Open Covers, Finite Subcovers, and Compact Sets | Real Analysis
Vložit
- čas přidán 28. 02. 2023
- We introduce coverings of sets, finite subcovers, and compact sets in the context of real analysis. These concepts will be critical in our continuing discussion of the topology of the reals. The definition of a compact set, in particular, is surprisingly fundamental, and we will provide and prove equivalent definitions of compactness in other videos. For now, we say a set A is compact if every open cover of the set A contains a finite subcover. #realanalysis
Open Sets: • Intro to Open Sets (wi...
Closed Sets: • All About Closed Sets ...
Identifying Open, Closed, and Compact Sets: • Identifying Open, Clos...
All About Compact Sets: (coming soon)
Real Analysis Course: • Real Analysis
Real Analysis exercises: • Real Analysis Exercises
◉Textbooks I Like◉
Graph Theory: amzn.to/3JHQtZj
Real Analysis: amzn.to/3CMdgjI
Proofs and Set Theory: amzn.to/367VBXP (available for free online)
Statistics: amzn.to/3tsaEER
Abstract Algebra: amzn.to/3IjoZaO
Discrete Math: amzn.to/3qfhoUn
Number Theory: amzn.to/3JqpOQd
★DONATE★
◆ Support Wrath of Math on Patreon for early access to new videos and other exclusive benefits: / wrathofmathlessons
◆ Donate on PayPal: www.paypal.me/wrathofmath
Thanks to Petar, dric, Rolf Waefler, Robert Rennie, Barbara Sharrock, Joshua Gray, Karl Kristiansen, Katy, Mohamad Nossier, and Shadow Master for their generous support on Patreon!
Thanks to Crayon Angel, my favorite musician in the world, who upon my request gave me permission to use his music in my math lessons: crayonangel.bandcamp.com/
Follow Wrath of Math on...
● Instagram: / wrathofmathedu
● Facebook: / wrathofmath
● Twitter: / wrathofmathedu
My Math Rap channel: / @mathbars2020
Please please keep making these! The number of creators who make quality undergrad maths content is VERY VERY small. Your videos have been so helpful for my first year :)
This is a great video, sucks that higher level math doesn't do well on youtube. Thank you
Thank you for the quality content! Im loving these Real Analysis videos.❤
Thank you for watching! More real analysis videos to come this summer, I find they take the longest to make!
It is really helpful! Thank you!
Taking my first real analysis class and this video really helped with understanding covers! Thank you so much.
Glad it helped - best of luck!
Amazing video
You really explain very well. Thanks a lot.
Thank you!
so clear and helpful!!
Glad it was helpful!
Thank you so much,Sir!
Glad to help!
Thank you for your great video ❤
My pleasure, thanks for watching!
Thank you so much , you make everything easy.
Glad to help, thanks for watching!
So helpful thank you!!
Glad to help - thanks for watching!
Thank you! You saved my life
It is my duty! Thanks for watching!
THANKS THIS HELPED ME
Glad to hear it, thanks for watching!
I really appreciate your videos. This real analysis series with the book I'm reading by Jay Cummings is of a great match!please up load more!!!!
So glad they've been helpful - thanks for watching! My playlist is mostly based on Jay's book and the Stephen Abbott text - at least so far, so it's definitely a good match! I intend on making many more this summer. Also, Jay Cummings appeared in my TI-108 documentary, if you're curious and have an hour+ to spare: czcams.com/video/xrmqoKchspo/video.html&pp=ygUFdGkxMDg%3D
legend
Thanku for this genius explanation
I do my best, thanks for watching!
Brooooooooo.u justttnailedddd ittt
can you also go over the Heine-Borel Theorem? love how you explain things
I certainly will! Which equivalence are you looking for, the open cover definition of compact sets and closed/boundedness?
yeah and also, K is closed/bounded and every open cover for K has a finite subcover
@@WrathofMath please make it before my exam in may plz, open covering!!!!!
Really helpful ❤️
Glad it was helpful!
Wrath of Math hits the nitty-gritty. Awesome! 😃
Can't wait for 1000 videos from now when my Real Analysis playlist is done!
Do you have something on heine boral theorem ?
interesting video. not monotonous. I understand hurray!!
Awesome - thanks for watching!
Hello, thank you very much for making the video, it helped me a lot. And please excuse my not good english. And i have a question on the third example that is about open cover of [0,2]. You wrote union between two sets and i think then it would make it (-0.1, 2.4) which is not family of sets, then i think it can't be the cover. But if you write it differently like {"the family of sets", (-0.1, 0.1), (1.8, 2.4)} then i think it can be a cover of [0,2]. (but i am not that sure of it and if i am wrong please correct me, thank you)
In R
Hey, I have a question
At 10:20 You said wasn't (-2,1) compact because every one of its open covers did not have a finite subcover, So what is "every open cover" in that example and how do they not contain finite subcovers?
In that example we looked at one open cover which happened to have a finite subcover. We did not establish anything about "all open covers" of (-2,1) which would be necessary to establish compactness. I don't explicitly show you an open cover without a finite subcover because the previous example showed how an an open cover of an open interval could be constructed so that it has no finite subcovers, showing open intervals are not compact.
@@WrathofMath just explain why it makes that cover not compact because i still dont get why its not compact
@@matsobanemarksmokhudu2584 It's not compact because it didnt satisfy the definition. The definition states an "if and only if" and so he used the first example to prove it's not compact
What literature would you recommend as an alternative for this video? [I learn better from reading than watching videos.]
You say "any index set" but it seems only countable sets are used. What would happen if we specified a, say subset of the reals as the index set. Take the unit interval of R as the index for example. What then? (No this is not any homework problem.)
Painful to listen to without turning the volume down lol
I try to make the volume loud and clear so anyone on a phone can hear easily. I notice on my phone sometimes the videos are a little hard to hear even on max volume. But I know I’ve gone overboard a couple of times haha, I hope the video was helpful otherwise.
@@WrathofMath I already passed analysis qual it’s fine lol
@@WrathofMaththat is the case for me as well, there are times where i can barely hear anything despite the volume being on max, thanks for being that considerate