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Showing the proof for the number of sheets was a bit sloppy. It is not clear to me how Elements of fundamental groups of different spaces can translate with each other. Using pi as both the covering map and the fundamental group is also confusing. 25:07

Wow ! So glad I found this

Great Lecture. Thank You !!!

Awesome video! I think I finally understand those subtle details in the definition of probability measure with your help :)

Brilliant !!! Thank You !!

Thank you Sir!

Thank you Sir ❤.

I will enjoy this during the summer. Thanks a lot!! wish you the best and your work will change lives for sure, at least mine ;)

Also homotopy theory would love to learn this course with you

Please do a lecture series on k theory and morse theory

Hey professor which course are you going to cover after this one

Thank you Sir❤

very good, keep it up!

ready to enjoy the lectures during summer. Thanks a lot!

very nice work

Theorem 5: The Euler characteristic of a topological space is a topological invariant. Proof: Let X be a topological space, and let χ(X) be its Euler characteristic, defined as: χ(X) = Σ_i (-1)^i β_i where β_i is the i-th Betti number of X, which counts the number of i-dimensional "holes" in X. To prove that χ(X) is a topological invariant, we need to show that it remains unchanged under continuous deformations of X, such as stretching, twisting, or bending, but not tearing or gluing. Consider a continuous map f : X → Y between two topological spaces X and Y. The induced homomorphisms on the homology groups of X and Y satisfy the following property: f_* : H_i(X) → H_i(Y) is a group homomorphism for each i Moreover, the alternating sum of the ranks of these homomorphisms is equal to the Euler characteristic: Σ_i (-1)^i rank(f_*) = χ(X) - χ(Y) Now, if f is a homeomorphism, i.e., a continuous bijection with a continuous inverse, then the induced homomorphisms f_* are isomorphisms, and their ranks are equal to the Betti numbers of X and Y: rank(f_*) = β_i(X) = β_i(Y) for each i Therefore, we have: Σ_i (-1)^i rank(f_*) = Σ_i (-1)^i β_i(X) - Σ_i (-1)^i β_i(Y) = χ(X) - χ(Y) = 0 This implies that χ(X) = χ(Y) whenever X and Y are homeomorphic, i.e., χ is a topological invariant. This proof highlights the fundamental role of the Euler characteristic in capturing the essential topological properties of a space, and suggests that the concept of zero or nothingness may be intimately connected to the deep structure of space and time.

Nice series I just found out keep it up man

Thanks for your videos

wer

great video :)

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Best topology lectures I 've ever seen .... ,👍

18:49 lecture starts

Thank you for the great lecture

Exceptional

underrated

Important note around 4:00: X = U ∪ V.

15:00 Another way to look at it: If you continuously shrink R, the image of the circle shrinks to a smaller and smaller circle around the constant term. At some point, this shrinking loop of winding number n will pass 0 - in fact, it will pass it n times.

Your videos have helped me a lot with my topology course in college. Your explanations are great!! Thanks.. 😀

There’s a bug . At 51:30, it jumps back to 16:40 and repeat it over

Amazing lecture series. Very clear explanation. Thank you so much. I subscribed this channel. 🙏

I believe at 1:36:15, it should be intersection rather than union ... but the conclusion still holds.

My friend Victor reccomended me your channel 👍🏼 Also, looking forward to reading your book 🥚🦔

Sir this series extremely helpful! You're doing absolutely great work . This is gem of lectures. I am feeling greatful to find your videos. Please continue this series sir. You're work is so inspirational