Talk:Homeomorphism

message: "The third requirement, that f−1 be continuous, is essential. Consider for instance the function f: [0, 2\pi) \to S^1 (the unit circle in \mathbb{R}^2) defined by f(\phi) = (cos(\phi), sin(\phi)). This function is bijective and continuous, but not a homeomorphism (S^1 is compact but [0, 2\pi) is not). The function f-1 is not continuous at the point (1, 0), because although f-1 maps (1, 0) to 0, any neighbourhood of this point also includes points that the function maps close to 2\pi, but the points it maps to numbers in between lie outside the neighbourhood.[2]"
I quote the "Notes" of the article. It should probably note that this is only true if the neighborhood is not the entire set. 174.124.170.209 (talk) 03:10, 27 February 2015 (UTC)
thread_title: A technicality

message: It wasn't until now that I even knew what Homeomorphism was, the problem is that upon reading  the article, I still don't know what Homeomorphism is other then some thing about how homeomorphism is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.  Excuse my ignorance, but it seems to me like some one who knew already what Homeomorphism was would understand this Article perfectly, but a Luddite like me has absolutely no idea what this article is even beginning to speak of.   — Preceding unsigned comment added by Robotwalla (talk • contribs) 19:28, 16 August 2011 (UTC)
thread_title: Unintelligible to newcomers

message: The trefoil knot being deformed into the torus in 4-space is not an example of a homeomorphism, such a deformation is referred to as an ambient isotopy.   Any torus embedded in 3-space so as to appear knotted is still a torus, and so is indeed "homeomorphic to a torus" in some tautological sense of the word.
Being knotted is only a property of the embedding, and not a property of the object itself.  See, e.g. Knots and Links by Dale Rolfsen. 137.82.36.10 01:57, 19 September 2007 (UTC)
thread_title: Knot example is not an example

message: Isn't the correct spelling homoeomorphism rather than homeomorphism? --fil
Not in everyday use.
Charles Matthews 15:20, 9 Feb 2004 (UTC)
replies: text: Ok I don't use this term every day, and I also thought homeomorphic is the correct spelling; but my spell checker (ispell-emacs) suggests homoeomorphic instead of homeomorphic. Maybe one should point out the alternative spelling. --fil
On the whole, I'd stick with the way humans spell it.
Charles Matthews 16:28, 9 Feb 2004 (UTC)

text: (Sorry to re-open an old thread.)  Some people also leave off the first "o" in oedema and oesophagus.  Might this be a English vs American thing?  (People outside the US are humans too...)  Alternatively, I notice that "homoeomorphism" is defined by several online dictinaries along the lines of "A near similarity of crystalline forms between unlike chemical compounds".  If there are indeed two distinct words, that would explain the spell checker's response. LachlanA (talk) 02:49, 7 May 2008 (UTC)

text: Yeah, but any metric space is automatically a topological space. Oleg Alexandrov (talk) 15:30, 18 July 2006 (UTC)
replies: text: oh thts fine in tht case.know a little about metric spaces, dont know anything about topology :-) --HellFire 10:56, 19 July 2006 (UTC)
thread_title: Correct spelling?

message: The animation is good example of homotopy not of homeomorphism. I suggest it to be placed there, and find another image for exemplifying homeomorphism.
SurDin 12:50, 7 January 2007 (UTC)
replies: text: I changed the caption a bit to make it clear that homeomorphic != homotopy. Maybe we need a counter example of a homeomorphism which is not a homotopy, perhaps two linked ring and two disjoint rings? --Salix alba (talk) 22:20, 12 March 2007 (UTC)
replies: text:  Every homeomorphism is a homotopy, but the converse is false. Two continuous  maps  f, g : X \to Y  are called homotopic if there exists a continuous map H : X \times [0,1] \to Y such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. Now, two spaces X and Y are called homotopic if there exist continuous maps f : X \to Y and g : Y \to X such that the maps g \circ f : X \to X and  f \circ g : Y \to Y are homotopic to the identity maps. Well, assume that f and g are homeomorphisms, and in fact assume that g = f^{-1}. Then g \circ f = f^{-1} \circ f = \text{id}_X and f \circ g = f \circ f^{-1} = \text{id}_Y. These composites aren't just homotopic to the identity maps, they are the identity maps! Thus two homeomorphic spaces are automatically homotopic. For a reference see pages 5 - 6 of Howard Osborn's Vector Bundles, Foundations and Stiefel-Whitney Classes, Volume 1, Academic Press, 1982. Dharma6662000 (talk) 17:19, 25 August 2008 (UTC)
thread_title: Animation

message: I am a bit concerned by the following sentence:
The primage of certain sets which are actual open in the relative topology of the half-open interval are not open in the more natural topology of the circle (they are half-open intervals).
Is primage some kind of technical term in topology? I can't find a definition for it other than an additional payment made to the master of a freight vessel. It kind of looks like maybe a typo for pre-image, except that my reading of the context suggests they're just images (given that the function goes from the half-open interval to the circle, and not vice versa). In any case, it should be plural to match the are later in the sentence (I mean in are not open).
While I am here, is actual open an actual technical term, or just a German-inspired use of an adjective as an adverb?
Rdbenham (talk) 14:03, 6 April 2009 (UTC)
replies: text: Primage is a typo for preimage. Note that the sentence refers to the inverse of the original mapping, that's why the domain and range are ostensibly reversed. You are also right about the plural and "actual open". — Emil J. 14:24, 6 April 2009 (UTC)
thread_title:  WTF is a "primage"? 

message: The Poincare quote in the introduction is really unnecessary and immature writing. If we include it, we should also include a "Webster's dictionary defines a homeomorphism as..." line.  —Preceding unsigned comment added by 76.226.148.207 (talk) 17:50, 4 October 2009 (UTC)
thread_title:  Poincare Quote 

message: The morph is very illustrative.  But also very distracting!  Can somebody make it so that it can be turned off or hidden?  After watching it ten times I think I get the idea, and I want to concentrate on the text!  It is beginning to make me hate coffee!
CountMacula (talk) 10:51, 15 May 2011 (UTC)
thread_title:  Can the Doughnut/Mug Morph Be Turned Off? 

message: Objects which seem like they are homeomorphic but aren't would improve this article. For example, what are some continuous mappings that are not bicontinuous? Does it have to do with cadlag?
Crasshopper (talk) 18:40, 30 August 2011 (UTC)
thread_title:  Can we add some non-examples? 

message: Can we get an example of a function that is continuous but not bicontinuous?
Crasshopper (talk) 15:59, 30 September 2011 (UTC)
thread_title:  Continuous but not Bicontinuous 

message: 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? — Preceding unsigned comment added by Lufnuf (talk • contribs) 09:55, 26 November 2014‎
:Please, sign your posts in talk page with four tildes (~~~~).
:The two mugs are not the same, but the first one is deformed to the second one by moving up the interior surface of the mug and expanding accordingly the matter of the mug. D.Lazard (talk) 10:24, 26 November 2014 (UTC)
thread_title:  Animated example with donut 

message: I am surprised that the notation X \cong Y for two topological spaces being homeomorphic is not included anywhere in the article. Unless anyone has a good reason otherwise, I suggest it be included. The appropriate place would be to append the sentence "If such a function exists, we say X and Y are homeomorphic" with ", and we denote this by X \cong Y". Joel Brennan (talk) 13:06, 5 April 2019 (UTC)
replies: text: This would need a source asserting that this is a standard notation in topology; that is that this notation is used in mathematical texts without being defined each time. D.Lazard (talk) 15:07, 5 April 2019 (UTC)
thread_title:  Notation for homeomorphism 

message: The graph of a differentiable function is homeomorphic to the domain of the function.
 A differentiable parametrization of a curve is a homeomorphism between the domain of the parametrization and the curve.
In both cases it should suffice to have just continuity rather than differentiability.—회기-로 (talk) 20:03, 25 May 2021 (UTC)
thread_title:  Why should differentiability matter? 

message: A discussion is taking place to address the redirect :Homeo. The discussion will occur at Wikipedia:Redirects for discussion/Log/2021 September 5#Homeo until a consensus is reached, and readers of this page are welcome to contribute to the discussion.  signed, Rosguill talk 16:59, 5 September 2021 (UTC)
thread_title:  "Homeo" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: The animation is an example of an isotopy, and isotopy equivalent is stronger than homeomorphic.
The previous discussion headed "Animation" suggests that at some point someone edited the caption of the animation to make this clear, but the caption as it currently stands is misleading.
You could change the last few words to "a coffee cup and a donut are not only homeomorphic but also isotopy equivalent." Or you could remove the animation and find something that is a better illustration of a homeomorphism. 50.207.106.34 (talk) 18:32, 9 February 2024 (UTC)
replies: text: Also please notice that this same diagram on the "Homotopy" page is labeled as an isotopy. 50.207.106.34 (talk) 21:28, 9 February 2024 (UTC)
thread_title:  Coffee mug example misleading/Animation 

message: I recommend editing the end of the intro section by changing the terminology for "a homeomorphism that is a continuous deformation" from homotopy to isotopy, because while "f is a homeomorphism that is a continuous deformation" does imply "f is a homotopy", the other direction is not always true; "f is a homotopy" does not imply "f is a homeomorphism that is a continuous deformation". Homotopies are often continuous deformations where there is no homeomorphism between f0 and f1. (Consider any homotopy of a nullhomotopic loop to a point.)
This statement on the page reads as a definition, and definitions should always be if AND only if, so using the term "isotopy" would be better. 50.207.106.34 (talk) 18:37, 9 February 2024 (UTC)
replies: text: I have already answered at user talk:D.Lazard. In short, an isotopy requires an ambient space for being defined, and there is no ambient space here. Also, the last sentence of the lead must not be understood as definition of an homotopy, since the definition of an homotopy is given in the linked article. Also, such a definition would be out of scope here. So, this last sentence must be read as a property. D.Lazard (talk) 19:08, 9 February 2024 (UTC)
replies: text: A homotopy also requires an ambient space, although we are not always explicit about identifying it, as the notion of *manifolds* being homotopy equivalent is extended from the definition of *functions* being homotopic by considering this embedding into the ambient space as the function. See Hatcher's Algebraic Topology page 3: "It is true in general that two spaces X and Y are homotopy equivalent if and only if there exists a third space Z containing both X and Y as deformation retracts." 
The fact that this can be read as a property is why I described it as misleading rather than incorrect, but it would still be better described as isotopy.
50.207.106.34 (talk) 21:27, 9 February 2024 (UTC)

text: Again, an isotopy requires an embedding space, and is not defined for stand-alone spaces. D.Lazard (talk) 21:57, 9 February 2024 (UTC)
replies: text: Then perhaps remove the term homotopy entirely, which also requires an ambient space/embedding space. 2001:5A8:4140:B900:C8E9:7B03:55AC:3D1F (talk) 22:45, 9 February 2024 (UTC)
replies: text: Homotopy does not requires an ambient space. However, the lead was confusing in another way, by saying thet a homomorphism is (often) a continuous deformation, while the correct formulation is that it results often from a continuous deformation. I have fixed this. D.Lazard (talk) 12:37, 10 February 2024 (UTC)
thread_title:  End of intro section misleading