What is a closed set ?

[@punditgi]

I am totally open to Dr Peyam's explanation of closed sets. Well done!

[@cybersecurityguy]

I'm taking this course as a prerequisite for general relativity. Amazing videos.

[@drpeyam]

Thank you!!

[@aneeshsrinivas9088]

[a,b] be like, all your limits ℝ belong to us.

[@MrWater2]

Thank you! And please continue teaching us with so much passion! Amazing teacher and videos!

[@raminrasouli191]

Thanks. This video could be a very good start for topology.

[@mathwithjanine]

Such an informative video! Thank you Dr. Peyam!!

[@iabervon]

I found it a bit odd that open is defined by balls but closed is defined by sequences. So: a point x is a limit point of E if, for all r>0, there is a point in E in B(x, r). The proof that this definition is equivalent is like half the proof that the complement of an open set is closed and vice versa.

[@drpeyam]

I completely agree! The sequence definition is more practical, that’s why I started with it

[@emanuelvendramini2045]

Excelent class of sets! Thank u man

[@jamesbentonticer4706]

If you can find time in between your teaching and research, you should write a math text book. Maybe DE's.

[@masoudsakha]

Thanks for the great video I have a question: In some references, there is a difference between the set of all limit (accumulation or cluster) points A' and the closure of a set "A bar". They define Closure of A (A bar)= union of A' and A I think A_bar=A' when we do not have isolated points.

[@claracherif6228]

thank youu dr peyam🥰🥰🥰

[@mfonpeter124]

You are a life saver. Thanks so much

[@drpeyam]

Happy to help!

[@danieldulchevsky]

I'm taking Topology this semester at ASU; I didn't know you taught here!

[@drpeyam]

Woooow what a small world! I was there last year :) Enjoy your topology course!

[@dgrandlapinblanc]

Ok. Thank you very much.

[@Happy_Abe]

If the complement of an open set is not open then what about R? R is open and its complement, the empty set, is open too.

[@drpeyam]

I said not *necessarily* open! It could of course be that the complement of an open set is open but it isn’t always the case

[@suayhossien]

Sets are not doors

[@suayhossien]

Remember complement of open is CLOSED not NOT OPEN. Not open and closed are different bro bro

[@suayhossien]

Keep in mind one criteria for a topology on a set is that both the everything and nothing are elements of the topology, strictly via definitions this implies they are both closed sets as well as they are complements of one another. Sort of also in def of connected , it doesn’t assume the sets that form serparation need be closed it only assumes open but then by def they are both open and Thus clopen lol

[@Happy_Abe]

@@drpeyam oh my bad, thanks for the clarification!!!

[@morgengabe1]

One that is non-trivially openable and clopenable

[@eliyasne9695]

very intriguing!

[@jeffreyanderson1249]

What about not a ball but a sphere? Is that open?

[@ahmedmghabat7982]

Hi Dr Peyam Is N*(set of naturals without zero) closed in real metric?

[@drpeyam]

Yes, a convergent sequence of natural numbers is eventually constant, hence converges to a natural number. Same thing if you remove 0

[@suayhossien]

Equal your closure ?

[@suayhossien]

Complement of an open set ?

[@suayhossien]

Include all your own limo points?

[@Galileo2pi]

Crack!!!