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I tried to correct this yesterday by changing "subset" to "bounded subset" and "connected" to "path-connected", but it's still wrong. It is possible to construct a bounded simply connected subset of the plane with disconnected complement. --Zundark, 2002 Feb 19
Thanks for finding the bug. I mindlessly copied it from Mathworld. Maybe we need X to be open? AxelBoldt
Never trust Mathworld. I think
may be correct, but we need to check it before adding it to the article. --Zundark, 2002 Feb 19
There was the statement in the article (before I deleted it) that SU(N) is simply connected. But in general it is not! Since for some N it has nontrivial fundamental group. The only case is N=2 (when it is 3-sphere). Am I right? —Preceding unsigned comment added by 193.124.167.14 (talk) 13:37, 30 April 2009 (UTC)[reply]
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.
(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?
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))
The point here is not that we don't know the fundamental group of the 3-sphere (we certainly do). It is the characterisation of the 3-sphere by topological invariants - which goes the other way.
Charles Matthews 22:23, 2 Mar 2004 (UTC)
(Charles beat me to it, but I'll post my reply anyway, as I've typed it out.) What the Clay Mathematics Institute says is correct. It's easy to prove that the 3-sphere is a simply connected compact 3-manifold without boundary, but the Poincaré Conjecture says that these properties characterize the 3-sphere - that is, any space with all these properties is (homeomorphic to) a 3-sphere. --Zundark 22:31, 2 Mar 2004 (UTC)
Ahh, you answered the question I was about to ask while I was typing it. Someone should probably (and I hope I'm right on this) put up a page at characterize/characterization explaining the term, it would have helped my confusion. Every time Charles opens his mouth I feel like a babbling idiot... guess now I know how the non-math majors feel when I open mine. :) -FM (FunnyMan3595 22:40, 2 Mar 2004 (UTC))
"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)[reply]
Is this term really in use? it is usually not simply conneced At least it should say "rearly", but it is safe to remove it.
Tosha 18:40, 8 May 2004 (UTC)[reply]
I knew the term - perhaps it is just old-fashioned.
Charles Matthews 08:20, 9 May 2004 (UTC)[reply]
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.
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)
No link with the previous discussion: is the Mandelbrot space simply connected? It would be a nice example in the appropriate section.
--Bernard Helmstetter 16:16, 25 Dec 2004 (UTC)
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
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)[reply]
The introduction seems to include those, while in the example section, they seem to be excluded, ipso facto.
So, what's the deal? Are both definitions in use? Should we say that the term is usually reserved for CW complexes, or complex domains?
RandomP 16:42, 3 June 2006 (UTC)[reply]
I think the assumption that the space is path-connected should be removed. A space is simply-connected if any closed curve in it is null-homotopic. Thus for instance a union of two disjoint closed discs in the plane is simply-connected, but not connected. Standard literature, especially on Lie groups, makes this distinction, often talking about a "connected, simply-connected" Lie group.
user:AMAN I think even if we remove the " path connected" constrain from the definition, then also the way rest of the definition goes itself will imply that it is path connected. So if simply connected region itself will be path connected. A union of two disjoint discs also cant be simply connected as by definition " if every simple closed curve in D encloses only points that are in D, it is simply connected". — Preceding unsigned comment added by 202.78.172.162 (talk) 12:33, 30 November 2012 (UTC)[reply]
Shouldn't Connected space be mentioned at least once in this article? --Abdull 15:08, 5 June 2006 (UTC)[reply]
It is not obvious to me why R² minus (0,0) is not simply connected, but R³ minus (0,0,0) is. Maybe someone can prove it? --Abdull 11:04, 6 June 2006 (UTC)[reply]
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)[reply]
The cylinder is homeomorphic to S1XR so can't be simply connected (it is a simple result that the fundamental group of XXY (where X and Y are topological spaces), is isomorphic to the direct product of the fundamental group of X with that of Y). The circle is not simply connected (well known); its fundamental group is the integers.
Topology Expert (talk) 06:57, 8 November 2008 (UTC)[reply]
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)[reply]
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)[reply]
The following sentences are contradictory:
> In complex analysis: an open subset 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)[reply]
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)[reply]
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)[reply]