What does it mean to "cut" something with a sphere? Since a sphere is a three-dimensional object, does this entail taking a chunk out of whatever we're cutting?
@lawrenceyan Essentially, the result of "cutting" a surface with a sphere is simply the set of points in the surface that are within the sphere. In the case of a manifold, we can always choose a sphere small enough such that there are no points on the surface for which cutting does not yield a disk, whatever that means (based on the slide, I'd guess it means there are no fragments of the cut surface that are connected only by a point, or that are connected to more than one fragment by the same face).
I wonder what classes should one take if one want's to understand the rigorous mathematical definition of a manifold and how it is equivalent to this sphere-cutting definition.
Is someone able to explain what the "with border" is trying to explain in this image?
@IsabelCDaniels I was also confused but I found this slide from an equivalent graphics course from CMU that's pretty helpful
What exactly does "yield a disk" entail? For example, in the pyramid example, isn't that more a 3-d cut of a sphere?
Maybe the with border label means that surfaces whose boundaries look like the surfaces on the left are allowed to be considered manifold even though cutting with a sphere may not yield a disk?
@et-yao my math-major roommate has explained to me that topology doesn't care about distances, so my understanding of what the whole 'yield a disk' idea entails is that we're looking at situations where a cut gives something that could be considered a disk if you just looked at it in a 2D way, removing the depth-distance that makes it look less like a disk. In the pyramid example you cited, looking at the cut shape from above would look like a disk, without the added depth of the diagram on this slide, so it is manifold. For the non-manifold examples, there is no way in which you could look at the cut shapes and see them as a disk, because they're missing chunks.
I haven't taken a topology class, but from what I know you're right about "distances" not mattering per se. I usually imagine these objects as made out of play-doh, and topologically equivalent if you can mold it without closing or creating new holes. For the pyramid, you kinda just slightly tilt the flaps up and you'll have a disk; with the borders, it's just that some of the disk is cut off. For the non-manifolds, the issue is that you have a 2D object arranged as flaps. You aren't allowed to mold the flaps together (like how you can't fill in holes), and you can't squish it without breaking the 2D part.