Totally Bounded
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- čas přidán 20. 08. 2024
- In this video, we define total boundedness, which is a beautiful concept from topology, and we show that any compact space must be totally bounded. Enjoy!
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Dr Peyam works very hard and energetically on the field of functional analysis and topology
The thing that's hard to think about is how a set can be bounded, but not totally bounded. (That is, why we need this.) I think you need a metric space with infinite area a finite distance from a point, which is weird, but I can't think of any reason it couldn't happen.
I really like how much effort you put into a video to make the students understand the concepts lucidly! Thanks a lot professor!
Interesting...the colorful thumbnail was mesmerizing
Great content, as usual
Well explained, good, keep it up.
Really nice sir, make video on JEE ADVANCED asked question ,love from india
You're a good person Dr Peyam.
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Sir please make video PRINCIPLE OF INCLUSION AND EXCLUSION
Ok. Strange thing this notion of countable and uncountable. Thanks.
Could I have prove the fact that F is dense in E by showing that the closure of F is E?
Yeah but it’s similar
@@drpeyamThank you
Who 😢are you ? You make it look so easy 😢. OMG wish you were my professor 😮. Thanks I found this 😊
You are a magician
Dr. Peyam, can you please suggest non trivial counter examples (bounded but not totally bounded, etc) ?
I don’t think such an example exists. If a set is bounded you can cover it with a large ball, hence finitely many balls of any radius you want
@@drpeyam Makes sense. Then totally bounded but not bounded ? I cannot think of one. If no such example exists, there both would be equivalent, wouldn't they ?
A counterexample for bounded but not totally bounded should be the unit ball in an infinite-dimensional Banach space; take l^{\infty} for instance.
Is there any set that is bounded but not totally bounded?
Dr peyam looks like actor Mark Ruffalo
Why is F countable if the radius can take on any real valued number?
No? The radii of F are rational numbers, of the form 1/n
@@drpeyam oh, I was under the impression that radii can be any real valued number. Seemed that way in the beginning by r>0
At the beginning yes, but at the end I talk about something different
@@drpeyam thank you, sorry for misunderstanding
At 13:31, we know that F is dense but how do we know if it is a subset of E? It seems like the centers of the balls could all be outside of the set ( or at least most of them be ). We need F to be a dense subset for E to be separable, but we only seem to have density guaranteed.
Dude... This is literally a set of points which are all in F. If this is not a subset of E, I don't know what is.
Why this👇👇👇👇 comment is 4 months old
Magic 🪄