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
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 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]?
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 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.
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.
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."
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 ;)
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.
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.
You are aware that these are automatically generated by UA-cam? If you want precise Korean subtitles, please add them on my GitHub, link in the description :)
I rarely comment on UA-cam, but gotta say these are so clean and enjoyable to listen to. Please continue to make more of these series!
Glad you like them!
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
my nigga its a 8 min video u dont all those timestamps
Brilliant, Your videos are excellent !
8:08 In the def. of compact the use of "all sequences" is misleading for me. Is it better to use "every sequences" ?
You are right. I meant "for every" sequences :)
very nice video. enjoyed it. thamks
Loved this & Heine-Borel one!
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?
"All covers have a finite subcover" is different from "there is a finite cover".
@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]?
@@malikialgeriankabyleswag4200 The community forum is a better place for that :)
Thank you!
You're welcome!
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?)
In my topology series, I will talk about this :)
@@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.
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.
Yeah, you can argue like this.
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."
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 ;)
Good one
If set is both open and closed we can call it clopen
Yeah, you can do that.
For functional analysis
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.
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.
You are aware that these are automatically generated by UA-cam? If you want precise Korean subtitles, please add them on my GitHub, link in the description :)