Compactness has to to defined using open covers and subcovers. Of course, the characterization discussed in the video is true, but only valid for R^n, as stated by Heine-Borel theorem. Great video, thanks 😊
I saw other videos. Those are more tecnical, talk about Hilbert spaces, show counterexamples. But for me this video is optimal: brings the point without extra details. Good news: I am not a mathematitian, airplane/powerplant designer, etc. So, even though my understanding of compactness is incomplete, this will not cause harm.
how can a set be closed and not bounded? does anyone have any example, because if there isn't any example, the condition of bounded would be useless then!
Closed means the complement is open, and open just means it's a neighbourhood for each of its points. For example, the integers as a subset of the reals. The complement of the integers (the reals excluding the integers) is open, because it consists of a union of open sets, and is thus a neighbourhood of each of its points. However the integers are clearly not bounded since they have no upper or lower bound. Likewise you could construct an uncountable number of closed but unbounded sets by simply centering the integers on any real number of your choosing - so there are at least as many closed but unbounded sets as there are reals. Of course there are many more closed but unbounded sets, but these are probably the easiest examples.
I got the definition, but I have a question. "Most of the time" we deal with open set topology. It is "concidered nice and useful". Why do we suddnly switch to closed sets here? What are the benefits?
Compactness has to to defined using open covers and subcovers. Of course, the characterization discussed in the video is true, but only valid for R^n, as stated by Heine-Borel theorem. Great video, thanks 😊
Exactly. Every open cover has a finite subcover.
I saw other videos. Those are more tecnical, talk about Hilbert spaces, show counterexamples. But for me this video is optimal: brings the point without extra details.
Good news: I am not a mathematitian, airplane/powerplant designer, etc. So, even though my understanding of compactness is incomplete, this will not cause harm.
Seriously, you need to speak at a lower voice😅
Use a playback speed of 1.5x.
how can a set be closed and not bounded? does anyone have any example, because if there isn't any example, the condition of bounded would be useless then!
Closed means the complement is open, and open just means it's a neighbourhood for each of its points. For example, the integers as a subset of the reals. The complement of the integers (the reals excluding the integers) is open, because it consists of a union of open sets, and is thus a neighbourhood of each of its points. However the integers are clearly not bounded since they have no upper or lower bound.
Likewise you could construct an uncountable number of closed but unbounded sets by simply centering the integers on any real number of your choosing - so there are at least as many closed but unbounded sets as there are reals. Of course there are many more closed but unbounded sets, but these are probably the easiest examples.
R^n is closed but not bounded.
Really great video. It really explains with visual aid what finite subcover is
This is not true in general, but it is true for Euclidean space...
Yes, and this video is looking at a Euclidean space
I got the definition, but I have a question.
"Most of the time" we deal with open set topology. It is "concidered nice and useful".
Why do we suddnly switch to closed sets here?
What are the benefits?
Thank's very useful !!!
Thanks you !
Thank you!
How can we prove an algebraically production set is a closed set?
not good ，pls tell open and bound set is not compact.
Thanks for this video 💥
Thank you
Compact? More like "Completely where it's at!" This was a great video, and I'm really glad that you decided to make it.