What are Metric Spaces and Limit Points?

[@adokwuondoma37]

Wow... great content Sir. Though I felt lost from the beginning until I had to watch at half the original speed and pauses in between. Thanks for sharing.

[@JoelRosenfeld]

I’m glad you liked it! This one did go a bit fast. Too much in one video maybe

[@jackbrolin7709]

Production is getting better and better! These awesome videos shine much needed light on topics only lightly touched in my analysis class.

[@JoelRosenfeld]

I’m glad you like it! It means a lot to me.

[@mattlydon8873]

I get super excited when I see you have a new video out. I'm working through some analysis courses this term, and your videos are consistently fantastic resources.

[@JoelRosenfeld]

I'm happy to hear it! It's exactly what I hope for.

[@123rikin]

This is such good timing considering I studied this just yesterday! Great video 🎉

[@JoelRosenfeld]

That’s awesome! More is coming! Happy to have you here

[@Cyclonus-fc1xx]

Great video, as always, I'm not anywhere near having finished working through Rudin, but for the Topology part of Analysis I definitely prefer Pugh's Real Mathematical Analysis treatment of it, as he uses visual representations much more often, which really helps with something like Topology otherwise it gets way too abstract

[@JoelRosenfeld]

I can relate to that. My professor in undergrad was adamant that we do not visualize the proofs, but rather that we stick strictly with the definitions. I tried my best to do as he said for that year, but it wasn't until graduate school when I started letting myself relax and use visuals that everything started to click for me.

[@JoelRosenfeld]

By the way, love the profile name and profile picture. Lifelong TF fan since the 80s.

[@Cyclonus-fc1xx]

@@JoelRosenfeld Haha, that show was my childhood

[@Cyclonus-fc1xx]

@@JoelRosenfeld Yeah, the definitions get more and more cryptic until you have to visualize them if you don't want to be staring at them for 1 hour to extract the meaning, in the long term I just am able to remember something I've visualized much better than just another definition in a pool of hundreds

[@JoelRosenfeld]

@@Cyclonus-fc1xx They aren't kidding when they say a picture is worth a thousand words. Working as a professional now, I found it's much more valuable to have the picture in my head to get me started. I can always shore up the rigor later on.

[@yingzhaolian5592]

Nice video. It would be great if you could talk a bit about the classification of the topological space. I find it hard to get a good intuition about how their difference can impact the proof or convergence.

[@JoelRosenfeld]

Hey Yingzhao! Happy to see you still hanging around here! Do you have something in particular in mind? Convergence is entirely dependent on the topology. For things like operator theory, where we have Strong Operator Topology, Norm Topology, and so forth, convergence can impact whether you see things like the convergence of the spectra of the operators.

[@yingzhaolian5592]

@@JoelRosenfeld Hey Joel, great to get your reply. Tbh, normed space is sufficient for my research ;) It is still good to learn more from mathematician :) To put the question more precise: when we learn operator theory, convergence in different topology have a clear "geometric" intuition. But every time when I read books or watch videos about topology, where those T1, T2.... topology come up, I fail to get an intuition about how those difference definition impact the convergence. Sure, It is clear how it looks like when we try to draw the open set on blackboard based on the definition. But it is unclear to me how these "shape" of open set will eventually affect the convergence. Basically, I am in a state that I am not so sure whether I understand them deep enough as I cannot "see" them in my brain.

[@anerudhan]

Did you flip your camera around or did you just start wearing your watch on your right hand instead of left?

[@JoelRosenfeld]

Nothing flipped. I usually wear a regular mechanical watch on my right and an Apple Watch on my left. Usually both, but sometimes one at a time. I kinda like watches lol

[@dans5595]

isn't the concept of open and closed sets, as you described it, closely analogous to open and closed intervals on the Reals?

[@JoelRosenfeld]

Yes, that is the idea. Open and closed sets are made to generalize open and closed intervals on the line. Maurice Frechet had two false starts in generalizing these concepts before he landed on a metric as we see it today

[@hansisbrucker813]

I learned so much today. Btw, can I say that I *love* that Spider-Man poster behind you 😁

[@JoelRosenfeld]

Glad you like the video! Yeah I’m a big spidey fan lol

[@dasdawfda9913]

Will be doing a real analysis course this semester and I'm already saving all this videos, I believe they will be quite useful. Thanks! Also, we'll be using understanding analysis by abbot, do you know it?

[@JoelRosenfeld]

I hope you find them helpful when you start taking the class. I never actually studied out of Abbot’s book. From what I understand, it sticks to the real line and studies analysis for that perspective. It’s a bit less general than Rudin or Apostol, but the motivation is that sticking to real numbers helps build intuition better. It’s also historically motivated, since most of topology was figured out for the real line before we had metric spaces and more general topologies.

[@darkpikachu_.]

cool, i just started reading topology from munkres, thanks for the content

[@JoelRosenfeld]

I'm glad you liked it! I fondly remember reading Munkres when I was a student. It always felt like a collection of clever tricks to avoid using a metric lol. Quick question for you. How was the pacing for the video? I felt like this one kinda got away from me, but it's hard to tell when you've been working at something for a couple of days.

[@Unemployed-Math-Major]

@@JoelRosenfeld I just started taking analysis I and I find it little hard to understand everything without pausing here and there...I love the animation btw.

[@JoelRosenfeld]

@@Unemployed-Math-Major that’s good to know, thanks! As long as you were able to follow eventually, I guess it wasn’t too far off. I’ll have to keep working on the pacing.

[@okoyoso]

Is there standard notation for the set of limit points? I've been using lim(E)

[@JoelRosenfeld]

The prime notation is pretty standard. It started with Cantor, I think.