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.
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.
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
I am totally open to Dr Peyam's explanation of closed sets. Well done!
I'm taking this course as a prerequisite for general relativity. Amazing videos.
Thank you!!
Thanks. This video could be a very good start for topology.
Excelent class of sets! Thank u man
Thank you! And please continue teaching us with so much passion! Amazing teacher and videos!
Such an informative video! Thank you Dr. Peyam!!
If you can find time in between your teaching and research, you should write a math text book. Maybe DE's.
very intriguing!
[a,b] be like, all your limits ℝ belong to us.
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.
Ok. Thank you very much.
You are a life saver. Thanks so much
Happy to help!
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.
I completely agree! The sequence definition is more practical, that’s why I started with it
I'm taking Topology this semester at ASU; I didn't know you taught here!
Woooow what a small world! I was there last year :) Enjoy your topology course!
One that is non-trivially openable and clopenable
Hi Dr Peyam
Is N*(set of naturals without zero) closed in real metric?
Yes, a convergent sequence of natural numbers is eventually constant, hence converges to a natural number. Same thing if you remove 0
What about not a ball but a sphere? Is that open?
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.
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
Sets are not doors
Remember complement of open is CLOSED not NOT OPEN. Not open and closed are different bro bro
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
@@drpeyam oh my bad, thanks for the clarification!!!
Crack!!!
Equal your closure ?
Include all your own limo points?
Complement of an open set ?