Real Analysis 13 | Open, Closed and Compact Sets

[@holt6890]

I rarely comment on KZbin, but gotta say these are so clean and enjoyable to listen to. Please continue to make more of these series!

[@brightsideofmaths]

Glad you like them!

[@qiaohuizhou6960]

0:15 epsilon neighborhood 1:02 another notion if needn’t quantify the neighborhood 2:15 2 is not the neighborhood of [-2,2] 2:54 definition of open set 3:51 boundary points cannot be in the open set 4:03 definition of closed set 4:29 open set is not the opposite of closed set 4:45 example : empty set and R are both open and close 5:45 (-2,2] is neither close or open 6:00 criterion to check the closeness with the help of sequence 7:40 definition of compact set The compact set requires more than closeness which leads to the Heine Borel theorem

[@urpapi6969]

my nigga its a 8 min video u dont all those timestamps

[@ashu5587]

Brilliant, Your videos are excellent !

[@NachiketJhalaRA]

8:08 In the def. of compact the use of "all sequences" is misleading for me. Is it better to use "every sequences" ?

[@brightsideofmaths]

You are right. I meant "for every" sequences :)

[@Taalam_quran]

6:10 We can find a neighborhood inside [-2,2] Why is it not open?

[@brightsideofmaths]

It has to work for any point in [-2,2]! We can discuss more in the community forum :)

[@VectorSpace33]

@2:19 Because you cant find a neighborhood around the points -2 and 2.

[@parianhatami]

Loved this & Heine-Borel one!

[@mathsandsciencechannel]

very nice video. enjoyed it. thamks

[@scollyer.tuition]

Are you planning to talk about the more topological "open cover" approach to compactness in this series, or will it be purely stuff that works in R^n (i.e sequential compactness, closed and boundedness or whatever?)

[@brightsideofmaths]

In my topology series, I will talk about this :)

[@scollyer.tuition]

@@brightsideofmaths That sounds good. I'd be very interested in seeing your approach to topology. You seem able to present the key points very clearly without getting lost in trivial details.

[@nicholaszamboni2676]

Thank you!

[@brightsideofmaths]

You're welcome!

[@malikialgeriankabyleswag4200]

Please Im stuck. Any open interval in the Real line covers itself, so it has a finite collection of open covers which cover it. So shouldnt every open interval be compact? Shouldnt any bounded interval be compact?

[@brightsideofmaths]

"All covers have a finite subcover" is different from "there is a finite cover".

[@malikialgeriankabyleswag4200]

@brightsideofmaths Thanks for everything you do. Is it possible if you have the time, to provide an example of an infinite cover of (0,1) that cannot be reduced to a finite subcover, but the same cover also covering [0,1] being reduced to a finite subcover of [0,1]?

[@brightsideofmaths]

@@malikialgeriankabyleswag4200 The community forum is a better place for that :)

[@tlli5066]

A short question dr. Großmann, the reason of R being closed. Do I understand this correctly of R is also closed: the complementary set of R in R is the empty set. hence by definition of closed set in case of empty set: for all x element of empty set, there exist an positive real epsilon so that M is a Ball(x) is an element of the empty set. But since the proposition x being elements of empty set is wrong, we have the logic "false -> true or false" is always true (principle of explosion). Therefore R is closed too.

[@brightsideofmaths]

Yeah, you can argue like this.

[@RangQuid]

It will be more general if you use the definition of compactness as "Every open cover of the subset has a finite subcover." Otherwise there is a little bit of overlap between Bolzano-Weierstrass theorem and Heine-Borel theorem in your real analysis series. Since H-B states that sequential compactness is equivalent to closed and bounded in R^n , denote this statement by Q, then B-W is an immediate consequence of Q. This is why I prefer B-W to be the statement Q, then H-B to be the statement that compactness is equivalent to closed and bounded in R^n."

[@brightsideofmaths]

For metric spaces the general compactness definition is not needed since it's equivalent to the sequentially compactness. And since the sequence definition fits nicely into the topics of the series, I chose this one :) The overlap with BW and HB is unavoidable because in R all the compactness notions are equivalent anyway ;)

[@watsonchipeta3471]

Good one

[@hyperduality2838]

Open is dual to closed, inclusion is dual to exclusion. Convergence (syntropy, homology) is dual to divergence (entropy, co-homology) -- the 4th law of thermodynamics! Increasing the number of states or dimensions is an entropic process -- co-homology. Integration (syntropy) is dual to differentiation (entropy). "Always two there are" -- Yoda.

[@Maria-yx4se]

If set is both open and closed we can call it clopen

[@brightsideofmaths]

Yeah, you can do that.

[@minglee5164]

For functional analysis

[@HomoSiliconiens]

I am native Korean. Do NOT use google machine translation. Your Korean subtitle does not make any sense to native Koreans. For example, the title of this video "Real Analysis 13 | Open, Closed and Compact Sets" is translated to Korean "실제 분석 13 | 개방형, 폐쇄형 및 소형 세트" Its reverse translation is "Actual analysis 13, Open Type, Closed Type and Small Set." it does not make any sense to native Koreans.

[@brightsideofmaths]

You are aware that these are automatically generated by KZbin? If you want precise Korean subtitles, please add them on my GitHub, link in the description :)