What a great explanation, thanks !

Looking forward to the next videos

6:12 I am really confused why X tilda = Union of Intersection of that set. We also have "for all k", so shouldn't it be equal to Intersection of Union of Intersection of the set?

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Thank you very much for your work here. You would deserve much more views and like! (Maybe only a little more volume at your agreeable voice 😉)

Thank you!

I think the definition of N_r is wrong. It should be the union of two sets which you defined.Not the intersection.

thank you!

hmm, not sure about your proof for R^omega in the box topology - I think we do have to show that B is open. Also, I'm only seeing a theorem in Munkres that X is connected if and only if the only subsets of X that are both open and closed in X are the empty set and X itself, but I think in your video you present it without closed being a necessary condition.

i have a question. G is not a sequence, so what does it mean to say μ(G) --> M?

im wondering if you could perhaps make a video on how to identify the triangle inequality in weird situations and apply it? like an "advanced triangle inequality" video. i struggle with recognizing how to use it when it's not directly identifiable, like in p. 122 or p.125 in this chapter of metric spaces in munkres. thank you! (:

Nice ASMR

great video, once again!! thank you!! it takes me such a long time to read munkres and i feel concerned when i don't understand a small detail; you help me identify and comprehend the main storyline.

i cannot thank you enough for your videos. my professor is extremely unclear, and i find munkres difficult to parse (and am frequently wanting more concrete examples, counterexamples, or images - as an example, i was not sure about what kind of objects his notation represented in his definitions in this section). your videos are a lifesaver in my topology course this fall!!

I don't think the proof for part 1 is correct, you used the same x in two different variables.

Please explain exercises 3.26.1 please help tomorrow is seminar😢

Hola! Pregunta: en el minuto 05:00 ¿no es lo mismo decir que el complemento de tal conjunto B es abierto y eso probaría que B es cerrado? Digo, porque la elección de x0 es arbitraria y siempre vas a poder hallar un abierto Wx que es disjunto de B. Gracias!

I want to thank you about your great videos I found many things that are not found anywhere else only here ❤

We can not take the case Ai's are countable except two uncountable sets A1 and A2 since the measure of the union will still 1 but the sum will be 2 so we must take one uncountable set not two .. ok why ? Is the reasons are the formula of the segma algebra & Ai's are disjoint ? I proved this using contapositive l said let Ai's are countable except two sets A1 and A2 where their completens are countable.. Since Ai's are disjoint then A1 is a subset of the completent of the union of Ai's (with A2) And if we take the complement of each side then we will replace each side I mean the union will be subset of the complement of A1 wich is a contradiction ( uncountable subset of countable is a contradiction)... the union is uncountable since it has A2 which is uncountabl and the complement of A1 is a countable from the formula of the segma algebra ..ok is it a logic reason or proof ? So there exist no case which say Ai's are countable except two ? If yes what about if we take all A are uncountable? .. we will have the same problem What about this case ?

Hi why in proving countable union of measure le set is measurable you suppose the sets are disjoint?

nice

Thank you for the video. I had a question on the first part of the problem. I thought the wording countable implied that we also needed to prove that the basis was a countable one as well. How would you suggest we go about doing that part? Thank you!

Can you please make a video on extension?

Well described , really that brought a clear idea about dictionary order topology. But I am struggling to understand why the dictionary order topology restricted to I=[0,1] is not same as the subspace topology on I×I obtained from the dictionary order topology on R×R...Can you please explain? I can understand that for inherited subspace topology there may be half-open intervals as it is just the intersection of an interval of R×R with the unit square... Is it for like the basis only consists of open intervals for this type of order topology ? Am I right?

Urysohn’s lemma can also be used to prove the Portmanteau theorem.

Sound is very low...thanks

Finally a video where I get the intuition behind how the simple functions are defined. I was never able to understand the intuition till now. Thank you so much!

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in the first question's example...why is Y convex in X??

Very useful video and nicely put together. Gives the viewer a better intuition of the countability axioms by going through an example.

5:45 why it is inf Lim not the opposite Lim inf¿

@ 4:48 when we are considering other case, why are we doing "there exist for some..." when in first case we just start that E_i are countable for all i? Is it just there to eventually force the form (U E_i)^c is countable? Since, in the first case, we already forced (U E_i) is countable by starting all E_i are countable? Proving two case since for something to be in A: (something is subset of X AND something is countable) OR (smthing is subset of X AND smthing-complement is countable} where, in our case, that Something = (U E_i). Lastly, in your video, are we to assume fancy A is subset of Powerset of X? so Since we claimed E in A earlier. E is also subset of X. So union of subsets of X is still subset of X so (U E_i) is subset of X, satisfying first part of each 2 cases? I ask this because I think you only briefly mentioned in the last video fact A is subset of P(X)... sorry for my English.

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Hello teacher, can you provide proof regarding the real analysis course?

Great work thank you🫶

Thanks for the video! This is super helpful! I have learnt this interesting problem! Please do upload more such videos and also request you to upload some interesting problems from functional analysis!

This channel is gold!!!

The way you produce the vedio is very elegance I think you are french Good for you lady Amazing explanation

Such a well explained video❤

hello problemathic girl i am sanjay from india i am doing my Bachelors in technology in electronics and communication engineering, i am in senior year now, but i have always had a deep love for theoretical mathematics since my childhood, i am planing to get a masters in mathematics after my engineering degree, your videos are very inspiring and helpful and please keep amazing content your work is great

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tnx

Amazing , just going through it for my phd interview at IISC

Please keep uploading u r soon going to become a legend in youtube higher mathematics🙌

Wonderful❤❤

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got lost a bit! when we say x belongs to exactly one N_r, uniquess is proven but what about existence? how do we know N_r is not empty? when proving that uniqueness, how are the inequalities ( r<= x <= 1 or x < r ) used? why does s<>r imply z-r and z-s belong to the same equivalence class?

hey, love the channel. btw the lemma as written says E is a subset of A, shouldn't that be E is a member of A instead?

I think you missed intervals of the type (a,b] when defining h-intervals. Also could you please explain why we need *disjoint* unions for the algebra?