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My calculus professor told me today that a set consisting of two simply connected regions that are disconnected from each other (say, two disks, each of radius $\frac 12$, centered at $(-1, 0)$ and $(1, 0)$ respectively) is still considered simply connected because every possible closed curve in the set is still contractable to a point, even though it's not possible for a path to go from one region to the other.

However, Wikipedia (and this other question here) say that a simply connected region is always connected by definition, which is what I thought initially as well.

Is there some quirk in the definition that I'm missing that caused me to misunderstand the term, or is my professor using the term incorrectly?

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Joe Z.
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    He's wrong. Simply connected is a stronger version of path-connected, which is a stronger version of connected... – ah11950 Mar 27 '14 at 22:20
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    How can you have a closed curve in a disconnected space? The curve connects it... by a path. – user2345215 Mar 27 '14 at 22:21
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    The closed curve would be on one connected component of the space, is how the professor appeared to explain it. – Joe Z. Mar 27 '14 at 22:22
  • In the example I gave above, the circle of radius $\frac 14$ centered around $(1, 0)$ would be considered a closed curve in the set. – Joe Z. Mar 27 '14 at 22:23
  • That sounds like a union of simply connected spaces. Maybe he's just using a different definition from the standard one. – user2345215 Mar 27 '14 at 22:25
  • In this picture, my calculus professor would consider set $B$ to be disconnected but also simply connected. – Joe Z. Mar 27 '14 at 22:27
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    I have encountered that use before, that a space (at least a nice enough one, like an open subset of $\mathbb{R}^n$) was called simply connected if every component was simply connected. I can't remember where I encountered it, however. – Daniel Fischer Mar 27 '14 at 22:37
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    This is semantics. Either use is close enough to the other to imply very similar things. If your professor defines simply connected in a way for his convenience, that is the definition of simply connected up to the end of his class. – PVAL-inactive Mar 27 '14 at 22:48
  • So I had to take this course again the following term, and the professor teaching it that time gave us the same question, but this time the set that was not connected was indeed not simply connected either. – Joe Z. Feb 24 '15 at 20:11
  • Maybe it's worth pointing out that in the theory of algebraic groups, one often encounters statements that start "Let $G$ be a connected, simply connected algebraic group ...". It is not redundant there, as over general base fields or rings (the whole point of algebraic groups is to be able to make statements beyond $\mathbb R$ and $\mathbb C$), both "simply connected" and "connected" are defined differently. – Torsten Schoeneberg Aug 12 '21 at 15:58

3 Answers3

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The concept of "simply connected" really only makes sense for a path-connected space. That said, there are a few definitions that get thrown around, all of which are equivalent for path-connected spaces:

Definition 1 A simply connected space is a space in which every closed curve is homotopic to a point.

Definition 2 A simply connected space is a path-connected space in which every closed curve is homotopic to a point.

Definition 3 A simply connected space is a pointed space in which every path that starts and ends at the distinguished point is homotopic to a constant map via a homotopy through paths that all start and end at the distinguished point.

The definition used most often in algebraic topology (where this concept gets the most use) is Definition $3$, so I'd say this definition is the "correct" one. However, because the concept is so simple, in calculus or real analysis it's often cumbersome to have to introduce concepts like pointed space, so the definition is shortened to definition $1$ or $2$.

According to definition $1$, a disconnected space with simply connected components is simply connected. According to definition $2$, of course, it's not. According to definition $3$, the space has to have a distiguished point, and simple connectedness is determined solely by the path component containing that point. So under the actual definition, you could have one component be a violently non-simply-connected space like a Hawaiian earring, but as long as the component with the distinguished point is simply connected, the whole space is.

Allow me to reiterate that when we talk about simply connected spaces, in general we only want to worry about path-connected spaces, and the difference in all these definitions really boils down to how we want to handle the exceptions.

MartianInvader
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  • I don't agree. The common definition in algebraic topology definitely includes the hypothesis of path-connectedness: cf [Hatcher, before Prop. 1.6], [Massey, 1991, Ch. II, before Corollary 4.3], [May, 1999, Ch. 2 §8]... – Zhen Lin Mar 27 '14 at 22:57
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This is only an issue of terminology, but it's fair to say your professor is making a mistake about the terminology. The property that the disconnected set $B$ has doesn't have a very short name: it's that its fundamental group is trivial at every base point. That just means that every closed curve in $B$ can be contracted to a point, as your professor says. Call this property $P$. The reason that $P$ takes such a long time to name is that it's not important relative to being simply connected. A set is simply connected if it not only has $P$ but also admits a continuous path connecting any two points. Since $B$ is not path connected, we don't say it's simply connected even though it has property $P$.

For those who know some topology: the reason simple connectedness is more important than property $P$ is that it's saying $\pi_i$ is trivial for all $i\leq 1$. This generalizes immediately to the notion of $n$-connectedness, and $n$-connected spaces are fundamental in algebraic topology, for example in obstruction theory, the Hurewicz map between homotopy and homology, and the construction of Postnikov towers. Part of the significance is that such spaces are weakly equivalent to CW-complexes with one $0$-cell and no $1,2,...,n-1$-cells.

Kevin Carlson
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Forgive me for my previous post.

A path connected space is connected (http://topospaces.subwiki.org/wiki/Path-connected_implies_connected), and a simply connected space is path connected (by definition). Thus, since a disconnected space is not connected, a disconnected space cannot be simply connected. Hope this helps.