Compactness
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- Опубліковано 3 жов 2024
- The single, most important concept in topology and analysis: Compactness. This is explained via covers, which I'll define as well. There are tons of applications of this concept, which you can find in the playlist below
Topology Playlist: • Topology
Other Compactness Video: • Compactness
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At the end of 2020 I started watching your videos. The best is that I watch just for entertaining. I don't have any exams or studying maths
colorful thumbnails are the most enthralling ones, I can't resist!
I was struggling with concept of compactness in Real Analysis classes, where my teacher won't stop naming it. This month I have my finals, so thank you very much for this video and every single other video! :D I will watch it later today
Just when mathematics is about to make me crazy your channel keeps me sane , you surprise me by how easy you make maths
This is maths sounding history class and we all listen wide open... thank you so much!
I was listening to a finite subcover band recently. They'd done every song I could think of, until I started looking for songs that were only digital downloads. It turned out that they'd only necessary done a song if it came out on a compact disc...
Oh my god 😂 Badum-tssss
I enjoy this way of learning math, because it is varied rather than strictly progressive -- and I never have to take an exam!
In other words, over time your videos are something like an open cover for the open set of things I want to learn.
That is perfect for my randomly wandering mind! My wishes are scattered and so is the cover. THANK YOU
(But the property of being compact seems strange to me.)
Agree .
Compactness in analysis has many important implications (it is equivalent to a set being closed and bounded) such as the intermediate value theorem.
I think this is my favorite video on the channel, it does a brilliant job at building the intuition for the definition of compactness.
Wow, thank you!
I finally found a video that makes it more clear....Thanks man
Today Dr Peyam taught me that inside jokes are compact.
Thank you very much as my confusion about compactness is cleared just because of diagrammatic explaination
One of the best channels!!!
Amazing explanation sir💯💯
Best explanation about compact sets🔥🙌
whenever i would struggle with something in my analysis class i just come to your channel and everything becomes clear :) thank you so much
🥰
Dr.Peyam thank you very very much for the efforts you put in your videos , you have a great ability of delivering the information is a simple way .
If there was rating i would give 5 star with the best comment i can give. Damn this nija is crazy good at this . Its like a play . I enjoy watching
You inspire me. What a remarkable video. Helped me a lot to gain the intuiton. Thank you so much
Thanks for your super helpful lectures, Dr. Peyam. I like how you always give many positive as well as counter-examples to every concept. Even though I have learned some of these concepts in my undergraduate days, your videos often provide new and intuitive angles.
Thanks so much!!
So in order to show that something ISN'T compact, all you have to do is find a cleaver patern of open covers that requires an infinite amount of them to cover the original set? If this is true, I'd assume it's easier to show a set isn't compact than to show it is compact.
(Ignoring the Heine-Borel theorem since that makes describing/identifying compact sets extremely easy)
Yep you got it! Much easier to show non compactness
@@drpeyam awesome! I think I'm getting it now. I'll have to rewatch these videos when it's not crazy late so hopefully this information sticks better 😁
how is your comment from 3 months ago
@@francaisdeuxbaguetteiii7316 he has the videos unlisted in a playlist.
@@GhostyOcean wait rlly?
your videos are so much fun and so educational
Very great visual explaination..
Hey Peyam, as I learn more about maths, ln(x) and e^x functions became more impressive to me. Its pretty thrilling that those functions are opposite of each other. Can you prove those functions are opposite of each other, maybe using calculus or infinite sums (though that one seems impossible)?
ln(x) is per definition the inverse function of e^x. The inverse function of an inverse function is the function itself, thats why e^x is the inverse of ln(x). There is nothing to prove, it's a definition thing
Thank you so nuch, Dr. Peyam.
Utilizing this concept, I demonstrated that it is possible to prove the existence of a boundary of a ball by means of the information that it is coverable in all of its parts. However, it turns out that the axiom of the choice is used and it may be not been considered as valid as it would be without it.
As Always one of the best!!! thanks man
I finally understand the concept of compact, thank you so much!!! Now I can keep preparing my finals.🥺🥺
Mathematicians contradict on which symbol represents a contradiction. Love the two swords, I didn't knew about them
Great video! Too late. Real analysis is done. Gone. Forgotten. I look forward to your next installments. Hiene-Borel?
Awesome video my friend! I had so many light bulb moments watching this, Thank You!
Happy new mathematical year Dr.
While the concept is quite abstract, your clear explanations in this great video have made me understand this difficult topic . Thank you very much Sir !
nice and clear video , could you please upload a video about locally compact
Very elegant
Ok. Thank you very much. Interesting this story of compactness.
Thank you sir, your teaching method is very good!!!!
Very well explained, crystal clear
You make me feel smart😂😂
Sir Peyam
Can u show me the provement of the Ramanujan infinite series i.e, 1+2+3+.......+∞= -1/12
Explain it in a detailed manner
Have a look at this: ua-cam.com/video/jcKRGpMiVTw/v-deo.html&ab_channel=Mathologer
No, I want a special explanation from Peyam sir
It's my humble request
This was very helpful thank you!
This was very helpful, thanks!
"beautiful U" LOL I love it. Thanks for the great explanation though!
R^2 has big balls and I cannot lie, all you mathematicians can’t deny, when a set walks in with an open cover in your face you get sprung
You talked about finite sub covers, I am wondering from data science perspective that is there a criteria to select sub cover. What I mean is in order to maximize the set area and reduce the number of sub-covers is there a formula which can determine the number of sub covers in the family of beautiful U ? (In the case when we do not know the subsets and want to efficiently make subsets). Thank you for the videos they always get me thinking. #drpeyam#lovemaths
great lecture thanks :)) easy to understand
Favourite cover band! Great!! 🤣
Hahaha
Thanks!
Thanks for the super thanks 😊
Suppose E is the unit circle (without the disk) and U is {open disk of radius 1 centered at angle 2πφn on the unit circle} where n ranges over all integers. The union of U is the punctured open disk of radius 2. However, the union of any finite subset of U (of which there are some that cover E) is smaller than the punctured open disk.
Ok, and what’s your point?
@@drpeyam Just an example of that the union of a finite subcover is not the union of the whole cover.
Oh I see, thank you!
I can't unsee the smiley face starting at 1:56.
Comparing the notation here to the other compactness video, there seems to be a slight inconsistency in set notation.
Specifically you removed the outer braces when taking the big union (or big intersection) of a family (of sets),
but not when taking the union of a family of sets defined via 'set builder notation'.
For example you would write U_i∈I K_i instead of U_i∈I { K_i} , where i is in some index set I.
However in this video, in set builder notation you would preserved the outer braces, e.g. U { K_i : i∈I }.
So it appears that U_i∈I K_i = U { K_i : i∈I } , and that seems slightly inconsistent as far as use of braces.
Big Union is being used differently.
It’s just notation, since a different book is used, both mean the same thing
This topic remembers me partition of sets, nostalgia! 😅
Sir,
Can you please make a video on partition of unity
I don't get why it isn't enough to say that a set is compact if it has a finite open cover. Why do we need it to be a subcover of another open cover?
Because then (0,1) is compact because {(0,1)} is an open cover
@@drpeyam thank you!
ty
Can you make a video on quotient topology? Like gluing, etc.
I wish!
In the definition of a finite subcover, can the elements themselves be infinite?
Of course, otherwise your set would be finite
1. Closed subsets of compact need be compact
2. Compact subsets of hausdorff need be closed
3. Subset of R is compact IFF it is closed and bounded in the metric.
4. In metric space compact and sequenctial compact are the same? This one I’m shaky on. The result that is.
5. In a metric space, subset of separable
Is still separable. Ok now steering away from compactness lol
Now I know what is compactness
Love u sir .
but how do you prove compactness from the definition?
You can’t prove a definition
I MEANT Prove compactness of a set from the definition. Like how do you go about constructing a finite open cover given an open cover.
I know u cant prove a definition. But How do we use this definition for provint compactness?
@dr peyam. Perhaps I should word my statement better. This definition seems kind of useless in practice because it’s hard to check every arbitrary open cover and check that it contains a finite subcover. Because in some sense there’s too many open covers to check. How do you actually use this definition to prove any non trivial sets are compact?
Cleanliness is next to godliness, and compactness is next to finiteness.
I like that!!!
Am I the only math major who watches this lol I’m being tested on point set topology in February so this helps me haha
After I watched this video, I went searching around the web trying to improve my understanding of what compactness really means.
I found the following comment that gave an example of how the attribute of compactness could be used, at
math.stackexchange.com/questions/2575862/understanding-compactness-and-how-it-relates-to-finiteness
"Compactness and finiteness are related because, for instance, every function from a finite set into the reals has a maximum and a minimum and every continuous function from a compact space into the reals also has a maximum and a minimum."
I thought of the following example which is consistent with that remark:
On the non-compact interval (0,1] if we take the continuous function 1/x, the function has a minimum at f(1)=1, but no maximum.
But the same 1/x function could not be used on [0,1], which is compact, because the function is not continuous on that closed interval.
(I hope the above is correct)
So the thing is that 1/x isn’t defined at 0, so I would change your example to x^2 on [0,1) (no max) and [0,1] (max)
@@drpeyam Yes, I see. Thanks!
Nice science or math idk
Math lol