Talk:Topological manifold

message: The article asserts that the disjoint union of n manifolds of the same dimension is a manifold, which is false in general (consider the line (x,y) with x = 0 and the line (x,y) with x > 0, y = 0, their union is certainly not a manifold).  I've removed this line from the article.  I'm not sure of all the necessary conditions though.  I do know that being compact, disjoint and same of the dimension are sufficient conditions though.
It has to be the topological coproduct, and not the set coproduct. This is an example of a coproduct in set which is not a coproduct in top. — Preceding unsigned comment added by 216.186.225.19 (talk) 23:59, 1 May 2022 (UTC)
thread_title: Disjoint Union of manifolds

message: Can anybody explain to me briefly what should be in this entry in distinction to the manifold entry. Then I may be able to clean up a little bit. Hottiger 11:03, 14 April 2006 (UTC)
replies: text: Besides: The example fo the hausdorff condition should clarify the topology imho. Hottiger 11:03, 14 April 2006 (UTC)

text: That is, what seemed to be the initil intention. The result is that the initial explaining image in the manifold entry explains what is a 2-dimensional Riemannian manifold (there are ankles shown). In my opinion, the entry isn't good neither for non-mathematicians nor for mathematicians. By the way, one should not even try to explain what a scheme is in a manifold entry. Hottiger 17:43, 4 October 2006 (UTC)
replies: text:  I like the idea that the manifolds entry is a general, non-technical article. It should cover, in broad terms, many classes of manifold: topological, smooth, infinite-dimensional, even non-Hausdorff. The present article, on the other hand, is one of its most basic important technical incarnations: finite dimensional Hausdorff second countable continuous manifolds. Although every wikipedia article should begin in as elementary a way as possible, this article should lead quite quickly to careful definitions and examples. It should be a prototype for articles on other classes of manifolds, and a springboard for discussing such issues as the classification problem for manifolds, including the piecewise linear and smooth categories, and the algebraic and differential topology of manifolds. At present it doesn't do that. Also, the examples section should focus on examples with topological significance (such as projective space, flag manifolds and Stieffel manifolds), with a more comprehensive list being moved to an article on smooth manifolds. I've commented about this on several other pages. Let me know what you think, as these ideas are all interlinked. Geometry guy 21:50, 8 February 2007 (UTC)
thread_title: Separation from manifold entry

message: See Talk:Manifold#Merging from topological manifold Oleg Alexandrov (talk) 15:26, 2 May 2006 (UTC)
thread_title:  Merging to [[manifold]] 

message: Can someone clarify what exactly is meant by the composition link? There are three separate definitions for mathematics-related composition on the disambig page, and so the link as it is not very helpful. All we need to know is what definition the author was using for the word. SingCal 06:58, 27 July 2006 (UTC)
replies: text: Function composition. I disamb'ed the link; thanks for your note. -- Jitse Niesen (talk) 09:34, 27 July 2006 (UTC)
thread_title:  composition link 

message: I had just seen an article: 5-manifold. I wondered if it should be listed under, "See Also" If so, it looks like it needs a bit of work.Brian Pearson 07:15, 21 November 2006 (UTC)
thread_title:  5-manifold? 

message: :A topological space X is locally Euclidean if every point in X has a neighborhood which is homeomorphic to an open subset of Euclidean space Rn. The integer n, called the dimension of X, must be the same for all points of X.
The second phrase is problematic in at least these ways:
:1. Quantifiers which appear after the objects they quantify are ambiguous and bad style.
:2. The way this phrase is written makes it seem that it is obvious that the dimension of X is well-defined. But this is a highly non-trivial theorem and I don't know if it is true for locally Euclidean spaces as opposed to topological manifolds.
Requiring every point to have a neighborhood homeomorphic to Rn itself is mathematically clearer and probably what the author had in mind.
--80.238.205.119 23:10, 22 July 2007 (UTC)
thread_title:  Cleanup definition 

message: Second countability (along with the Hausdorff and locally Euclidean condition), allow manifolds to be imbedded in finite-dimensional Euclidean space (this is useful in applications). In many branches of mathematics, spaces are assumed to be paracompact (these branches do deal with manifolds). Therefore, spaces like the Long line should not be manifolds and second countability should be a condition (since this implies paracompactness, normality and other such useful topological properties). Even if the main definition in this article does not include second countability as an assumption for manifolds, there should be a detailed discussion of manifolds for which second countability is an assumption.
Topology Expert (talk) 13:29, 18 September 2008 (UTC)
replies: text: A clarification to the article would be to simply define a topological space to be an n-manifold if it is Hausdorff, second countable and locally homeomorphic to R^n. This would probably meet the needs of most readers with a sidenote that the definition may vary slightly this being one of the most commonly used considering eg. applications.
 80.221.7.131 (talk) 18:59, 10 January 2009 (UTC)
thread_title:  Why not second countability? 

message: In the section "classification of manifolds", it says
The 3-dimensional case may be solved. Thurston's geometrization conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman sketched a proof of this conjecture in 2003 which (as of 2006) appears to be essentially correct.
Is this a joke? I thought wikipedia was supposed to be on the current affairs and I come up here to find an more detailed explanation whenever my book doesn't have enough detail. But 2006??? This is only a minor problem because everyone knows Perelman's proof is correct as of 2009, but I've seen lot's of "as of year" stretching all the way back to 2003. Breath of the Dying (talk) 22:08, 4 October 2009 (UTC)
thread_title:  Old information 

message: http://www.maths.ed.ac.uk/~aar/haupt/ks76.pdf (in the references) is a dead link (404).  —Preceding unsigned comment added by 77.187.29.157 (talk) 20:31, 12 February 2010 (UTC)
thread_title:  Dead link 

message: I think it needs to at least say "homeomorphic to a connected open subset of Euclidean space Rn."  Or perhaps to an open ball in Rn.  Or perhaps it should simply use the usual definition and say "homeomorphic to Rn (or even better to En)" as in Hocking and Young (p278).  If I've missed a point here, please correct me, otherwise I'll change it in a few days to read "homeomorphic to Euclidean space En, or equivalently, to a connected open set or open ball in En," or something like that. Eleuther (talk) 10:58, 13 March 2012 (UTC)
replies: text: I think connectedness is not actually required: if the homeomorphism is to an arbitrary open set, we can always take an open ball in this set, and a ball is homeomorphic to Rn. This should confirm what I'm saying. --Nicelbole (talk) 09:21, 21 January 2015 (UTC)
replies: text: However, the now-used definition "homeomorphic to real n-space Rn" clashes with the first sentence of "Coordinate charts", which says "By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of \mathbb R^n ." Maybe it should be noted somewhere that both definitions are equivalent. IttalracS (talk) 15:11, 16 November 2020 (UTC)
thread_title:  Formal Definition seems incorrect