What is a Manifold? Lesson 6: Topological Manifolds

Topological manifolds! Finally! I had two false starts with this lesson, but now it is fine, I think.

[omarelmofty]

Great explanation! Thanks very much! I don't think anyone else has given such a good summary of manifolds all together!

[ahmedamr]

I am an engineering phd student studying dynamical systems and your lectures are extremely valuable in helping me get comfortable with manifolds. Thanks a lot!!!!

[tursinbayoteev]

At 2:28, I find that refinement is a union of the "open balls" which in previous lectures was mentioned as a cover. At the same time, it is a union of the subsets.

[samtux]

Love this series. Lots of examples and counterexamples. Ansews to questions in the comments.

[shubhamgothwal]

The best part I loved in the lecture was assigning coordinates to random topological space which is locally homeomorphic to R^n spaces. :)

The only thing that bugged me up was CLOPEN. We considered the points on the boundary of the sphere. But if it's a member of topology T_x, then it should be open. How can we get those boundary points? Could you kindly clear this doubt of mine? I do know that every set is open and closed relative to itself. But how is it both open and closed in the whole space?

[joebloggsgogglebox]

At 48:50 you say that the T-shaped space is not a topological manifold, but then you say that most topological manifolds are like that. What do you mean?

[tursinbayoteev]

At 21:10, what is the difference between the different Cartesian planes if the third dimension is not considered? Or the orientations of the planes matter?

[tursinbayoteev]

At 33:07, the surface of the sphere is locally Euclidean.

[shubhamgothwal]

1) Does a manifold have a unique dimension, say R3 or the dimension of a manifold can vary from one open set to another?

2) Also, you've already answered the point that why we should map the charts to different Euclidean spaces by saying that one chart may map using normal co-ordinates and other using cylindrical co-ordinates. Can you give or suggest an example from a book for visual clarification?

3) Why can't we take derivatives directly on a manifold?

Thank you in advance.

[jaylee]

Hi, I have 2 questions regarding compactness and the example at 48:02 well.. it may be too obvious but please confirm with me.

1) infinite is a notion of indefinitely increasing(or decreasing) values, which means if there exists a certain point that stops increasing then it is finite. so in that sense, if a set S is open, then the set S innately implies that we have to define limit to cover the whole points on the set. But if a set is closed, likewise a sphere set, there exists an exit point so it is compact and we need a finite number of covers.

2) example at 48:02, a function must take 1 input and output 1 value but at the junction, we are outputting 3 values from one input point so on that junction point function cannot be defined.

[xpmrz]

At around 35:00, you said that the biggest open set we can find that covers that point is the entire sphere minus the south pole. Did you mean the biggest open set covering that point which is also homeomorphic to the plane? Since the entire sphere is also an open set, it is certainly the biggest open set that covers that point. Also, can the stereographic projection work for every two points on the sphere? You use the north pole and south pole term, which I imagine to be opposite of each other to work, but then we have to define the notion of "oppositeness" first. I think it is not necessary because we can always cut the sphere with a plane that separates any two chosen points, and then work out the projection. If I understand correctly, what you essentially do in the last part of the video is proving the sphere is indeed a topological manifold.

A minor correction at 35:23, I think it should be U_p=X-{Q}, since Q is a point and X is a set, you can't have a subtraction between the 2.

[taraspokalchuk]

specific examples abouts specific homeo's from specific sets on a sphere to the plane would be really helpful

[charlescrawford]

The stereographic projection seems kinda weird. Seems like the interior of the circle that gets a whole hemisphere area would be more dense than the exterior which is infinite but also gets a hemisphere area. So, there would be some kind of "weight" that differentiates the two areas.

[Sonyash]

48:25 could you please explain in more detail why we can't create a homeomorphism between the structure and R2. you said it's because a line segment is not an open set in R2, but why is that? isn't it open on the sides?

[tursinbayoteev]

At 48:36, why is the line segment in IR^2 not open set? Is it closed because of containing boundaries?

[jnk]

Very interesting…clearly explaination…
I don’t have to attend classes!

[shubhamgothwal]

Can you suggest few books for reading the topics in these lecture series for clearing the doubts?

For example, the stereographic projection of sphere onto the real plane is a bit confusing and it would be great if you could suggest the books or sources for further clarification of the topics.

Thank you.

[ThomasImpelluso]

At 21:09, you say that each "f" that comes out of X goes to a different copy of R2. I get that. But is that true even though the "f"'s apply to subsets that OVERLAP (intersect). I would imagine that the intersections invalidate your comment.

A few mintues later you say that some of the topological spaces share points. And that is a problem that you will address next. Is this addressing my concern?

[jilltalks]

How the sphere is becoming compact?? I.can still find a cover which is overlapping like that for r2

[zassSRK]

is there a property similar to compactness that checks if all covers have a countable subcover?