can exhibit a set of points in 3-space which is homeomorphic to any and all Mobius strips. That is a triviality. However, we cannot build a Klein bottle in 3-space. So there really is an significant and purely mathematical difference here.

''Side note: Mobius is actually spelt with an umlaut, two dots over the o. The preferred anglicization is to change this to an oe, but sometimes people just drop the o.  It's not so much an incorrect spelling as a non-conventional one. (Of course "Mobious" is not an accepted spelling under either scheme.)''

"Mobious" was a typo. Anything that is a typo in something unsigned can pretty safely be edited out. In fact, if this is to be a useful resource for anyone, the whole page should be radically edited. Among possible pages of a similar length, the average Joe on the street would not get very much edification out of it.

The discussion of the non-existence in our physical universe of Mobius strips (as well as lines, planes, and every other object studied in mathematics) should then be placed in a separate page that discusses mathematical ontology in relation to physical ontology, or something along those lines, since it comes up in discussion of more than one sub-discipline of mathematics, although not very much in the practice of mathematics proper.

-- CalvinOstrum

Better yet the discussion of the non-existence in our physical universe of Moebius (Mobius, whatever) strips (and everything else) could be edited out altogether. It adds very little substantive value to the understanding of any branch of mathematics, certainly not topology as I understand it.

Just my 2 cents.

Untitled
This is a question which pertains to the following as quoted from the article:


 * "The term topology is also used for a system of open sets used to define topological spaces, but this article will focus on the branch of mathematics".

Where can one find out about this definition which concerns "a system of open sets used to define topological spaces"? (Ericross)


 * In the topological space article. --Zundark 19:19 May 2, 2003 (UTC)

What's "equivariant jazz"? -- The Anome

I don't know, I took it out for now. --AxelBoldt

How do you define a limit of a sequence of points in a topological space? --Karl Palmen


 * A point x is a limit of a sequence (xn) if and only if the sequence is eventually in every neighbourhood of x. (That is, for every neighbourhood U of x, there are only finitely many n for which xn is not in U.) --Zundark, 2002 Jan 21

I've added several links to pages for terms that are referred to in the article and defined in the Topology Glossary. Many of these pages are nothing but redirects to that glossary, which people were told to be familiar with before reading past a certain point. I intend to write articles for most of these subjects that give more than just the definitions in the glossary but say something about what the concepts are good for and how they are used -- encylopedia stuff rather than dictionary stuff. In the meantime, they provide hotlinks to the glossary for anybody that thought that they were familiar with all of the terms but actually were not, which I think is useful. -- Toby Bartels

I wonder whether we should move all the material about "Topology - the structure" and the examples of topological spaces to topological space and reserve this article for the branch of mathematics? AxelBoldt, Thursday, June 6, 2002

I've often wondered the same thing, but haven't expressed it since I'm not up to doing the change myself yet. But I would approve. &mdash; Toby Bartels, Saturday, June 8, 2002


 * Topological space was probably the first page on Wikipedia that I created. I always thought that most of the content from Topology should be moved there, but I never got around to doing it. --Zundark, Sunday, June 9, 2002

Axel wondered what to do with the "useful theorems". I say, keep 'em here. I haven't looked at Topological_space yet, but what you have here now seems to belong here to me. &mdash; Toby Bartels, Tuesday, June 11, 2002

I just removed this:


 * Applications in science - The topology of the universe We do not yet know the global topology of the universe, and in fact may never be able to know what it is. However, some scientists, called cosmologists, are trying to measure cosmic topology using data from ground-based and space-based telescopes. Results from the MAP telescope may give an answer by 2002 or 2003. Or maybe not...

This doesn't fit in the topology article; I think it should probably go to manifold or pseudo-Riemannian manifold or differential geometry; this article is mainly about point-set topology which is unrelated. AxelBoldt 11:52 Aug 28, 2002 (PDT)

It's not that this article is unrelated, but that it's more general. It's not just the pseudo-Riemannian structure or even the differentiable structure that we don't know, but the topological structure itself is unknown. (Of course, this sense of "topology" is discussed at Topological space, not here.) I do agree that the paragraph doesn't belong in this article, but it would make a reasonable stub for an article Topology of the universe, which could even be listed here with a "See also". I think that I will make that stub now. &mdash; Toby 05:15 Sep 17, 2002 (UTC) -- Does the term genus need to be used in the article someplace. A layman would say hole. A donut is genus one. A sphere is genus zero. Two16

I don't know that it needs to be here, but it should show up in algebraic topology and surface. -- Toby 05:41 Feb 9, 2003 (UTC)

I removed
 * and in fact the term "topologically equivalent" is mainly used when explaining topology to non-topologists.

on the grounds that I see analysts using this term (in a situation where you might have several types of equivalence running around, all called "&laquo;adverb&raquo; equivalent"). -- Toby 04:39 Mar 5, 2003 (UTC)

Yeah, and in hindsight it had a condescending air, too. Thanks. MightCould

True, but I didn't want to say that ^_^. But I forgot to mention that I like the elementary introduction -- that's the hardest part of writing this sort of article. -- Toby 10:19 Mar 8, 2003 (UTC)

I removed:
 * If you do, we have, for most fonts, the class {a,b,d,e,g,o,p,q} of letters with a hole, the class {c,f,h,k,l,m,n,r,s,t,u,v,w,x,y,z} of letters without a hole, and the class {i,j} of letters consisting of two pieces.

on the grounds that this heavily depends on assuming that the lines have non-zero width. I thought of just saying "for most fonts if you assume non-zero width" instead of just "for most fonts", but that could make the exercise seem a lot simpler than it really is.It's a much more involved problem to assume that their widths are zero, and that is when the font choice (especially serifs) comes into question. How can we rephrase this so as not to make the exercise seem too trivial? Do we need to give the answer at all? Or could there be a better exercise to use instead? -- Toby 17:34 Mar 19, 2003 (UTC)

"If you do" referred to "you consider the lines of each letter to have nonzero width", so it seems correct. Your second objection amounts to: "this is a rather simple problem, more complicated versions exist". I do not see why you delete this; you are welcome to discuss a more complicated version as well. - Patrick 17:55 Mar 19, 2003 (UTC)

Then I simply didn't understand the sentence; I thought it was if you do the problem. Let me try to come up with something clear, and you change it if it's bad. -- Toby 05:06 Mar 20, 2003 (UTC)


 * Okay, it is very clear now. - Patrick 09:50 Mar 20, 2003 (UTC)

Great! -- sorry that I took so long to get back to this. -- Toby 02:36 Apr 13, 2003 (UTC)

I think this page is long overdue some revision - too slanted towards algebraic topology, for one thing. Charles Matthews 16:27, 27 Apr 2004 (UTC)

Recent edit saying letter 'g' has two holes - I must be missing something (or a font...). Charles Matthews 10:31, 23 May 2004 (UTC)


 * You are thinking of a g like this: $$g\,$$. But serif fonts (like Times New Roman) typically have a g like this: $$\mathrm{g}\,$$. --Zundark 13:03, 23 May 2004 (UTC)

Well, that should perhaps be made clear; we can't rely on people reading the page in any given font. Charles Matthews 13:07, 23 May 2004 (UTC)


 * I would prefer to get rid of the whole paragraph. I don't see how it's helpful, and differences in fonts make it potentially confusing. --Zundark 13:20, 23 May 2004 (UTC)


 * An image of the alphabet in an applicable font could be added, or another set of examples can be presented.--Patrick 22:34, 23 May 2004 (UTC

Assessment comment
Substituted at 20:54, 4 May 2016 (UTC)

GeoPhysical TopoLOGY, the movement in space
The article is severely lacking in context, topology is defined as the movement of systems (physical) in place or space, Geophysical Topology is the study of the transformation of landforms or structure throughout natural processes, applied commonly in planetary geophysics, volcanology, erosion and environmental systems, and related to any other physical transformation in any space, especially the time series modeling of processes or systems. — Preceding unsigned comment added by 67.52.86.218 (talk) 06:35, 7 April 2013 (UTC)
 * That's topoGRAPHY, isn't it? — Arthur Rubin  (talk) 08:22, 7 April 2013 (UTC)
 * Agreed, this is called geomorphology or perhaps changes in geophysical topography. The notion of topology is used in cartography; see for instance, Topological map and Geospatial topology. Both these links are on the Topology (disambiguation) page, so I think we have them covered.. --Mark viking (talk) 15:42, 7 April 2013 (UTC)

Topology
An ip-editor had written "This is of course nonsense." as the sections final sentence. It was was recently reverted as possible vandalism. The ip-editor was right. The whole section is utter nonsense. YohanN7 (talk) 21:10, 22 April 2014 (UTC)

Now removed. For reference, what was in the article is displayed below.YohanN7 (talk) 21:13, 22 April 2014 (UTC)

Humanities
In fields as different as cultural geography, political theory and psychoanalysis, an increasing use of topology has attempted to consider space using metaphors of proximity, closeness, neighbourhood and transformation rather than simply distance. With its emphasis on holes and sutures and its attention for qualitative traits rather than quantitative properties of geometrical structures such as lengths, degrees, and areas, topology has offered crucial tools for critical investigations. This has allowed for consideration of complex structures in their totality, addressing elements of contiguity and transformation and producing new ways of representing and problematising subjectivity formations. Applications of topological structures like the Möbius strip have been used to illustrate a number of phenomena, including the formation of the unconscious in Lacanian psychoanalysis or the spatial conceptualisation of universalism in Islamic political theology.



YohanN7 (talk) 21:13, 22 April 2014 (UTC)

Topology -Humanities
I can't see why this section has been removed. It makes perfectly sense to me, as it gives readers an idea of the practical application of topology in the Humanities (a reference that was regrettably missing in the page), providing also some useful reference, in the hope that new editors could integrate (rather than delete) the section. It seems to me that the ip-editor's comment that the section and the last sentence are 'nonsense' is completely unjustified. As far as I know, Lacan was well-known to use topology in his psychoanalitic theory of the unconscious. — Preceding unsigned comment added by Sesamo12 (talk • contribs) 08:38, 26 May 2014 (UTC)
 * Lacan may have used what Lacan called "topology". But no one really understands Lacan, so no one knows what he meant, assuming he did in fact mean anything.  It seems very unlikely that it had much to do with the topic of this article. --Trovatore (talk) 08:57, 26 May 2014 (UTC)

I can hardly agree with the content and the form of this comment: 'no one really understands Lacan'. This is quite a superficial remark, and convinces me even more that the the removal of that section was unjustified and unhelpful. To point to the practical applications of topology in the humatities, including cultural geography, urban studies, etc. would help give a broader picture on topology, enriching this page, which by the way is the general page on topology as such, not topology AND math. Anyway, regardeless your thoughts on Lacan, it is against the spirit of wikipedia to censure things, removing sections only because according to you they express a different take or meaning on the topic of the page. Please, leave wikipedia being a place for the commons, and the commoning, where people can debate and confront their veiws, rather than deleting parts and contributions bluntly.--Sesamo12 (talk) 13:24, 26 May 2014 (UTC)
 * I would be happy for the article to discuss "practical applications of topology in the humanities" &mdash; if there were any. It's not impossible that there are, I suppose, though I don't know of any.  A few postmodern mystifiers misusing the term, though, doesn't count. --Trovatore (talk) 18:47, 26 May 2014 (UTC)


 * very arrogant comment indeed by Trovatore --Sesamo12 (talk) 21:32, 26 May 2014 (UTC)


 * The article cited used the Moebius strip merely as an analogy. What would be required for such a section would be a scholarly reliable source explaining how topology is used (or misused a la Sokal) in those subjects, not a random paper that happens to use a concept from topology as a metaphor.  Deltahedron (talk) 21:37, 27 May 2014 (UTC)


 * Whether one believes that Lacan's work is an important bridge between topology and the humanities or that his application of topology is only pseudo-scientific posturing, his work is notable and discussed in reliable sources. The sections Jacques Lacan and Jacques Lacan reference multiple RS regarding his topological writings and their reception; there is an entire book, Lacan: Topologically Speaking, of commentary and criticism of his topological writings. I don't think it would be undue weight to mention (1) his work and (2) that it is a controversial application of topology.
 * In the broader context of humanities, a simple Google search shows a number of applications of topology to humanities, usually in the form of network connectivity relations that don't have a notion is distance, or are independent of it, e.g.,, , and . How these should be treated in this article depends on whether there is a good secondary source, like a review article, on the role of topology in these subjects. I have not yet found a general review of topology in the humanities, but there are such for topics such as corpus linguistics, see and . --Mark viking (talk) 22:20, 27 May 2014 (UTC)
 * How is it an application of topology at all? It's what he calls topology.  Where does he do any actual math?  Topology in the sense of this article is mathematics; any application of topology to the humanities is ipso facto an application of mathematics to the humanities.  There are applications of mathematics to the humanities, but if you can't see that it's mathematics, then it's also not topology. --Trovatore (talk) 22:47, 27 May 2014 (UTC)

Animated example with donut
I thing that intermediate stages in animation of going from mug to doughnut is not a correct represantaition of continuous deformation, therefore are misleading. Compaire the Mugs 1. with closed top and 2. with open top. Are they same?


 * Yes. The cup part of the mug is an indentation, and is topologically the same as (homeomorphic to) a solid.  For example, consider a "fat" hemisphere that opens upward, like a cup.  That can be mapped downward into a "fat" pancake by a continuous bijection, and then mapped back upward to the hemisphere by the inverse bijection.  The part of the cup that is topologically equivalent to the hole in the donut is the handle. Rick Norwood (talk) 12:56, 26 November 2014 (UTC)


 * PS Why is this header red? Does anyone know.  And why are some headers black and others blue.  The formatting looks the same. Rick Norwood (talk) 13:01, 26 November 2014 (UTC)
 * It's red because it's a link to an article which doesn't exist; blue would be a link to an article which does exist, and black is not a link. — Arthur Rubin (talk) 13:33, 26 November 2014 (UTC)

Thanks!Rick Norwood (talk) 15:05, 26 November 2014 (UTC)


 * Just for clarification. Those two mugs (with closed and open top) have different serfice areas. This is visualy apparent. Does it matter? 109.252.68.49 (talk) 19:43, 26 November 2014 (UTC)
 * No, surface area doesn't matter. Two spheres of different sizes have different surface area, but they can certainly be continuously deformed into each other. -GTBacchus(talk) 19:48, 26 November 2014 (UTC)

In fact, the power of topology is that it applies in many situations where measures like length, area, and volume do not apply. It measures something more fundamental. Rick Norwood (talk) 20:53, 26 November 2014 (UTC)

First sentence defines topology as "...topological spaces."
Topology... is the mathematical study of shapes and topological spaces.

From page on 'Topological space': "In topology..."

Is there cyclic defining going on here? — Preceding unsigned comment added by 128.143.102.247 (talk) 16:45, 7 May 2015 (UTC)
 * No, since the definition of a "topological space" is not defined using the term "topology". Paul August &#9742; 16:49, 7 May 2015 (UTC)

Still, we should avoid the appearance of a circular definition.Rick Norwood (talk) 11:50, 8 May 2015 (UTC)
 * In general I don't think we need to worry too much about circularities in WP "definitions". The complaint at natural number and integer, in particular, was kind of a silly one, and in my opinion did not need to be addressed.  The thing to keep in mind is that a WP definition is not like a mathematical definition.  It is not intended to completely specify a class of thing, but just to let the reader know what the article is about.
 * This one is equally silly except for the the repetition topology/topological in the sentence itself. That repetition is not a logical problem, just a stylistic one, but it would be nice to address that.  I have no immediate suggestion as to how to do it. --Trovatore (talk) 15:34, 8 May 2015 (UTC)

General vs point-set topology
OK, this is not a new issue, but I wonder if we can finally clarify it, maybe not once-and-for-all, but at least clearer than it's been.

I happened to notice this passage from the introductory section: However, point-set topology, which is linked, is simply a redirect to general topology.
 * General topology establishes the foundational aspects of topology and investigates properties of topological spaces and concepts inherent to topological spaces. It includes point-set topology, which is the foundational topology used in all other branches (including topics like compactness and connectedness).

I know that in past discussions some editors have tried to draw a distinction between general topology and point-set topology, but I've never gotten a sense that there's really a clear agreed one to draw. Some editors have based it on the "level", but I've never been sure which topic is supposed to be "more advanced".

I guess you could argue that "general" topology includes pointfree topology, whereas point-set topology doesn't (if there are no points, how can there be pointsets?) but pointfree topology being a relatively small field, I have my doubts that most workers who talk about "general topology" or "point-set topology" are really making that particular distinction. I think the terms are used mostly as synonyms, with "general" being perhaps slightly more "modern" terminology.

I suppose my suggested fix for this particular passage is pretty simple; something like this:
 * General topology, also called point-set topology, establishes the foundational aspects of topology and investigates properties of topological spaces and concepts inherent to topological spaces. It defines the basic notions used in all other branches of topology (including concepts like compactness and connectedness).

So I know, kind of a lot of words for a change I could just have made boldly, but if there really is a distinction that's generally understood in the field between general and point-set, then maybe someone could point it out. In the meantime, I think I'll go ahead and put in my suggested change. --Trovatore (talk) 08:44, 6 September 2016 (UTC)


 * Back in the stone age–when I was in graduate school–general topology and point-set topology were considered synonyms. More particularly, "general topology" was used to differentiate the field from "algebraic topology" (other flavors of topology were never brought up in this dichotomy, even though there were some specialists in those areas around). The usual locution was "General (point-set) topology ..." with the parenthetical remark, in a lower voice, thrown in for clarification. So, I support your change. --Bill Cherowitzo (talk) 21:17, 6 September 2016 (UTC)

Imperfect topology not mentioned here
Imperfect topology, under some critical conditions, allows breaking.
 * we can avoid that with QFT, but mention it! — Preceding unsigned comment added by 2A02:587:410E:4500:AD6F:A2A0:C913:453C (talk) 17:23, 5 November 2016 (UTC)
 * Could you explain what you're talking about, please? --Trovatore (talk) 03:19, 6 November 2016 (UTC)
 * The editor might be referring to quantum field theory.

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