This video is incredible. I'm struggle a lot with Real Analysis because textbooks and professors refuse to break down the fundamental meanings of each symbol and definition. They just expect us to immediately know the implicit meaning of every concept. I wish more explanations like yours existed. This is how real people come to understand math notation.
Isn’t your definition at 9:47 wrong? A closed set is usually defined as a set whose complement is open. But, for example, (1,3] is neither open nor closed. And yet, the ‘openness’ criterion fails, but it certainly is not closed, as its complement is neither open nor closed as well. Also, the empty set is open AND closed, same as R in the set of reals, and the open set definition holds, but the closed set definition also does.
I feel like your definition of the epsilon neighborhood at 1:40 is different than the one you use at 7:40. Am I making a mistake? At 1:40 you say that all elements y in the epsilon neighborhood of x are defined to be an element of X that is a distance less than epsilon from x, but in your definition of open sets at 7:40 you use the counterexample of a point on the edge to show what is not in an open set. However, since the epsilon neighborhood was defined to be a subset of X, the epsilon neighborhood of x would always be a subset of X, which means the points on the edge of the set would also be contained in the open set.
Plain English definition of an open set := "A set (X) is open if every and each of its elements (x) has any neighborhood (of elements) of any positive distance (epsilon) that is greater than zero (no matter how infinitesimal the positive distance could be) such that the neighborhood is contained (improperly contained) in the set."
Plain English definition of a Closed Set := A set is closed if none of its limit points/boundary/sphere points has any neighborhood of any positive distance that is greater than zero such that the neighborhood is contained in the set. In other words, all boundary points should not have any contained-in-set neighborhood of distance that is greater than zero.
This video is incredible. I'm struggle a lot with Real Analysis because textbooks and professors refuse to break down the fundamental meanings of each symbol and definition. They just expect us to immediately know the implicit meaning of every concept. I wish more explanations like yours existed. This is how real people come to understand math notation.
Thank you so much sir, I finally understand this concept after a year and half, thanks to this video.
year and a half?!
Thank you so much for this video. I have difficulty understanding the notation in all of this, but your visual explanations finally made it all click.
thanks so much for creating this video. a lifesaver!
Isn’t your definition at 9:47 wrong? A closed set is usually defined as a set whose complement is open. But, for example, (1,3] is neither open nor closed. And yet, the ‘openness’ criterion fails, but it certainly is not closed, as its complement is neither open nor closed as well. Also, the empty set is open AND closed, same as R in the set of reals, and the open set definition holds, but the closed set definition also does.
Yes, that is the rigorous definition of closed set. It is the complement of an open set.
Super clear and concise explanation, thanks.
I feel like your definition of the epsilon neighborhood at 1:40 is different than the one you use at 7:40. Am I making a mistake?
At 1:40 you say that all elements y in the epsilon neighborhood of x are defined to be an element of X that is a distance less than epsilon from x, but in your definition of open sets at 7:40 you use the counterexample of a point on the edge to show what is not in an open set. However, since the epsilon neighborhood was defined to be a subset of X, the epsilon neighborhood of x would always be a subset of X, which means the points on the edge of the set would also be contained in the open set.
Yeah he should have used a subset of the whole set, say U subset of X, and then it would have worked correctly I think
Plain English definition of an open set := "A set (X) is open if every and each of its elements (x) has any neighborhood (of elements) of any positive distance (epsilon) that is greater than zero (no matter how infinitesimal the positive distance could be) such that the neighborhood is contained (improperly contained) in the set."
Super intuitive . Thanks a lot. Keep up the good work sir
Uumm ... could you help a bit ......... what do you do when you asked to ...show that the union of a finite collection of closed sets is closed
Hidden gem.
An epsilon neighborhood is an epsilon ball?
Great video! Woule the closed set consist of x in [0,1] that are not equal to 0 or 1?
I am uni student from myamar i find difficulty in sets. Thank u for teaching.
Great video . Keep it up 👍
Intuitive. Thank you.
Love you
Thanks 🙏
I appreciate you
👍
Great
Plain English definition of a Closed Set := A set is closed if none of its limit points/boundary/sphere points has any neighborhood of any positive distance that is greater than zero such that the neighborhood is contained in the set. In other words, all boundary points should not have any contained-in-set neighborhood of distance that is greater than zero.
A set is closed if the individual epsilon neighborhoods of every point on the boundary is not contained in the set
very fucking lit
clap clap clap
tell u a joke, i dont use many symbols. then he wrote down a sentence only contained symbol lol