To be simply-connected means to be path-connected and able to continuously shrink a closed curve while remaining in the domain. According to wikipedia, a circle is not simply connected, but a disk is. Why is that? EDIT: I didn't realize that a circle is just the perimeter, and a disk is a circle that is filled in. Thanks to all who answered!
$\begingroup$ I think in general "a disk" means the open disk, a circle that is filled in is the closed disk $\endgroup$
Commented May 25, 2016 at 2:54Imagine you have a rubber band and you lay it down on top of a circle drawn on a piece of paper. You're asked to stretch/shrink the rubber band until it's crumpled up at a single point; however, the rubber band must stay on top of the circle at all times, and you can't cut it. Intuitively this does not seem possible. This corresponds to the topological fact that $\pi_1(S^1)=\mathbb$. This means there are loops in $S^1$ (the circle) which cannot be continuously shrunk to a point. Hence the circle is not simply connected.
Now if you're allowed to move the rubber band on top of the circle but also inside the circle (i.e., in a disk), it's easy to crumple it up to one point - just push every point on the rubber band in a straight line toward the point where you want it to end up. No matter how the rubber band is initially arranged within the disk, you can still crumple it to a point. This corresponds to the topological fact that $\pi_1(D^2)=0$. That is, every loop in $D^2$ (the disk) can be continuously shrunk to a point (via a straight-line homotopy as described with the rubber band). Hence the disk is simply connected.
answered May 25, 2016 at 2:34 20.9k 1 1 gold badge 24 24 silver badges 42 42 bronze badges$\begingroup$ One thing I'd like to add; the reason why a circle's fundamental group is the set of all integers (instead of all real numbers in [0,1]/<0,1>) is because a fundamental group is only interested in loops, not any possible path between any two points on the circle. A path between points halfway around the circle isn't a loop. So the fundamental group is only left with the option of counting how many times you want to go around (full rotations), and what direction (+/-), thus you're left with integers. $\endgroup$
Commented Jul 11, 2019 at 20:41$\begingroup$ Also, one thing that confused me a bit about the "crumple up the rubber band" metaphor is that if you did it by hand, you'd be crumpling up the rubber band in 2-D space (at least), not in S1 space. I would guess that crumpling up the rubber band in S1 space would be like trying to compress the rubber band around the circle (without moving it) in 1 dimension (like a compression wave / sound waves); in that case you would never get it in a single point because it's a loop and has no end? Do I have that right? I wonder if there's any spaces where compression is a way to reduce a path to a point $\endgroup$