Talk:Simply connected space

message: I smell inconsistency: a hollow ball (in 3D) can't be simply connected any more than a circle (in 2D).  Whether or not they have a hole depends on your definition of hole.
:Yes, it depends on your definition of a hole, which is why we give a formal of definition of simply connected which is unambiguous. A hollow ball (a sphere) is simply connected - it's intuitively obvious that a circle on a sphere can be contracted to a point. --Zundark 15:52 Nov 28, 2002 (UTC)
(I'm here to look up these concepts because the descriptions are infinitely better than those in the books of our math library's geometry/topology section, but there seems to be an imperfection here.) 
The definition given is over my head: why involve the circle?
If "simply connected" is meant to express the notion "completely filled up",
isn't it better to define it as a connected subspace of which the complement's interior is also connected, or something like that?
:Your definition does not give an intrinsic property of the space, it gives a property of the embedding of the space in some larger space. The idea of "simply connected" is that there is a path from every point to every other point (this is path-connectedness) and that there is essentially only one such path (thus "simply" connected), in the sense that any two paths from A to B can be deformed into one another. --Zundark 15:52 Nov 28, 2002 (UTC)
----
Zundark, if you're right about the 3-sphere, then the bit on the Poincaré Conjecture at Clay Mathematics Institute needs to be corrected.  It says, "In topology, a sphere with a two-dimensional surface is essentially characterized by the fact it is simply connected. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question has been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds."  I don't know enough about the topic to correct it.  -FM (FunnyMan3595 22:20, 2 Mar 2004 (UTC)
thread_title:  a hollow ball is? 

message: "Multiply Connected" seemed to be in general use up until about 1980: the term is definitely used but definitely "old fashioned" RodVance (talk) 02:36, 25 April 2014 (UTC)
replies: text: Probably old-fashioned. I've seen the term in some old books. BTW it would be nice if the article explained the rationale or etymology behind the term "simply connected". My guess is that the connection between any two points is "simple" since the connecting path is unique up to homotopy, and in "multiply" connected spaces there exists several essentially different paths. Can anyone confirm this?Punainen Nörtti 13:37, 27 March 2007 (UTC)
replies: text: I think that's right. The article already says this at the end of the second paragraph in the Formal definition and equivalent formulations section. --Zundark 15:04, 27 March 2007 (UTC)
thread_title: multiply connected???

message: I don't like the explanation of simple connectedness given in the introduction. It attempts to explain the concept without being formal, but is imprecise, long and confusing.
* Why give an explanation of connected space at all? It doesn't belong here. What's more, it is not rigourous. Imagine the object {M \in R^3, d(O,M) <= 1 and d(O,M) \ne 0.5}, a ball with a sphere missing inside. The inside ball can't move. It is true it can still spin, but this could be prevented easily by changing a little the shape of the object.
* The illustration with a string is too long and imprecise. It could be misleading, because people will think they can only insert the string at the boundary of the object (imagine a 3D object in 3D space). "Feeding some extra string" is also imprecise.
I think we should give a more mathematical definition in the introduction to begin with. To aid non mathematically inclined people, we could give a few diagrams of simply connected and non-simply connected spaces, and show how a path can be deformed in an example.
--Bernard Helmstetter 16:16, 25 Dec 2004 (UTC)
replies: text: I don't understand that section either. It seems to me that it tries to examine whether the complement of the space is simply connected. But again, my English is not perfect. \Mike(z) 17:33, 15 May 2005 (UTC)
thread_title:  The explanation in the introduction is confusing 

message: I just rewrote the intro (had been "insert a string in the object", now is "imagine object as aquarium with diver, magic loop of string"). I hope it's better. I'm not quite sure how to convey the idea that the string will never get hung up on obstacles (for example, think of a 3D donut glued to a flat piece of plywood; the union is simply connected), without making too big a deal of it and making the reader wonder, "So when are obstacles important?"
I removed the "Naming" section, which had been
:* In topology, a connection is often referred to as a handle. This is probably a reference to the way that a (singly-connected) beaker can be topologically turned into a (doubly-connected) teacup by the addition of a handle.
:* In theoretical physics, an additional connection is known as a wormhole.
The article never defines connection, although I assume a connection is something like an equivalence class of paths between two fixed points that can deform into each other. Except I'm fairly certain that a "wormhole" is really another name for a handle-shaped hole, and not for the connection itself. A wormhole would be a connection in the complementary space, right? Anyway, the section seems unnecessary and doesn't make sense as written, so, I commented it out. --Quuxplusone 23:13, 8 May 2006 (UTC)
thread_title:  New intro 

message: Shouldn't Connected space be mentioned at least once in this article? --Abdull 15:08, 5 June 2006 (UTC)
thread_title:  Connected space 

message: I think it is.  If so, it ought to be listed as such in the "Examples" section of this article.
I answered my own question.  The answer is no, as there are definitely two different paths along a cylinder between any two points on that cylinder that can't be deformed into each other. Kevin Lamoreau 02:23, 14 September 2006 (UTC)
thread_title:  Is a [[cylinder (geometry)|cylinder]] simply connected? 

message: While the introduction is not incorrect, the standard definition of simply-connected space is given in terms of contractibility of loops in a pointed space (rather than paths between two points). Thus a space is simply connected if all loops (passing through a fixed point in the space) can be continuously deformed and contracted to a point. More formally, the first homotopy group of the space is trivial. -  Subh83 (talk | contribs) 17:50, 11 December 2013 (UTC)
thread_title:  introduction 

message: The statement "In Lie theory, simple connectedness is prerequisite for working of important Baker–Campbell–Hausdorff formula." is made. This is either misleading or vacuous: the Campbell Baker Hausdorff formula is defined for small enough neighbourhoods of the identity (small enough that the matrix log series coverges in the adjoint representation of the group and a few other requirements), and the neighbourhoods one talks about are "automatically" simply connected by dint of the fundamental local homeomorphism of a Lie group with open neighbourhoods of the origin ℝ^N that the group gets through its charts. I mean, one could use open, non-simply connected neighbourhoods if one really wanted to as long as they were small enough in the sense above, and of course one can always carve out a simply connected open neighbourhood of the origin from the non-simply connected one and restrict discussion to the latter neighbourhood. This is what I mean by "the statement is vacuous": it is not needed to prove the CBH formula. What the comment's poster might be thinking of is that simple connectedness is prerequisite for the lifting of a homomorphism between two Lie groups' Lie algebras to a homomorphism between the Lie groups themselves: the preimage of the homomorphism must be a simply connected Lie group: see chapter 9 of Stillwell, "Naive Lie Theory" for a readable explanation. --RodVance (talk) 02:49, 25 April 2014 (UTC)
replies: text: Agreed. I deleted the line. Sean Eberhard (talk) 12:55, 14 April 2015 (UTC)
thread_title:  Lie Theory 

message: The following sentences are contradictory:
> In complex analysis: an open subset X\subseteq\mathbb{C} is simply connected if and only if both X and its complement in the Riemann sphere are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one, furnishes a nice example of an unbounded, connected, open subset of the plane whose complement is not connected.
-- Svennik (talk) 13:03, 13 September 2019 (UTC)
replies: text:  Actually, I see now that the example given is simply connected in the Riemann Sphere (it does not include the point at \infty). Pictured on a sphere, the "width" of the band approaches zero as the band approaches the point at infinity. As a result, the complement is (just about) connected. --Svennik (talk) 13:10, 13 September 2019 (UTC)
thread_title:  Gives a characterisation of simply connected subsets of <math>\mathbb C</math> and then contradicts it 

message: Most importantly, what does "path" mean?
How come a path can't change what its endpoints are but also the definition is something about shrinking a circle down to a point? nhinchey (talk) 00:33, 28 May 2021 (UTC)
replies: text: I have added a missing link to Path (topology) in the lead section and another missing link to Loop (topology). This doesn't really help much with the general level of this article, I'm afraid. The article isn't really helping by using different definitions in the lead section and in the first paragraph of . However, that section does mention the definition using homotopic paths given in the lead section as an equivalent definition. The article used to be more approachable, I think. See for example . – Tea2min (talk) 06:56, 28 May 2021 (UTC)
thread_title:  as a layperson, this article is unreadable 

message: The image at the forefront of this article probably shouldn't be a counterexample, even if it almost certainly has a place elsewhere in the article. Not only does this lead to a visual conflict and possible confusion while reading, but it also causes problems when this page shows up in search results, as is often the case when someone looks for information on this topic. The problem is exemplified by this google search result as the image shows up by itself, sans clarifying caption, with the erroneous title of "simply connected space," which it explicitly is not. I have given similar feedback to google on this issue of the image being an incorrect search result, but it seems like the problem is mostly on this end. The article should probably be mostly about what the subject is and only parenthetically what it isn't. ZephyrCubic (talk) 18:56, 29 March 2023 (UTC)
thread_title:  image ambiguity, especially in search results