Weird Topological Spaces // Connected vs Path Connected vs Simply Connected
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- Опубліковано 13 чер 2024
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What exactly does it mean for a space to be connected? In this video we will contrast the notions of connected, path connected and simply connected for a range of topological spaces such as the topologists sine curve, the torus, and the fascinating Alexander's Horned Sphere.
0:00 Topologist's Sine Curve
1:25 Definition of Connected
3:10 Definition of Path Connected
4:06 Topologist's Sine Curve again
7:24 Simple Connected
9:37 Alexander's Horned Sphere
11:55 Brilliant.org/TreforBazett
The animation of Alexander's Horned Sphere was created by Gian Todesco: en.wikipedia.org/wiki/Alexand...
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I have heard that topology proofs are often "hand-wavey", but didn't think I would witness this so soon! Loved it.
Ha! Ok, I'm definitely leaning into that reputation because I shared the big idea as opposed to the formal proof in at least 3 spots in this video. BUT! Topology absolutely can be completely rigorous and the arguments I've made here definitely can and should be formalized, just sort of beyond the target audience of this video:)
as a math undergrad student i believe that if hell exists it would have to be a rigorous topology class with no handwaving
@@kkanden haha I might be teaching this class this year:D
@@kkanden Currently studying that for my masters: it's called algebraic topology and it's fascinating but absolutely insane.
@@polelot5824 from the few instances ive seen of it it's insane, hats off to you!
I have red-green colorblindness and the path in the “path connectedness” section is very difficult for me to see.
If you added a thin black or white border around that path, it would be easier to see without changing the original color palette you chose.
Also, I’m taking diff. geometry next semester so hopefully I can find some good explainers to help study!
I like your shirt and the gradual morphology of a donut into a coffee mug. lol.
Absolutely top tier video. Recently been interested in exploring more topology beyond what is taught in analysis, particularly compactness and connectedness. This definitely got me excited! Thank you, Dr Bazett!
Oh nice, you are going to have lots of fun!
Alexander's Horn Sphere is an inspiration for my cognitive landscapes. Thank you so much for the imagery and your amazing content 😎🤓🌹
Thank you so much!
It's a lovely shape, I was introduced to it as Alexander's Wildly Horned Sphere - I think it deserves the "Wildly".
p.s. Dr Trefor has better graphics of it than were available when I was at university.
Thank you so much for bringing those concepts to me, presented with such enthusiasm and great examples and explanations. These objects are amazing to me! :D
I just discovered your channel and it's so amazing!!😍😍
Thanks for saving my calculus
Glad I could help!
I love definitions! Thanks for the video, Dr. Bazett.
Glad you enjoyed it!
Hello
In the Arfken's book ( Mathematical methods for Physicists, on the 9th page is written:
From the definition of a_ij as cosine of the angle between the positive x_i direction and positive x_j direction we may write:
a_ij = partial(x_i)/partail(x_j)
Please explain the connection between partial derivative and direction cosine. which is used for rotation of coordinate system.
There is no video about it on youtube.
Thank you
Wow this is so great ! can you do something like an introductory course in real analysis ?
One thing that confused me a bit was how there was a jump from "simply connected" meaning that any two paths between a given two points being possible to transform into each other, to the term having something to do with transforming two loops into each other or into a point.
After thinking about it, it becomes clear that the original definition implies that the latter condition would be true, because if you choose the same point as the start and end point, then a connecting path will either be a point or a cycle, and so any two cycles must be possible to transform into each other or into a point.
This would already be enough to prove the tools used in the video, but what's less obvious to me is whether the cycle criterion is actually equivalent to, and not only implied by, the path definition. I strongly believe that they're equivalent, but I don't really have the tools to prove it rigorously.
It is true the equivalents of the two notions isn't immediate. Well, one direction is (the path definition implies the loop definition as loops are special cases of paths).
Thoughts on taking differential equations, linear algebra, and calc 3 in the same semester? I will be a first year college student who loves math and is currently learning vector calc via UA-cam and ur playlists.
That's definitely a lot, but it is doable. They are all fairly separate from each other, but if you were to do only two I'd do lin alg and calc 3, then save odes for the next semester. Just make sure you have time to be cranking out problems for each class every week.
A Topological space is simply connected iff it is path connected and the first homotopy group (obviously for homology groups too) is non-trivial .
There is an isomorphism between first homotopy group and first homology group , this is called Hurewicz theorem .
I thought the union of the vertical line and the sin(1/x) curve must be path disconnected simply because the limiting y-value of the sin(1/x) curve does not exist. So, no matter which y-value at x = 0 you choose, the function f must be discontinuous because the limits for 0- and 0+ are different.
please make a series on numerical methods like newton rapshon.pls
2:40 With this definition of (dis)connectedness, it's not obvious why for example the union of half open intervals [0,1)u(1,2] would be disconnected, although this set should clearly be disconnected as 1 is still not in the set and the outside boundary points 0 and 2 should not change the connectedness property. I guess using (some version of) the theorem of adding limit points you mention at 5:18, this set is indeed disconnected, but that theorem is not obvious to me as it seems to directly contradict the definition which refers to open sets. I think some explanation where this theorem comes from would have been helpful.
The goal here is to establish the intuition of the definition alone. So you are right, it is non-trivial to show that intervals like (0,2) are connected and this is typically the first major proof done after establishing the definition.
The definition that I remember learning is that a set X is disconnected if you can separate it into two disjoint sets A and B such that (Cl(A))intersect(B) and (A)intersect(Cl(B)) are both empty, where for any S, Cl(S) (usually notated with an overline) is the set S with all its limit points included, and connected otherwise.
For example, [0,1)u(1,2] is disconnected because you can set A=[0,1) and B=(1,2], and both sets I described above turn out to be empty.
The definition given for connectedness in the video is entirely correct, but one needs to be careful in that when we are saying that a subset S of an ambient topological space X is connected, we mean that S is a connected topological space with respect to the subspace topology (where the open sets of S are exactly the sets occurring as the intersection of an open subset of X with S). This is equivalent to the definition that @lucashoffses9019 gave, because saying that B is disjoint from Cl(A) in the ambient space is equivalent to saying that B is open in the subspace topology, and similarly for A. For the example you discussed, note that [0, 1) and (1, 2] are both open subsets of [0, 1) U (1, 2] with respect to the subspace topology.
Amazing and beautiful 😍
10:04 It's highly nonobvious to me why Alexander's horned sphere is simply connected. Intuitively, it seems to me like to contract a closed curve around the 'main bulb' of the sphere to a point, you'd need to work the curve around an infinite set of horns and you'd run into a similar issue as when you try to extract a belt from passing through the middle of the sphere's exterior.
I'm sure there's a rigorous topological way of proving that it's simply connected, but is there an intuitive way to think about it?
A (not rigorous) intuitive way is that at each stage of the animation, it is just a 2-sphere. As you grow out the horns you are just stretching that sphere a bit, and even in the limit you never get a loop that jumps over one of those "pinch points" so it couldn't be contracted back.
@@DrTrefor To me, it's tough to intuit why this behaviour carries over into the limit of infinitely fine horns when the simply-connectedness of the exterior (much more understandably) does not - after all, the simply-connectedness of the exterior only does change in the infinite limit.
It feels to me that any curve passing around the surface of the horned sphere should be able to be 'lifted' a finite distance into the exterior - an ever-decreasing distance when you get into the horns, but a finite distance nonetheless - and so the smooth deformation of the loop within the horned sphere to the point (showing simple connectedness of the sphere) feels like it should naturally lead to a smooth deformation of a loop in the exterior to a point.
@@DrTrefor Agree, you can sort of add some rigour by defining each branching as being smaller by some ratio (say 1/2) then you can look at how far each point on the sphere is from the original centre So if the first stretch adds a length of k you get that all though the path can have an infinite number of twists it has a length that is the sum of stretches of size k + k/2 + k/4 + ... so the total path length back to the centre is 2k. So any 2 points only get further away within the sphere by at most 4k.
>"It feels to me that any curve passing around the surface of the horned sphere should be able to be 'lifted' a finite distance into the exterior - an ever-decreasing distance when you get into the horns, but a finite distance nonetheless "
I wonder if there's an extra layer of infinity that can't be hurdled by the ever increasing surface distance created by the multiplying horns. The same sort of happenstance as toricelli's trumpet where the finite volume produces an infinite surface area.
The 5:00 part really could have used some zooming animation on the plot.
10:53 When you defined "simply connected", you talked about a path connecting 2 points, not a loop. Why this change?
Oh these two notions are equivalent, but the loops were easier to show visually I thought.
10:41 …yet
When you say the outside of the horned sphere is not simply connected, do you mean the surface or the space around it.
The space around it. The surface is equivalent to a sphere, I.e simply connected. Then there is an outside and an inside, and I’m referring to the outside.
@@DrTrefor Ahh, ok, that explains it a bit more clearly. So the actual sphere surface of the ball is still equivalent to just a plain old sphere; a path *on* the sphere *would* be connected. *But,* the reciprocal surface of the space _around_ the ball which is right up against its surface is no longer two-dimensional due to the infinite branching curves and, thus, *that* surface, the "surface" of the medium _around_ the ball, can no longer be considered simply connected as it could when it was merely two-dimensional.
And, iirc, that would be because the infinite branching horns makes the volume of the ball object fractal, which means it has non-integer dimensions; some value _between_ 3 and 4. And the surface is still the same 2D sphere equivalent, so the extra dimension must be borrowed from the surface of the surrounding space. That would mean the surface of the surrounding space has become a degenerate surface with dimension between 1 and 2, correct?
I am about to watch this vid so a quick question does this mean that we are connected to everything
coplex step FD