A Set is Closed if and only if it contains all of it's Limit Points If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my channel by becoming a member / @themathsorcerer Thank you:)
Hey, The Math Sorcerer I have a question. At (2:10), you wrote there exist x is not in A, s.t x is a limit point. x is a limit point of what? Do you mean limit point of A?
It's weird, never seen anyone do any concrete examples of this, they just go through the proof. Could you please do an example? For example for the subset {(x, y) in R^2 | x^2+xy+y^2 = 4}.
Im studying limit point of an interval, this method also seems applicable when it comes to an interval (the proof given by my tutor was too complicated
I remember a related concept of "adherent points". A set is closed iff it contains all its adherent points. The proof is almost the same. I think the only real difference is when it comes to the singleton set, {x}.
@@TheMathSorcerer One thing that was frustrating for me as a student was when I went to the library to get a supplimental explanation for a concept that I was having trouble understanding from the lectures and the textbook, I found that other authors would give "equlivant" definitions of the same concepts. What was a definition in our textbook was a theorem in the library book and vice versa. A => B => C => A. We go around in a circle. Pick your definition, A, B, or C. Then prove the other two.
@@OleJoe omg yes!!!!!!!! I totally agree this happens so much especially in proof based mathematics. It's constant and it is unfortunate. I remember this happening with my definition of limsup and liminf, ahh so many different definitions:)
hey! im just starting my studies in basic topology (with baby rudin). and i cant visualize what is a limit point. Do you have any intuitive way of understanding this concept? thanks
I think of a *limit point of a set A* as a point that is "really REALLY close" to the set A; for example, in the case of a metric space, say X, you'd have that a limit point x of the subset A of X follows that d(x,A)=0 ("the distance of x to A" is the smallest possible). Remember that "the distance of a point y in X to the subset B of X" is d(y,B):=inf{d(y,z) : z is in B}
