Open Covers, Finite Subcovers, and Compact Sets | Real Analysis

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

All About Compact Sets: (coming soon)

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[lizzywhite]

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 :)

[austinhendrix]

Taking Analysis for the first time and I am not understanding it too well. I also work a full-time job and 2 part time jobs, so I don't have too much time to study. Your video has helped me grasp it in such a way I feel more comfortable working on some proofs! Thank you!

[swoyer]

This is a great video, sucks that higher level math doesn't do well on youtube. Thank you

[JesseReed-no]

Taking my first real analysis class and this video really helped with understanding covers! Thank you so much.

[KhaznahAlkhaldi]

Your videos really help me understand a compact set and open cover.
Thank you

[ashleyjuarez]

can you also go over the Heine-Borel Theorem? love how you explain things

[jayantsoni]

You really explain very well. Thanks a lot.

[jessicapriscilacerqueiraba]

continue de playlist plssss i wanna know more about compact sets

[begum]

Thank you for the quality content! Im loving these Real Analysis videos.❤

[wqr]

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!!!!

[zakhelexulu]

Thank you so much, you make everything easy.

[JRTO_X]

At 8:48 are you saying that the first example (0, 2) : {(1/k, 2-1/k)} will not have a finite subcover because if we restrict the indexing set of k from 1 to infinity then it will always be impossible for that set to be able to cover (0, 2). If other words k must go from 0 to infinity for {(1/k, 2-1/k)} to cover (0, 2)? Is that line of thinking correct? Many thanks for all the Real Analysis videos, you have been a great help with my degree 😄

[punditgi]

Wrath of Math hits the nitty-gritty. Awesome! 😃

[Dupamine]

Do you have something on heine boral theorem ?

[alondrachavez]

What literature would you recommend as an alternative for this video? [I learn better from reading than watching videos.]

[Түмэн-АмгаланАлимаа]

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)

[FlexThoseMuscles]

interesting video. not monotonous. I understand hurray!!

[sanjursan]

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.)

[okikiolaotitoloju]

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?

[coreymonsta]

Painful to listen to without turning the volume down lol