Real Analysis 13 | Open, Closed and Compact Sets

Please consider to support me if this video was helpful such that I can continue to produce them :)

🙏 Thanks to all supporters! They are mentioned in the credits of the video :)

This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

x

00:00 Intro
00:14 Recalling (epsilon-)neighbourhoods
01:33 Example: neighbourhoods
02:43 Definition open sets
04:00 Definition closed set
04:43 Examples
06:11 Criterion for checking closeness with the help of sequences
06:44 Example for the criterion
07:40 Definition compact sets
08:58 Credits

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[holt]

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

[qiaohuizhou]

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

[ashu]

Brilliant, Your videos are excellent !

[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?)

[NachiketJhalaRA]

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

[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."

[hyperduality]

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-yxse]

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

[tlli]

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.

[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.